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| ==Non-linear wave equations== | | ==Non-linear wave equations== |
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| <div class="MsoNormal" style="text-align: center"><center>
| | Nonlinear wave equations arise in physics from two major sources: relativity and [[elasticity]]. |
| ----
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| </center></div>
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| <center>'''Overview'''</center>
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| Let <math>R^{d+1}</math> be endowed with the Minkowski metric
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| <center><math>ds^2 = dx^2 - dt^2</math>.</center>
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| (In many papers, the opposite sign of the metric is used, but the difference is purely notational). We use the usual summation, raising, and lowering conventions. <br /> The D'Lambertian operator
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| <center><math>\Box := \partial_a \partial^a = \Delta - \partial_t^2</math></center>
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| is naturally associated to this metric, the same way that the Laplace-Beltrami operator is associated with a Riemannian metric.
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| Space and time have the same scaling for wave equations. We will often use D to denote an unspecified derivative in either the space or time directions.
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| All relativistic field equations in (classical) physics are variants of the free wave equation
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| <center><math>\Box f = 0,</math></center>
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| where <math>f</math> is either scalar or vector-valued. One can also consider add a mass term to obtain the Klein-Gordon equation
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| <center><math>\Box f = f</math>.</center>
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| In practice, this mass term makes absolutely no difference to the local well-posedness theory of an equation (since the mass term <math>f</math> is negligible for high frequencies), but often plays a key role in the global theory (because of the improved decay and dispersion properties, and because the Hamiltonian controls the low frequencies more effectively).
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| There are several ways to perturb this equation. There are linear perturbations, which include the addition of potential terms, connection terms, and drag terms, as well as the replacement of the flat Minkowski metric with a more general curved metric, or by placing obstacles or otherwise changing the topology of the domain manifold R^{n+1}. There is an extensive literature on all of these perturbations, but we shall not discuss them in depth, and concentrate instead on model examples of non-linear perturbations to the free wave equation. In the fullest generality, this would mean studying equations of the form
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| <center><math>F(f, Df, D^2f) = 0</math></center>
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| where <math>D</math> denotes differentation in space or time and the Taylor expansion of <math>F</math> to first order is the free wave or Klein-Gordon equation. Such fully non-linear equations, though, are very difficult to study, and have only really been analyzed in the one-dimensional case (in which case it can be subsumed into the general theory of 1+1-dimensional hyperbolic systems). In higher dimensions the only known tool to analyze this case is to differentiate the equation, turning it into a quasi-linear system. As such we do not discuss fully non-linear wave equations here. Instead, we consider three less general types of equations, which in increasing order of complexity are the [#semilinear semi-linear], [#dnlw semi-linear with derivatives], and [#Quasilinear quasi-linear] equations.
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| Non-linear wave equations are often the Euler-Lagrange equation for some variational problem, usually with a Lagrangian that resembles
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| <center><math>\int \partial_af \partial^af dx dt</math></center>
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| (this being the Lagrangian for the free wave equation). As such the equation usually comes with a divergence-free stress-energy tensor T^{<font face="Symbol">a</font> <font face="Symbol">b</font>}, which in turn leads to a conserved Hamiltonian E(<font face="Symbol">f</font>) on constant time slices (and other spacelike surfaces). There are a few other conserved quantites such as momentum and angular momentum, but these are rarely useful in the well-posedness theory. It is often worthwhile to study the behaviour of E(D<font face="Symbol">f</font>) where D is some differentiation operator of order one or greater, preferably corresponding to one or more Killing or conformal Killing vector fields. These are particularly useful in investigating the decay of energy at a point, or the distribution of energy for large times.
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| It is often profitable to study these equations using conformal transformations of spacetime. The Lorentz transformations, translations, scaling, and time reversal are the most obvious examples, but ''conformal compactification'' (mapping R^{d+1} conformally to a compact subset of S^d x R known as the ''Einstein diamond'') is also very useful, especially for global well-posedness and scattering theory. One can also blow up spacetime around a singularity in order to analyze the behaviour near that singularity better.
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| The one-dimensional case n=1 is special for several reasons. Firstly, there is the very convenient null co-ordinate system u = t+x, v = t-x which can be used to factorize \Box. Also, the stress-energy tensor often becomes trace-free, which leads to better conformal invariance properties. There are a vastly larger number of conformal transformations, indeed anything of the form (u,v) -> (<font face="Symbol">F</font>(u), <font face="Symbol">Y</font>(v)) is conformal. Also, the one-dimensional wave equation has no decay, local smoothing, or dispersion properties, and its solutions are essentially travelling waves. Finally, there are a much larger range of spaces beyond Sobolev spaces which are available for well-posedness theory, because the free wave evolution operator preserves all translation-invariant spaces. (In two and higher dimensions only L<sup>2</sup>-based spaces such as Sobolev spaces H^s are preserved, because waves can focus at a point (or defocus from a point)).
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| The higher-dimensional case is usually quite different from the one-dimensional case, although in spherically symmetric situations one can obtain similar behaviour, especially when viewed in the null co-ordinates u = t+r, v = t-r. Indeed one can think of spherically symmetric wave equations as one-dimensional wave equations with a singular drag term (n-1) <font face="Symbol">f</font>_r / r.
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| A very basic property of wave equations is finite speed of propagation: information only propagates at the speed of light (which we have normalized to 1) or slower. Also, singularities only propagate at the speed of light (even for Klein-Gordon equations). This allows one to localize space whenever time is localized. Because of this, there is usually no distinction between periodic and non-periodic wave equations. Another application is to convert local existence results for large data to that of small data (though in sub-critical situations this is often better achieved by scaling or similar arguments). Also, the behaviour of blowup at a point is only determined by the solution in the backwards light cone from that point; thus to avoid blowup one needs to show that the solution cannot concentrate into a backwards light cone. One can also use finite speed of propagation to truncate constant-in-space solutions (which evolve by some simple ODE) to obtain localized solutions. This is often useful to demonstrate blowup for various focussing equations.
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| The non-linear expressions which occur in non-linear wave equations often have a ''null form'' structure. Roughly speaking, this means that travelling waves exp(i (k.x +- |k|t)) do not self-interact, or only self-interact very weakly. When one has a null form present, the local and global well-posedness theory often improves substantially. There are several reasons for this. One is that null forms behave better under conformal compactification. Another is that null forms often have a nice representation in terms of conformal Killing vector fields. Finally, bilinear null forms enjoy much better estimates than other bilinear forms, as the interactions of parallel frequencies (which would normally be the worst case) is now zero.
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| An interesting variant of these equations occur when one has a coupled system of two fields u and v, with v propagating slower than u, e.g.
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| <center>\Box u = F(U, DU) <br /> \Box_s v = G(U, DU)</center>
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| where U = (u,v) and \Box_s = s<sup>2</sup> <font face="Symbol">D</font> - \partial_t<sup>2</sup> for some 0 < s < 1. This case occurs physically when u propagates at the speed of light and v propagates at some slower speed. In this case the null forms are not as useful, however the estimates tend to be more favourable (if the non-linearities F, G are "off-diagonal") since the light cone for u is always transverse to the light cone for v. One can of course generalize this to consider multiple speed (nonrelativistic) wave equations.
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| <center>'''Wave estimates'''</center>
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| Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms L^q_t L^r_x, or in X^{s,b} spaces, defined by
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| <center><tt><font size="10.0pt"><nowiki>|| u ||_{s,b} = || <</nowiki></font></tt><font face="Symbol">x</font><tt><font size="10.0pt">>^s <|</font></tt><font face="Symbol">x</font><tt><font size="10.0pt"><nowiki>|-|</nowiki></font></tt><font face="Symbol">t</font><tt><font size="10.0pt"><nowiki>|>^b \hat{u} ||_2</nowiki></font></tt></center>
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| Linear space-time estimates are known as [#linear Strichartz estimates]. They are especially useful for the [#semilinear semilinear NLW without derivatives], and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The X^{s,b} spaces are used primarily for [#bilinear bilinear estimates], although more recently [#multilinear multilinear estimates have begun to appear]. These spaces first appear in one-dimension in [[references.html#RaRe1982 RaRe1982]] and in higher dimensions in [[references.html#Be1983 Be1983]] in the context of propagation of singularities; they were used implicitly for LWP in [[references.html#KlMa1993 KlMa1993]], while the Schrodinger and KdV analogues were developed in [[references.html#Bo1993 Bo1993]], [[references.html#Bo1993b Bo1993b]].
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| <center>'''Linear estimates'''</center>
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| [Thanks to Chengbo Wang for some corrections – Ed.]
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| * Fixed-time estimates for free solutions <font face="Symbol">f</font><nowiki>:</nowiki>
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| ** (Energy estimate) If <font face="Symbol">f</font>(0) is in [H^s], then <font face="Symbol">f</font>(t) is also.
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| ** (Decay estimate) If <font face="Symbol">f</font>(0) has more than (d+1)/2 derivatives in [L^1], then ||<font face="Symbol">f</font>(t)||_<font face="Symbol">¥</font> decays like <t>^{-(d-1)/2}. One can obtain the endpoint of (d+1)/2 derivatives if one is willing to localize in frequency or use Hardy spaces and BMO.
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| ** One can interpolate between these estimates to get (L^p, L^p') estimates with the sharp loss of regularity [[references.html#Br1975 Br1975]]. This is useful for Strichartz estimates and for scattering theory.
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| * Strichartz estimates: A free \dot H^s solution is in L^q_t L^r_x if
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| ** (Scaling) d/2 - s = 1/q + d/r
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| ** (Parallel interactions) (d-1)/4 >= 1/q + (d-1)/2r
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| ** (Increase of integrability) q, r >= 2
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| ** (No double endpoints) (n,q,r) \neq (3, 2, infinity)
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| *** This estimate can be recovered for radial functions [[references.html#KlMa1993 KlMa1993]], or when a small amount of smoothing (either in the Sobolev sense, or in relaxing the integrability) in the radial variable [MacNkrNaOz-p].However in the general case one cannot recover the estimate even if one uses the BMO norm or attempts Littlewood-Paley frequency localization [[references.html#Mo1998 Mo1998]]
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| *** Actually even when n > 3, the (q,r) = (2,infty) estimate is slightly subtle; one has BMO and Besov space estimates but not directly L^infty estimates.However, the endpoint (q,r) = (2, 2(d-1)/(d-3)) is OK; see [[references.html#KeTa1998 KeTa1998]].
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| ** In the case s=1/2, d=3, q=r=4, a maximizer exists (e.g. with initial position zero and initial velocity given by the Cauchy distribution 1 / 1 + |x|^2), witih best constant (3pi/4)^{1/4} [Fc-p4]
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| ** These results extend globally outside of a convex obstacle [Bu-p], [[references.html#SmhSo1995 SmhSo1995]], [SmhSo-p], [Met-p]; see [So-p] for a survey of this issue and applications to nonlinear wave equations outside of an obstacle.
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| ** For data which is radial (or otherwise enjoys additional angular regularity) a much larger range of Strichartz estimates is possible (basically because the parallel interaction obstruction is substantially weakened); see [Stz-p4] for further discussion.
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| These estimates extend to some extent to the Klein-Gordon equation Box u = m^2 u.A useful heuristic to keep in mind is that this equation behaves like the Schrodinger equation <u>+</u>i u_t + 1/(2m) Delta u = 0 when the frequency <font face="Symbol">x</font> has magnitude less than m, but behaves like the wave equation for higher frequencies.Some basic Strichartz estimates here are in [[references.html#MsSrWa1980 MsSrWa1980]]; see for instance [Na-p], [MacNaOz-p], [MacNkrNaOz-p] for more recent treatments.
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| For inhomogeneous estimates it is known that a solution with zero initial data and forcing term containing s-1 derivatives in a dual
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| space L^Q’_t L^R’_x will lie in L^q_t L^r_x if both (q,r) and (Q,R) are admissible in the above sense, if s >= 0, and if one has the scaling condition
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| 1/Q + d/R + 1/q + d/r = d + s + 1.The “s-1” represents a smoothing effect of one derivative, though this full gain is only attainable if one uses
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| the energy exponents L^1_t L^2_x and L^infty_t L^2_x.It is possible to obtain inhomogeneous estimates in which only one of the exponents
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| are admissible; this phenomenon was first observed in [[references.html#Har1990 Har1990]], [[references.html#Ob1989 Ob1989]] (see also [[references.html#KeTa1998 KeTa1998]]).More recently in [Fc-p2], inhomogeneous
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| estimates are obtained with the above scaling condition assuming the weaker conditions 1 <= q,r <= infty and 1/q < (n-1)(1/2-1/r) or (q,r) = (infty,2)
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| and similarly for Q,R, and if the following additional conditions hold:
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| <span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>In d=1,2 no further conditions are required;
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| <span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>When d=3 r, R are required to be finite;
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| <span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>When d > 3, either 1/q + 1/Q < 1, (n-3)/r <= (n-1)/R, and (n-3)/R <= (n-1)/r, or 1/q + 1/Q = 1, (n-3)/r < (n-1)/R, (n-3)/R < (n-1)/r, r >= q, and R >= Q.
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| Strichartz estimates extend to situations in which there is a potential or when the metric is variable.For local-in-time estimates and smooth potentials or
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| metrics this is fairly straightforward (the potential can be treated by iterative methods, and the metric by parametrix methods).More interesting issues
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| arise for global-in-time estimates with smooth potentials/metrics or local-in-time estimates with rough potentials/metrics (the two types of results are linked by
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| scaling).For potentials of power-type decay, the global results are as follows:
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| <span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>For potentials of the form V = a/|x|^2 with d >= 3 and a > -(d-2)^2/4, one has global Strichartz estimates [BuPlStaTv-p]; a simplified proof and more general result dealing with inverse square-like potentials which are not too negative is in [BuPlStaTv-p2].The condition on a is necessary to avoid bound states.For potentials decaying slower than this, Strichartz estimates can fail (Duyckaerts?)
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| <span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>For potentials decaying an epsilon faster than 1/|x|^2, and assumed to be nonnegative, dispersive and Strichartz estimates were obtained when d=3 in [[references.html#GeVis2003 GeVis2003]].
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| (More results to be added in future).
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| <center>'''Bilinear estimates'''</center>
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| * Let d>1. If <font face="Symbol">f</font>, <font face="Symbol">y</font> are free \dot H^{s_1} and \dot H^{s_2} solutions respectively, then one can control <font face="Symbol">fy</font> in \dot X^{s,b} if and only if
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| ** (Scaling) s+b = s_1 + s_2 - (d-1)/2
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| ** (Parallel interactions) b >= (3-d)/4
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| ** (Lack of smoothing) s <= s_1, s_2
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| ** (Frequency cancellation) s_1 + s_2 >= 1/2
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| ** (No double endpoints) (s_1, b), (s_2, b) \neq ((d+1)/4, -(d-3)/4); (s_1+s_2, b) \neq (1/2, -(d-3)/4).
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| See [[references.html#FcKl2000 FcKl2000]]. Null forms can also be handled by identities such as
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| <center>2 Q_0(<font face="Symbol">f</font>, <font face="Symbol">y</font>) = Box(<font face="Symbol">f</font> <font face="Symbol">y</font>).</center>
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| * Some bilinear Strichartz estimates are also known. For instance, if s, q, r are as in the linear Strichartz estimates <font face="Symbol">f</font>, <font face="Symbol">y</font> are \dot H^{s-<font face="Symbol">a</font>} solutions, then
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| <center>D^{-2<font face="Symbol">a</font>} (<font face="Symbol">fy</font>) is in L^{q/2}_t L^{r/2}_x</center>
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| as long as 0 <= <font face="Symbol">a</font> <= d/2 - 2/q - d/r [[references.html#FcKl-p FcKl-p]]. Similar estimates for null forms also exist [[references.html#Pl2002 Pl2002]]; see also [[references.html#TaVa2000b TaVa2000b]], [Ta-p4].
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| <center>'''Multilinear estimates'''</center>
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| There are some isolated examples of multilinear estimates that cannot be obtained from linear and bilinear estimates and Holder's inequality. For instance, the inequality
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| <center><nowiki>|\int D^{-a}D_-(u_1 u_2) u_3 u_4 dx dt| <~ ||f_1||_{H^{n/2-a}} ||f_2||_2 ||f_3||_2 ||f_4||_2</nowiki></center>
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| is proven for d=2 and 3/4 < a < 1 in [Sb-p].
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| <div class="MsoNormal" style="text-align: center"><center>
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| ----
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| </center></div>
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| <center>'''Semilinear wave equations'''</center>
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| [Note: Many references needed here!]
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| Semi-linear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form
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| <center>\Box <font face="Symbol">f</font> = F(<font face="Symbol">f</font>) <br /> \Box <font face="Symbol">f</font> = <font face="Symbol">f</font> + F(<font face="Symbol">f</font>)</center>
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| respectively where F is a function only of <font face="Symbol">f</font> and not of its derivatives, which vanishes to more than first order. Typically F grows like |<font face="Symbol">f</font><nowiki>|^p for some power p. If F is the gradient of some function V, then we have a conserved Hamiltonian </nowiki>
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| <center>\int |<font face="Symbol">f</font>_t|<sup>2</sup> / 2 + |<font face="Symbol">Ñf</font><nowiki>|</nowiki><sup>2</sup>/2 + V(<font face="Symbol">f</font>)\ dx.</center>
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| For NLKG there is an additional term of |<font face="Symbol">f</font><nowiki>|</nowiki><sup>2</sup>/2 in the integrand, which is useful for controlling the low frequencies of <font face="Symbol">f</font>. If V is positive definite then we call the NLW defocussing; if V is negative definite we call the NLW focussing. The term "coercive" does not have a standard definition, but generally denotes a potential V which is positive for large values of <font face="Symbol">f</font>.
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| To analyze these equations in H^s we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that F is smooth, or that F is a p^th-power type non-linearity with p > [s]+1.
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| The scaling regularity is s_c = d/2 - 2/(p-1). Notable powers of p include the ''L''<sup>2</sup>''-critical'' power p_{L<sup>2</sup>} = 1 + 4/d, the H^{1/2}-critical or ''conformal'' power p_{H^{1/2}} = 1 + 4/(d-1), and the ''H^1-critical'' power p_{H^1} = 1 + 4/{d-2}. <br />
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| {| class="MsoNormalTable" style="width: 100.0%; mso-cellspacing: 1.5pt; mso-padding-alt: 0in 0in 0in 0in" width="100%" border="1"
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| |- style="mso-yfti-irow: 0; mso-yfti-firstrow: yes"
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| Dimension d
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| Strauss exponent (NLKG)
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| L<sup>2</sup>-critical exponent
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| Strauss exponent (NLW)
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| H^{1/2}-critical exponent
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| H^1-critical exponent
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| |- style="mso-yfti-irow: 1"
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| 1
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| 3.56155...
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| 5
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| infinity
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| infinity
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| N/A
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| |- style="mso-yfti-irow: 2"
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| 2
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| 2.41421...
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| | style="padding: .75pt .75pt .75pt .75pt" |
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| 3
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| 3.56155...
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| 5
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| infinity
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| |- style="mso-yfti-irow: 3"
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| 3
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| 2
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| 2.33333...
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| 2.41421...
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| | style="padding: .75pt .75pt .75pt .75pt" |
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| 3
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| 5
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| |- style="mso-yfti-irow: 4; mso-yfti-lastrow: yes"
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| 4
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| 1.78078...
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| | style="padding: .75pt .75pt .75pt .75pt" |
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| 2
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| | style="padding: .75pt .75pt .75pt .75pt" |
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| 2
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| 2.33333...
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| 3
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| |}
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| The following necessary conditions for LWP are known. Firstly, for focussing NLW/NLKG one has blowup in finite time for large data, as can be seen by the ODE method. One can scale this and obtain ill-posedness for any focussing NLW/NLKG in the supercritical regime s < s_c; this has been extended to the defocusing case in [CtCoTa-p2]. By using Lorentz scaling instead of isotropic scaling one can also obtain ill-posedness whenever s is below the conformal regularity
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| <center>s_{conf} = (d+1)/4 - 1/(p-1)</center>
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| in the focusing case; the defocusing case is still open. In the H^{1/2}-critical power or below, this condition is stronger than the scaling requirement.
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| | |
| * When d >= 2 and 1 < p < p_{H^{1/2}} with the focusing sign, blowup is known to occur when a certain Lyapunov functional is negative, and the rate of blowup is self-similar [[references.html#MeZaa2003 MeZaa2003]]; earlier results are in [[references.html#AntMe2001 AntMe2001]], [[references.html#CafFri1986 CafFri1986]], [[references.html#Al1995 Al1995]], [[references.html#KiLit1993 KiLit1993]], [[references.html#KiLit1993b KiLit1993b]].
| |
| | |
| To make sense of the non-linearity in the sense of distributions we need s >= 0 (indeed we have illposedness below this regularity by a high-to-low cascade, see [CtCoTa-p2]). In the one-dimensional case one also needs the condition 1/2 - s < 1/p to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.
| |
| | |
| Finally, in three dimensions one has ill-posedness when p=2 and s = s_{conf} = 0 [[references.html#Lb1993 Lb1993]]. <br />
| |
| | |
| * In dimensions d<=3 the above necessary conditions are also sufficient for LWP.
| |
| * For d>4 sufficiency is only known assuming the condition
| |
| | |
| <center>p (d/4-s) <= 1/2 ( (d+3)/2 - s) (*)</center>
| |
| | |
| and excluding the double endpoint when (*) holds with equality and s=s_{conf} [[references.html#Ta1999 Ta1999]]. The main tool is two-scale Strichartz estimates.
| |
| | |
| ** By using standard Strichartz estimates this was proven with (*) replaced by
| |
| | |
| <center>p ((d+1)/4-s) <= (d+1)/2d ( (d+3)/2 - s); (**)</center>
| |
| | |
| see [[references.html#KeTa1998 KeTa1998]] for the double endpoint when (**) holds with equality and s=s_{conf}, and [[references.html#LbSo1995 LbSo1995]] for all other cases. A slightly weaker result also appears in [[references.html#Kp1994 Kp1994]].
| |
| | |
| GWP and scattering for NLW is known for data with small H^{s_c} norm when p is at or above the H^{1/2}-critical power (and this has been extended to Besov spaces; see [Pl-p4]. This can be used to obtain self-similar solutions, see [MiaZg-p2]). One also has GWP in H^1 in the defocussing case when p is at or below the H^1-critical power. (At the critical power this result is due to [[references.html#Gl1992 Gl1992]]; see also [[references.html#SaSw1994 SaSw1994]]. For radial data this was shown in [[references.html#Sw1988 Sw1988]]). For more scattering results, see below.
| |
| | |
| For the defocussing NLKG, GWP in H^s, s < 1, is known in the following cases:
| |
| | |
| * d=3, p = 3, s > 3/4 [[references.html#KnPoVe-p2 KnPoVe-p2]]
| |
| * d=3, 3 <= p < 5, s > [4(p-1) + (5-p)(3p-3-4)]/[2(p-1)(7-p)] [MiaZgFg-p]
| |
| * d=3, 2 < p < 3, or n>=4, (d+1)^2/((d-1)^2+4) <= p < (d-1)/(d-3), and
| |
| | |
| <center>s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] / [2(p-1)(d+1-p(d-3))]</center>
| |
| | |
| [MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition s_conf > s_c and the condition (**).
| |
| | |
| * d=2, 3 <= p <= 5, s > (p-2)/(p-1) [Fo-p]; this is for the NLW instead of NLKG.
| |
| * d=2, p > 5, s > (p-1)/p [Fo-p]; this is for the NLW instead of NLKG.
| |
| | |
| GWP and blowup has also been studied for the NLW with a conformal factor
| |
| | |
| <center>Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p;</center>
| |
| | |
| the significance of this factor is that it behaves well under conformal compactification. See [[references.html#Aa2002 Aa2002]], [[references.html#BcKkZz2002 BcKkZz2002]], [[references.html#Gue2003 Gue2003]] for some recent results.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Scattering theory for semilinear NLW'''</center>
| |
| | |
| [Thanks to Kenji Nakanishi for many helpful additions to this section - Ed.]
| |
| | |
| The ''Strauss exponent''
| |
| | |
| <center>p_0(d) = [d + 2 + sqrt(d<sup>2</sup> + 12d + 4)]/2d</center>
| |
| | |
| plays a key role in the GWP and scattering theory. We have p_0(1) = [3+sqrt(17)]/2; p_0(2) = 1+sqrt(2); p_0(3) = 2; note that p_0(d-1) is always between the L<sup>2</sup> and H^{1/2} critical powers, and p_0(d) is always between the H^{1/2} and H^1 critical powers.
| |
| | |
| Another key power is
| |
| | |
| <center>p_*(d) = [d+2 + sqrt(d^2 + 8d)]/2(d-1)</center>
| |
| | |
| which lies between the L^2 critical power and p_0(d-1).
| |
| | |
| '''Caveats'''<nowiki>: the d=1,2 cases may be somewhat different from what is stated here (partly because some of the powers here are not well-defined). Also, in many of the NLW results one needs some additional decay at spatial infinity (e.g. finiteness of the conformal energy), except in the special H^1-critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the L</nowiki><sup>2</sup> norm (even with conservation of energy).
| |
| | |
| Scattering for small H^1 data for arbitrary NLW:
| |
| | |
| * Known for p_*(d) < p <= p_{H^{1/2}} [[references.html#Sr1981 Sr1981]].
| |
| * For p < p_0(d-1) one has blow-up [[references.html#Si1984 Si1984]].
| |
| * When d=3 this is extended to 5/2 < p <= p_{H^{1/2}}, but scattering fails for p<5/2 [Hi-p3]
| |
| * When d=4 this is extended to p_0(d-1) = 2 < p < 5/2, but scattering fails for p<2 [Hi-p3]
| |
| * An alternate argument based on conformal compactification but giving slightly different results are in [[references.html#BcKkZz1999 BcKkZz1999]]
| |
| | |
| Scattering for large H^1 data for defocussing NLW:
| |
| | |
| * Known for p_{H^{1/2}} < p <= p_{H^1} [[references.html#BaSa1998 BaSa1998]], [[references.html#BaGd1997 BaGd1997]] (GWP was established earlier in [[references.html#GiVl1987 GiVl1987]]).
| |
| * Known for p = p_{H^{1/2}}, d=3 [[references.html#BaeSgZz1990 BaeSgZz1990]]
| |
| * When d=3 this is extended to p_*(3) < p <= p_{H^{1/2}} [Hi-p3]
| |
| * When d=4 this is extended to p_*(4) < p < 5/2 [Hi-p3]
| |
| * For d>4 one expects scattering when p_0(d-1) < p <= p_{H^{1/2}}, but this is not known.
| |
| | |
| Scattering for small smooth compactly supported data for arbitrary NLW:
| |
| | |
| * GWP and scattering when p > p_0(d-1) [[references.html#GeLbSo1997 GeLbSo1997]]
| |
| ** For d=3 this is in [[references.html#Jo1979 Jo1979]]
| |
| * Blow-up for arbitrary nonzero data when p < p_0(d-1) [[references.html#Si1984 Si1984]] (see also [[references.html#Rm1987 Rm1987]], [[references.html#JiZz2003 JiZz2003]]
| |
| ** For d=4 this is in [[references.html#Gs1981b Gs1981b]]
| |
| ** For d=3 this is in [[references.html#Jo1979 Jo1979]]
| |
| * At the critical power p = p_0(d-1) there is blowup for non-negative non-trivial data [YoZgq-p2]
| |
| ** For d=2,3 and arbitrary nonzero data this is in [[references.html#Scf1985 Scf1985]]
| |
| ** For large data and arbitrary d this is in [[references.html#Lev1990 Lev1990]]
| |
| | |
| Scattering for small H^1 data for arbitrary NLKG:
| |
| | |
| * Decay estimates are known when p_0(d) < p <= p_{L<sup>2</sup>}[[references.html#MsSrWa1980 MsSrWa1980]], [[references.html#Br1984 Br1984]], [[references.html#Sr1981 Sr1981]], [[references.html#Pe1985 Pe1985]].
| |
| * Known when p_{L<sup>2</sup>} <= p <= p_{H^1} [[references.html#Na1999c Na1999c]], [[references.html#Na1999d Na1999d]], [Na-p5]. Indeed, one has existence of wave operators and asymptotic completeness in these cases.
| |
| | |
| Scattering for large H^1 data for defocussing NLKG:
| |
| | |
| * In this case one has an a priori L<sup>2</sup> bound and one does not need decay at spatial infinity.
| |
| * Scattering is known for p_{L<sup>2</sup>} < p <= p_{H^1} [[references.html#Na1999c Na1999c]], [[references.html#Na1999d Na1999d]], [Na-p5]
| |
| ** For d>2 and p not H^1-critical this is in [[references.html#Br1985 Br1985]] [[references.html#GiVl1985b GiVl1985b]]
| |
| ** The L<sup>2</sup>-critical case p = p_{L<sup>2</sup>} is an interesting open problem.
| |
| | |
| Scattering for small smooth compactly supported data for arbitrary NLKG:
| |
| | |
| * GWP and scattering for p > 1+2/d when d=1,2,3 [[references.html#LbSo1996 LbSo1996]]
| |
| ** When d=1,2 this can be obtained by energy estimates and decay estimates.
| |
| ** In principle this extends to higher dimensions but there is a difficulty with lack of smoothness in the nonlinearity.
| |
| * Blowup in the non-Hamiltonian case when p < 1+2/d [[references.html#KeTa1999 KeTa1999]]. The endpoint p=1+2/d remains open but one probably also has blow-up here.
| |
| ** Failure of scattering for p <= 1+2/d was shown in [[references.html#Gs1973 Gs1973]].
| |
| | |
| An interesting (and apparently under-explored) problem is what happens to these global existence and scattering results when there is an obstacle. For [#nlw-5_on_R^3 NLW-5 on R<sup>3</sup>] one has global regularity for convex obstacles [[references.html#SmhSo1995 SmhSo1995]], and for smooth non-linearities there is the [#gwp_qnlw general quasilinear theory]. If one adds a suitable damping term near the obstacle then one can recover some global existence results [[references.html#Nk2001 Nk2001]].
| |
| | |
| On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in [[references.html#BchNic1993 BchNic1993]], [[references.html#Nic1995 Nic1995]], [[references.html#BluSf2003 BluSf2003]]
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Non-relativistic limit of NLKG'''</center>
| |
| | |
| By inserting a parameter c (the speed of light), one can rewrite NLKG as
| |
| | |
| <center>u_tt/c<sup>2</sup> - <font face="Symbol">D</font> u + c<sup>2</sup> u + f(u) = 0.</center>
| |
| | |
| One can then ask for what happens in the non-relativistic limit c -> infinity (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, u should be localized near the double hyperboloid
| |
| | |
| <center><font face="Symbol">t</font> = +- c sqrt(c<sup>2</sup> + <font face="Symbol">x</font><sup>2</sup>).</center>
| |
| | |
| In the non-relativistic limit this becomes two paraboloids
| |
| | |
| <center><font face="Symbol">t</font> = +- (c<sup>2</sup> + <font face="Symbol">x</font><sup>2</sup>/2)</center>
| |
| | |
| and so one expects u to resolve as
| |
| | |
| <center>u = exp(i c<sup>2</sup> t) v_+ + exp(-i c<sup>2</sup> t) v_- <br /> u_t = ic<sup>2</sup> exp(ic<sup>2</sup> t) v_+ - ic<sup>2</sup> exp(ic<sup>2</sup> t) v_-</center>
| |
| | |
| where v_+, v_- solve some suitable NLS.
| |
| | |
| A special case arises if one assumes (u_t - ic<sup>2</sup> u) to be small at time zero (say o(c) in some Sobolev norm). Then one expects v_- to vanish and to get a scalar NLS. Many results of this nature exist, see [Mac-p], [[references.html#Nj1990 Nj1990]], [[references.html#Ts1984 Ts1984]], [MacNaOz-p], [Na-p]. In more general situations one expects v_+ and v_- to evolve by a coupled NLS; see [[references.html#MasNa2002 MasNa2002]].
| |
| | |
| Heuristically, the frequency << c portion of the evolution should evolve in a Schrodinger-type manner, while the frequency >> c portion of the evolution should evolve in a wave-type manner. (This is consistent with physical intuition, since frequency is proportional to momentum, and hence (in the nonrelativistic regime) to velocity).
| |
| | |
| A similar non-relativistic limit result holds for the [#mkg Maxwell-Klein-Gordon] system (in the Coulomb gauge), where the limiting equation is the coupled <br /> Schrodinger-Poisson system
| |
| | |
| <center>i v^+_t + <font face="Symbol">D</font> v/2 = u v^+ <br /> i v^-_t - <font face="Symbol">D</font> v/2 = u v^- <br /><font face="Symbol">D</font> u = - |v^+|<sup>2</sup> + |v^-|<sup>2</sup></center>
| |
| | |
| under reasonable H^1 hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields <font face="Symbol">f</font>, A, A_0 and the Schrodinger-Poisson fields u, v^+, v^- are
| |
| | |
| <center>A_0 ~ u <br /><font face="Symbol">f</font> ~ exp(ic<sup>2</sup> t) v^+ + exp(-ic<sup>2</sup> t) v^- <br /><font face="Symbol">f</font>_t ~ i M exp(ic<sup>2</sup>)v^+ - i M exp(-ic<sup>2</sup> t) v^-</center>
| |
| | |
| where M = sqrt(c^4 - c<sup>2</sup> <font face="Symbol">D</font>) (a variant of c<sup>2</sup>).
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Sine-Gordon'''</center>
| |
| | |
| [Contributions to this section are sorely needed!]
| |
| | |
| The sine Gordon equation
| |
| | |
| <center>Box u = sin(u)</center>
| |
| | |
| in R^{1+1} arises in the study of optical pulses, or from the Scott model of a continuum of pendula hanging from a wire. It is a completely integrable equation, and has many interesting solutions, including "breather" solutions.
| |
| | |
| Because the non-linearity is bounded, GWP is easily obtained for L<sup>2</sup> or even L^1 data.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quadratic NLW/NLKG'''</center>
| |
| | |
| [Thanks to Chengbo Wang for some corrections – Ed.]
| |
| | |
| * Scaling is s_c = d/2 - 2.
| |
| * For d>4 LWP is known for s >= d/2 - 2 by Strichartz estimates [[references.html#LbSo1995 LbSo1995]]. This is sharp by scaling arguments.
| |
| * For d=4 LWP is known for s >= ¼ by Strichartz estimates [[references.html#LbSo1995 LbSo1995]].This is sharp from Lorentz invariance (concentration) considerations.
| |
| * For d=3 LWP is known for s > 0 by Strichartz estimates [[references.html#LbSo1995 LbSo1995]].
| |
| ** One has ill-posedness for s=0 [[references.html#Lb1996 Lb1996]]. This is related to the failure of endpoint Strichartz when d=3.
| |
| * For d=1,2 LWP is known for s>= 0 by Strichartz estimates (or energy estimates and Sobolev in the d=1 case).
| |
| ** For s<0 one has rather severe ill-posedness generically, indeed cannot even interpret the non-linearity <font face="Symbol">f</font><sup>2</sup> as a distribution [CtCoTa-p2].
| |
| ** In the two-speed case ([#two-speed see Overview]) one can improve this to s>-1/4 for non-linearities of the form F = uv and G = uv [Tg-p].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Cubic NLW/NLKG on R'''</center>
| |
| | |
| * Scaling is s_c = -1/2.
| |
| * LWP for s >= 1/6 by energy estimates and Sobolev (solution is in L<sup>3</sup>_x).
| |
| ** For s<1/6 one has ill-posedness [CtCoTa-p2], indeed it is not even possible to make sense of solutions in the distributional sense.
| |
| * GWP for s>1/3 for defocussing NLKG [[references.html#Bo1999 Bo1999]]
| |
| ** For s >= 1 this is clear from energy conservation (for both NLKG and NLW).
| |
| ** Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| * ''Remark''<nowiki>: NLKG can be viewed as a symplectic flow with the symplectic form of H^{1/2}. NLW is similar but with the homogeneous H^{1/2}.</nowiki>
| |
| * Small global solutions to NLKG (either focusing or defocusing) have logarithmic phase corrections due to the critical nature of the nonlinearity (neither short-range nor long-range).However there is still an asymptotic development and an asymptotic completeness theory, see [[references.html#De2001 De2001]], [LbSf-p]
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Cubic NLW/NLKG on R<sup>2</sup>'''</center>
| |
| | |
| * Scaling is s_c = 0.
| |
| * LWP for s >= 1/4 by Strichartz estimates (see e.g. [[references.html#LbSo1995 LbSo1995]])
| |
| ** This is sharp by concentration examples in the focusing case; the defocusing case is still open.
| |
| * GWP for s>=1/2 for defocussing NLKG [[references.html#Bo1999 Bo1999]] and for defocussing NLW [Fo-p]
| |
| ** For s >= 1 this is clear from energy conservation (for both NLKG and NLW)..
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| * ''Remark''<nowiki>: This is a symplectic flow with the symplectic form of H^{1/2}, as in </nowiki>[#nlw-3_on_R the one-dimensional case].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Cubic NLW/NLKG on R<sup>3</sup>'''</center>
| |
| | |
| * Scaling is s_c = 1/2.
| |
| * LWP for s >= 1/2 by Strichartz estimates (see e.g. [[references.html#LbSo1995 LbSo1995]]; earlier references exist)
| |
| ** When s=1/2 the time of existence depends on the profile of the data and not just on the norm.
| |
| ** One can improve the critical space H^{1/2} to a slightly weaker Besov space [Pl-p2].
| |
| ** For s<1/2 one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
| |
| * GWP for s>3/4 [[references.html#KnPoVe-p2 KnPoVe-p2]] for defocussing NLKG.(An alternate proof is in [[references.html#GalPl2003 GalPl2003]]).
| |
| ** For s>=1 this is clear from energy conservation (for both NLKG and NLW).
| |
| ** One also has GWP and scattering for data with small H^{1/2} norm for general cubic non-linearities (and for either NLKG or NLW).
| |
| ** In the defocussing case one has scattering for large H^1 data [[references.html#BaeSgZz1990 BaeSgZz1990]], see also [Hi-p3].
| |
| ** Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| * For periodic defocussing NLKG there is a weak turbulence effect in H^s for s > 5 (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} [[references.html#Kuk1995b Kuk1995b]].In particular H^s cannot be a symplectic phase space for s > 5.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Cubic NLW/NLKG on R^4'''</center>
| |
| | |
| * Scaling is s_c = 1.
| |
| * LWP for s >= 1 by Strichartz estimates (see e.g. [[references.html#LbSo1995 LbSo1995]]; earlier references exist)
| |
| ** When s=1 the time of existence depends on the profile of the data and not just on the norm.
| |
| ** One has strong uniqueness in the energy class [Pl-p5], [[references.html#FurPlTer2001 FurPlTer2001]]. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.
| |
| ** For s<s_c one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
| |
| * GWP for s=1 in the defocussing case [[references.html#SaSw1994 SaSw1994]] (see also [[references.html#Gl1990 Gl1990]], [[references.html#Gl1992 Gl1992]], [[references.html#Sw1988 Sw1988]], [[references.html#Sw1992 Sw1992]], [[references.html#BaSa1998 BaSa1998]], [[references.html#BaGd1997 BaGd1997]]).
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quartic NLW/NLKG'''</center>
| |
| | |
| * Scaling is s_c = d/2 - 2/3.
| |
| * For d>2 LWP is known for s >= d/2 - 2/3 by Strichartz estimates. This is sharp by scaling argumentsin both the focusing and defocusing cases [CtCoTa-p2]
| |
| * For d=2 LWP is known for s>=5/12 by Strichartz estimates. This is sharp by concentration arguments in the focusing case; the defocusing case is open.
| |
| ** In the defocusing case one has GWP for s > 2/3 [Fo-p]
| |
| * For d=1 one has LWP for s>= 1/4 by energy estimates and Sobolev (solution is in L^4_x). Below this regularity one cannot even make sense of the solution as a distribution.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quintic NLW/NLKG on R'''</center>
| |
| | |
| * Scaling is s_c = 0.
| |
| * LWP for s >= 3/10 by energy estimates and Sobolev (solution is in L^5_x)
| |
| ** For s<3/10 one has ill-posedness [CtCoTa-p2], indeed it is not even possible to make sense of solutions in the distributional sense.
| |
| * GWP for s >= 1 in the defocussing case from energy conservation.
| |
| ** It is overwhelmingly likely that one can lower this s index.
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quintic NLW/NLKG on R<sup>2</sup>'''</center>
| |
| | |
| * Scaling is s_c = 1/2.
| |
| * LWP for s >= 1/2 by Strichartz estimates (see e.g. [[references.html#LbSo1995 LbSo1995]]; earlier references exist)
| |
| ** When s=1/2 the time of existence depends on the profile of the data and not just on the norm.
| |
| ** For s<s_c one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
| |
| * GWP for s > 3/4 for defocussing NLW/NLKG [Fo-p]
| |
| ** For s>= 1 this follows energy conservation.
| |
| ** One also has GWP and scattering for data with small H^{1/2} norm for general quintic non-linearities (and for either NLW or NLKG).
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quintic NLW/NLKG on R<sup>3</sup>'''</center>
| |
| | |
| * Scaling is s=1.
| |
| * LWP for s >= 1 by Strichartz estimates (see e.g. [[references.html#LbSo1995 LbSo1995]]; earlier references exist)
| |
| ** When s=1 the time of existence depends on the profile of the data and not just on the norm.
| |
| ** For s<s_c one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
| |
| * GWP for s=1 in the defocussing case [[references.html#Gl1990 Gl1990]], [[references.html#Gl1992 Gl1992]]. The main new ingredient is energy non-concentration [[references.html#Sw1988 Sw1988]], [[references.html#Sw1992 Sw1992]]
| |
| ** Further decay estimates and scattering were obtained in [[references.html#BaSa1998 BaSa1998]]; global Lipschitz dependence was obtained in [[references.html#BaGd1997 BaGd1997]].
| |
| ** For smooth data GWP and scattering was shown in [[references.html#Gl1992 Gl1992]]; see also [[references.html#SaSw1994 SaSw1994]]
| |
| ** For radial data GWP and scattering was shown in [[references.html#Sw1988 Sw1988]]
| |
| ** For data with small energy this was shown for general quintic non-linearities (and for either NLW or NLKG) in [[references.html#Ra1981 Ra1981]].
| |
| ** Global weak solutions can be constructed by general methods (e.g. [[references.html#Sr1989 Sr1989]], [[references.html#Sw1992 Sw1992]]); uniqueness was shown in [[references.html#Kt1992 Kt1992]]
| |
| ** In the focussing case there is blowup from large data by the ODE method.
| |
| ** When there is a convex obstacle GWP for smooth data is known [[references.html#SmhSo1995 SmhSo1995]].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Septic NLW/NLKG on R'''</center>
| |
| | |
| * Scaling is s_c = 1/6.
| |
| * LWP for s >= 5/14 by energy estimates and Sobolev (solution is in L^7_x)
| |
| ** For s < 5/14 it is not even possible to make sense of the solution as a distribution (note there are no smoothing effects for the 1D wave equation!) and so there is ill-posedness [CtCoTa-p2].
| |
| * GWP for s = 1 in the defocusing case by energy conservation. Presumably this is improvable.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Septic NLW/NLKG on R<sup>2</sup>'''</center>
| |
| | |
| * Scaling is s_c = 2/3.
| |
| * LWP for s >= s_c by Strichartz estimates [[references.html#LbSo1995 LbSo1995]]
| |
| ** For s<s_c one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
| |
| * GWP for s > 6/7 in the defocusing case [Fo-p]
| |
| ** For s >= 1 this follows from energy conservation.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Septic NLW/NLKG on R<sup>3</sup>'''</center>
| |
| | |
| * Scaling is s=7/6.
| |
| * LWP for s>=7/6 by Strichartz estimates (see e.g. [[references.html#LbSo1995 LbSo1995]]; earlier references exist)
| |
| ** When s=7/6 the time of existence depends on the profile of the data and not just on the norm.
| |
| ** For s<s_c one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case [CtCoTa-p2]
| |
| * Global existence of large smooth solutions is unknown in the defocussing case; in the focussing case one certainly has blowup by ODE methods.
| |
| ** In the energy class s=1, one has ill-posedness in the sense that the solution map is not uniformly continuous [[references.html#Leb2000 Leb2000]]; for higher dimensions see [[references.html#BrKum2000 BrKum2000]]. This is despite an a priori bound on the H^1 x L<sup>2</sup> norm in the defocussing case from energy conservation. A variant of this result appears in [CtCoTa-p2].
| |
| ** For small data one of course has GWP and scattering [[references.html#LbSo1995 LbSo1995]]
| |
| ** It is not known what happens to large smooth solutions in the defocusing case, even in the radial case. This can be viewed as an extremely simplified model problem for the global regularity issue for Navier-Stokes.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''NLW with derivatives'''</center>
| |
| | |
| The D-NLW equation is given by
| |
| | |
| <center><tt><font size="10.0pt">Box u = F(u, Du)</font></tt></center>
| |
| | |
| where u is scalar or vector valued, and F is at least quadratic. The D-NLKG equation is similar:
| |
| | |
| <center><tt><font size="10.0pt">Box u = u + F(u, Du).</font></tt></center>
| |
| | |
| From a LWP perspective the two equations are virtually equivalent, but the NLKG is slightly better behaved for the GWP (it decays like the NLW at one higher dimension).
| |
| | |
| Among the more intensively studied derivative NLWs are the [#dnlw-2 quadratic DNLWs] (which include the Yang-Mills and Maxwell-Klein-Gordon equations) and the [#ddnlw DDNLWs] (which include equations of wave maps type).
| |
| | |
| From energy estimates one can always obtain LWP in H^s for s > n/2 + 1. (or s > n/2 if the non-linearity is at most linear in Du). However, this is rarely best possible.
| |
| | |
| In many cases the non-linearity F has a ''null structure''. The precise meaning of this is hard to quantify exactly, but roughly speaking this means that if u is a plane wave then the highest order terms in the non-linearity vanish. In other words, the self-interaction of plane waves is relatively small. The presence of a null structure usually makes the LWP and GWP theory significantly better.
| |
| | |
| The GWP theory for small data is usually accomplished by vector fields methods (or similar methods which try to capture the decay, and proximity to the light cone, of the global solution), or via conformal compactification. The method of normal forms is also often useful, as it can eliminate the worst terms in a non-linearity. <br /> As a general principle, the small data GWP theory becomes better whenever the order of the non-linearity increases (because this makes the non-linearity even smaller) or when the dimension increases (because there is more decay). There is rarely any need to distinguish between u and Du in the small data GWP theory. In many cases the theory is robust enough to carry over to the [#Quasilinear quasilinear case].
| |
| | |
| For small smooth compactly supported data of size <font face="Symbol">e</font> and smooth non-linearities, the GWP theory for D-NLW is as follows.
| |
| | |
| * If the non-linearity is a null form, then one has GWP for d>=3; in fact one can take the data in a weighted Sobolev space H^{2,1} x H^{1,2} [[references.html#Cd1986 Cd1986]]
| |
| ** This is also true in the multi-speed (i.e. nonrelativistic) case [[references.html#KlSi1996 KlSi1996]] (see also [[references.html#Yk2000 Yk2000]]). Earlier related work appears in [[references.html#Ko1987 Ko1987]], [[references.html#Ko1989 Ko1989]]
| |
| *** In fact, one does not need the compact support condition, and can take data small in H^9 [SiTu-p].
| |
| ** Without the null structure, one has almost GWP in d=3 [[references.html#Kl1985b Kl1985b]], and this is sharp [[references.html#Jo1981 Jo1981]], [[references.html#Si1983 Si1983]]
| |
| ** With the null structure and outside a convex obstacle one has GWP in H^{2,1} x H^{1,2} [[references.html#KeSmhSo2000 KeSmhSo2000]], assuming the standard compatibility conditions on the data. Earlier work in this direction is in [[references.html#Dt1990 Dt1990]].
| |
| *** For radial data and obstacle this was obtained in [[references.html#Go1995 Go1995]]; see also [[references.html#Ha1995 Ha1995]].
| |
| *** GWP for small smooth data outside a star-shaped obstacle was shown for d>=6 and non-linearities quadratic in Du in [[references.html#ShbTs1984 ShbTs1984]], [[references.html#ShbTs1986 ShbTs1986]].
| |
| ** For d>3 or for cubic nonlinearities one has GWP regardless of the null structure [[refernces.html#KlPo1983 KlPo1983]], [[references.html#Sa1982 Sa1982]], [[references.html#Kl1985b Kl1985b]].
| |
| * For the d=2 case, the results are as follows.
| |
| ** One has existence for roughly <font face="Symbol">e</font>^{-2}without a null condition, but at least exp(C/<font face="Symbol">e</font><sup>2</sup>) with a null condition [[references.html#Al1999b Al1999]], [[references.html#Al1999b Al1999b]], [[references.html#Al2001 Al2001]], [[references.html#Al2001b Al2001b]]
| |
| *** For semilinear equations these are in [[references.html#Gd1993 Gd1993]]
| |
| *** For cubic nonlinearities these are in [[references.html#Hg1995 Hg1995]]; furthermore one has global existence assuming a "second null condition"
| |
| *** For spherically symmetric data this is in [[references.html#Lad1999 Lad1999]]; furthermore one has global existence assuming a "second null condition"
| |
| ** Earlier results are in [[references.html#Ky1993 Ky1993]]. Non-relativistic variants are in [[references.html#Hg1998 Hg1998]], [[references.html#HgKu2000 HgKu2000]]
| |
| | |
| For small smooth compactly supported data of size <font face="Symbol">e</font> and smooth non-linearities, the GWP theory for D-NLKG is as follows.
| |
| | |
| * One has GWP for d >= 2 [[references.html#SnTl1993 SnTl1993]], [[references.html#OzTyTs1996 OzTyTs1996]].
| |
| ** For d>=3 this was proven in [[references.html#Kl1985 Kl1985]] (by vector fields) and [[references.html#Sa1985 Sa1985]] (by normal forms).
| |
| ** For d=3 this result is extended to Klein-Gordon-Zakharov systems in [[references.html#OzTyTs1995 OzTyTs1995]]
| |
| * For d=1 one has GWP for quartic and higher non-linearities [[references.html#LbSo1996 LbSo1996]].
| |
| ** For cubic non-linearities one has almost global existence (for time exp(C/<font face="Symbol">e</font><sup>2</sup>)); see [[references.html#Ho1997 Ho1997]]. This is sharp [[references.html#KeTa1999 KeTa1999]], [Yo-p]. Explicit lower bounds on the constant C are in [[references.html#De1999 De1999]].
| |
| ** Using normal forms one can push the almost global existence to quadratic non-linearities [[references.html#MrTwSr1997 MrTwSr1997]].
| |
| ** For septic and higher nonlinearities one has global existence [[references.html#Mr1997 Mr1997]], [[references.html#Yag1994 Yag1994]]
| |
| ** For quadratic non-linearities of null form type one has existence for time roughly <font face="Symbol">e</font>^{-4} [[references.html#De1997 De1997]]
| |
| *** The compact support condition can be weakened substantially [[references.html#De1997 De1997]]. One can even just assume small H<sup>3</sup> data, but then the time of existence shrinks slightly, to <font face="Symbol">e</font>^{-4} log(1/<font face="Symbol">e</font>)^{-6} [[references.html#De1997b De1997b]].
| |
| *** Without the null form one cannot do better than about <font face="Symbol">e</font>^{-2}[[references.html#KeTa1999 KeTa1999]].
| |
| *** A necessary and sufficient condition on the nonlinearity has been obtained as to whether one can obtain global existence for small data [[references.html#De2001 De2001]]
| |
| | |
| For small smooth data of size <font face="Symbol">e</font> on the circle T, and smooth non-linearities, the GWP theory for D-NLKG is as follows.
| |
| | |
| * For quadratic non-linearities one has existence for time <font face="Symbol">e</font>^{-2}. [[references.html#De1998 De1998]].This was extended to higher dimensional tori and to quasilinear NLKG in [DeSze-p]
| |
| * For higher non-linearities of order r, one has existence for time <font face="Symbol">e</font>^{1-r} log(1/<font face="Symbol">e</font>)^{3-r}, and in some cases this bound is sharp. [[references.html#De1998 De1998]]
| |
| * One has a similar result (with more technical time of existence when the domain is a sphere, although now one must exclude a set of masses of measure zero. [DeSze-p]
| |
| | |
| For non-smooth non-linearities of order p, one has blow-up examples for D-NLKG from small smooth compactly supported data whenever p <= 1+2/d [[references.html#KeTa1999 KeTa1999]], although in certain cases (esp. coercive Hamiltonian systems) one still has GWP [[references.html#Ca1985 Ca1985]]. One also has failure of scattering for this range of powers [[references.html#Gl1981b Gl1981b]]. It would be interesting to see if one could obtain GWP for D-NLKG the p>1+2/d case (though in the non-smooth non-linearity case one probably is restricted to d=1,2,3 in order to keep the non-linearity sufficiently smooth).
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Damping DNLW'''</center>
| |
| | |
| The equation
| |
| | |
| <center>\Box u = u<sup>2</sup> u_t</center>
| |
| | |
| in three dimensions is known to be locally well-posed in any sub-critical regularity s>1, and has scattering in H<sup>3</sup> [Smh-p]. It would be interesting to see whether one has local well-posedness in the critical energy regularity H^1.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quadratic DNLW'''</center>
| |
| | |
| By Quadratic DNLW we mean equations with the schematic form
| |
| | |
| <center>\Box <font face="Symbol">f</font> = <font face="Symbol">f</font> D<font face="Symbol">f</font>.</center>
| |
| | |
| This equation has the same scaling as cubic NLW, but is more difficult technically because of the derivative in the non-linearity. In practice one can always add a cubic term <font face="Symbol">f</font><sup>3</sup> to the non-linearity without disrupting any of the well-posedness theory, as <font face="Symbol">f</font><sup>3</sup> is usually much easier to estimate than <font face="Symbol">f</font> D<font face="Symbol">f</font>.
| |
| | |
| Important examples of this type of equation include the [#mkg Maxwell-Klein-Gordon] and [#ym Yang-Mills] equations (in the Lorentz gauge, at least), as well as the simplified model equations for these equations. The [#ymh Yang-Mills-Higgs] equation is formed by coupling equations of this type to a semi-linear wave equation. The most interesting dimensions are 3 (for physical applications) and 4 (since the energy regularity is then critical). The two-dimensional case appears to be somewhat under-explored. The Yang-Mills and Maxwell-Klein-Gordon equations behave very similarly, but from a technical standpoint the latter is slightly easier to analyze.
| |
| | |
| In d dimensions, the critical regularity for this equation is s_c = d/2 - 1. However, there are no instances of Quadratic DNLW for which any sort of well-posedness is known at this critical regularity (except for those special cases where the equation can be algebraically transformed into a simpler equation such as NLW or the free wave equation).
| |
| | |
| Energy estimates give local well-posedness for s > s_c + 1. Using Strichartz estimates this can be improved to s > s_c + 3/4 in two dimensions and s > s_c + 1/2 in three and higher dimensions [[references.html#PoSi1993 PoSi1993]]; the point is that these regularity assumptions together with Strichartz allow one to put <font face="Symbol">f</font> into L<sup>2</sup>_t L^<font face="Symbol">¥</font>_x, hence in L^1_t L^<font face="Symbol">¥</font>_x, so that one can then use the energy method.
| |
| | |
| Using X^{s,\theta} estimates [[references.html#FcKl2000 FcKl2000]] instead of Strichartz estimates, one can improve this further to d > s_c + 1/4 in four dimensions and to the near-optimal s > s_c in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space B^{s_c}_{2,1} [Stz-p3], and local well-posedness for large Besov data.In four dimensions one has a similar result if one imposes one additional angular derivative of regularity [Stz-p2].
| |
| | |
| Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in H^1 [[references.html#Lb1993 Lb1993]], although one can recover well-posedness in the Besov space B^1_{2,1} [[references.html#Na1999 Na1999]], or when an epsilon of radial regularity is imposed [MacNkrNaOz-p]. It would be interesting to determine what the situation is in the other low dimensions.
| |
| | |
| If the non-linearity <font face="Symbol">f</font> D<font face="Symbol">f</font> has a null structure then one can improve upon the previous results. For instance, the model equations for the Yang-Mills and Maxwell-Klein-Gordon equations are locally well-posed for s > s_c in three [[references.html#KlMa1997 KlMa1997]] and higher [[references.html#KlTt1999 KlTt1999]] dimensions. It would be interesting to determine what happens in two dimensions for these equations (probably one can get down to s > s_c + 1/4). In one dimension the model equation trivially collapses to the free wave equation.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Two-speed quadratic DNLW'''</center>
| |
| | |
| * One can consider two-speed variants of quadratic DNLW ([#two-speed see Overview]), when both F and G have the form U DU.
| |
| * The Strichartz and energy estimates carry over without difficulty to this setting. The results obtained by X^{s,\theta} estimates change, however. The null forms are no longer as useful, however the estimates are usually more favourable because of the transversality of the two light cones. Of course, if F contains uDu or G contains vDv then one cannot do any better than the one-speed case.
| |
| * For d=3 one can obtain LWP for s >= 1 for non-linearities of the form F = uDv + vDu + vDv and G = uDu + vDu [[references.html#OzTyTs1999 OzTyTs1999]], [[references.html#Tg2000 Tg2000]].
| |
| ** When G contains uDv the relevant bilinear estimate generically fails by a logarithmic factor [[references.html#OzTyTs2000 OzTyTs2000]]
| |
| * For d=2 one can obtain LWP for s>1/2 for non-linearities of the form F = uDv + vDu + vDv and G = uDu + vDu + uDv [Tg-p].
| |
| * For d=1 one can obtain LWP for s>1/4 for non-linearities of the form F = uDv + vDu and G = uDv + vDu [Tg-p].
| |
| ** One can improve this to s>0 for non-linearities of the form F=D(uv + v<sup>2</sup>) and G = D(u<sup>2</sup> + uv) [Tg-p]. These expressions are better behaved than the previous ones, for instance their Fourier transform vanishes at the origin.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''The Yang-Mills equation'''</center>
| |
| | |
| Let A be a connection on R^{d+1} which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the ''Yang-Mills equation'' if it is a critical point for the Lagrangian functional
| |
| | |
| <center>\int F^{<font face="Symbol">a</font> <font face="Symbol">b</font>} F_{<font face="Symbol">a</font> <font face="Symbol">b</font>}</center>
| |
| | |
| where F:=dA + [A,A] is the curvature of the connection A. The Euler-Lagrange equations for this functional have the schematic form
| |
| | |
| <center>\Box A + <font face="Symbol">Ñ</font> (\div_{x,t} A) = [A, <font face="Symbol">Ñ</font> A] + [A, [A,A]]</center>
| |
| | |
| where \div_{x,t} A = \partial_<font face="Symbol">a</font> A^<font face="Symbol">a</font> is the spacetime divergence of A. A more succinct (but less tractable) formulation of this equation is
| |
| | |
| <center>\partial_<font face="Symbol">a</font> F^{<font face="Symbol">ab</font>} = 0.</center>
| |
| | |
| It is often convenient to split A into temporal and spatial components as A = (A_0, A_i).
| |
| | |
| As written, the Yang-Mills equation is under-determined because of the gauge invariance
| |
| | |
| <center>A -> U^{-1} dU + U^{-1} A U <br /> F -> U^{-1} F U</center>
| |
| | |
| in the equation, where U is an arbitrary function taking values in G. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones:
| |
| | |
| <center>Temporal gauge: A^0 = 0 <br /> Coulomb gauge: \partial_i A_i = 0 <br /> Lorentz gauge: div_{x,t} A = 0</center>
| |
| | |
| There are also several other useful gauges, such as the Cronstrom gauge [[references.html#Cs1980 Cs1980]] centered around a point in spacetime.
| |
| | |
| The Lorentz gauge has the advantage of being invariant under conformal transformations, but it appears that the Yang-Mills equation is not well-behaved in this gauge for rough data. (For smooth data one can obtain local well-posedness in this gauge by energy estimates). The Coulomb gauge is the simplest to work with technically, and in this gauge the bilinear expression [A, <font face="Symbol">Ñ</font> A] acquires a null structure [[references.html#KlMa1995 KlMa1995]] which allows for a satisfactory analysis of the equation. Unfortunately there are often difficulties in creating a global Coulomb gauge, and one often has to rely instead on local Coulomb gauges pieced together using finite speed of propagation; see [[references.html#KlMa1995 KlMa1995]]. The Temporal gauge is fairly close to the Coulomb gauge, and one can develop a parallel theory for this gauge. The temporal gauge has the advantage of being easy to establish globally, but the null form structure is less obvious (one needs to partition the connection into divergence-free and curl-free components). See e.g. [[references.html#Ta-p3 Ta-p3]].
| |
| | |
| <nowiki>In the Coulomb or Temporal gauges, one can create a model equation for the Yang-Mills system by ignoring cubic terms and any contribution from the "elliptic'' portion of the gauge (A_0 in the Coulomb gauge, or the curl-free portion of A_i in the Temporal gauge). The resulting model equation is </nowiki>
| |
| | |
| <center>\Box A = <font face="Symbol">Ñ</font>^{-1} Q(A,A) + Q(<font face="Symbol">Ñ</font>^{-1}A, A)</center>
| |
| | |
| where Q(A,A') is some null form such as
| |
| | |
| <center>Q(A,A') := \partial_i A \partial_j A' - \partial_j A \partial_i A'.</center>
| |
| | |
| The results known for the model equation are slightly better than those known for the actual Yang-Mills or Maxwell-Klein-Gordon equations.
| |
| | |
| The Yang-Mills equations come with a positive definite conserved Hamiltonian
| |
| | |
| <center>\int |F_{0,i}|<sup>2</sup> + |F_{i,j}|<sup>2</sup> dx</center>
| |
| | |
| which mostly controls the H^1 norm of A and the L<sup>2</sup> norm of A_t. However, there are some portions of the H^1 x L<sup>2</sup> norm which are not controlled by the Hamiltonian (in the Coulomb gauge, it is \partial_t A_0; in the Temporal gauge, it is the H^1 norm of the curl-free part of A_i). This causes some technical difficulties in the global well-posedness theory.
| |
| | |
| The Yang-Mills equations can also be coupled with a g-valued scalar field <font face="Symbol">f</font>, with the Lagrangian functional of the form
| |
| | |
| <center>\int F^{<font face="Symbol">a</font> <font face="Symbol">b</font>} F_{<font face="Symbol">ab</font>} + D_<font face="Symbol">a</font> <font face="Symbol">f</font> . D^<font face="Symbol">af</font> + V(<font face="Symbol">f</font>)</center>
| |
| | |
| where D_<font face="Symbol">a</font> = \partial_<font face="Symbol">a</font> + [A<sub><font face="Symbol">a</font></sub>, .] are covariant derivatives and V is some potential function (e.g. V(<font face="Symbol">f</font>) = |<font face="Symbol">f</font><nowiki>|^{k+1}). The corresponding Euler-Lagrange equations have the schematic form </nowiki>
| |
| | |
| <center>\Box A + <font face="Symbol">Ñ</font> (\div_{x,t} A) = [A, <font face="Symbol">Ñ</font> A] + [A, [A,A]] + [<font face="Symbol">f</font>, D <font face="Symbol">f</font>] <br /> D_<font face="Symbol">a</font> D^<font face="Symbol">a</font> <font face="Symbol">f</font> = V'(<font face="Symbol">f</font>)</center>
| |
| | |
| and are generally known as the ''Yang-Mills-Higgs system of equations''. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. [#mkg The Maxwell-Klein-Gordon system] is a special case of Yang-Mills-Higgs.
| |
| | |
| The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise.
| |
| | |
| Attention has mostly focussed on the [#YM_on_R^3 three] and [#YM_on_R^4 four dimensional] cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to A_tt = 0). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data [[references.html#CaSaTv1998 CaSaTv1998]] (see also [Biz-p]). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension [[references.html#BizTb2001 BizTb2001]], [Biz-p].
| |
| | |
| The Yang-Mills equations can also be coupled with a spinor field. In the U(1) case this becomes the Maxwell-Dirac equation.
| |
| | |
| The Yang-Mills equations in dimension n have many formal similarities with the wave maps equation at dimension n-2 (see e.g. [[references.html#CaSaTv1998 CaSaTv1998]] for a discussion)
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Yang-Mills on R<sup>2</sup>'''</center>
| |
| | |
| * Scaling is s_c = 0.
| |
| * One can use the method of descent and finite speed of propagation to infer R<sup>2</sup> results from [#YM_on_R^3 the R<sup>3</sup> results]. Thus, for instance, one has LWP for s>3/4 in the temporal gauge and GWP in the temporal gauge for s>= 1. These results are almost certainly non-optimal, however, and probably have much simpler proofs (for instance, one can obtain the LWP result from the general theory of DNLW without using any null form structure).
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Yang-Mills on R<sup>3</sup>'''</center>
| |
| | |
| [Thanks to Jacob Sterbenz for corrections - Ed.]
| |
| | |
| * Scaling is s_c = 1/2.
| |
| * LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small [[references.html#Ta-p3 Ta-p3]]. The main tools are bilinear estimates involving both X^{s,\theta} spaces and product Sobolev spaces.
| |
| ** Presumably the small data assumption can be removed, but the usual methods to do this fail because there are too many time derivatives in the non-linearity in the temporal gauge.
| |
| ** For s >= 1 in the Temporal or Coulomb gauges LWP for large data was shown in [[references.html#KlMa1995 KlMa1995]].
| |
| ** For s > 1 LWP for the Temporal, Coulomb, or Lorentz gauges follows from Strichartz estimates [[references.html#PoSi1993 PoSi1993]].
| |
| ** For s > 3/2 LWP for the Temporal, Coulomb, or Lorentz gauges follows from energy estimates [[references.html#EaMc1982 EaMc1982]].
| |
| ** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorentz gauge.
| |
| ** For the model equation LWP fails for s < 3/4 [MaStz-p]
| |
| ** The endpoint s=1/2 looks extremely difficult, even for Besov space variants.
| |
| * GWP is known for data with finite Hamiltonian (morally, this is for s >= 1) in the Coloumb or Temporal gauges [[references.html#KlMa1995 KlMa1995]].
| |
| ** For smooth data this was proven in [[references.html#EaMc1982 EaMc1982]].
| |
| *** This result was extended to curved space in [[references.html#CcSa1997 CcSa1997]]
| |
| ** It seems likely that one can improve this to something like s>7/8, in analogy with the [#mkg_on_R^3 theory for the Maxwell-Klein-Gordon equation].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''MKG and Yang-Mills in R^4'''</center>
| |
| | |
| * Scaling is s_c = 1.
| |
| * For the MKG equations in the Coulomb gauge, LWP is known for s > 1 [Sb-p5]. This is still not known for Yang-Mills.
| |
| ** For the model equations this is in [[references.html#KlTt1999 KlTt1999]]
| |
| *** For general quadratic DNLW this is only known for s > 5/4 (e.g. by the estimates in [[references.html#FcKl2000 FcKl2000]]). Strichartz estimates need s > 3/2 [[references.html#PoSi1993 PoSi1993]], while energy estimates need s > 2.
| |
| ** The latter two results (Strichartz and energy) easily extend to the actual MKG and YM equations in all three standard gauges.
| |
| * It is conjectured that one has global well-posedness results for small energy, but this is open.
| |
| ** For small smooth compactly supported data, one can obtain global existence from the [#gwp_qnlw general theory of quasi-linear equations].
| |
| ** For large data Yang-Mills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time [[references.html#BizTb2001 BizTb2001]], [Biz-p].
| |
| *** Further numerics suggests that the radius of the instanton in fact decays like C t / sqrt(log t) [[references.html#BizOvSi BizOvSi-p]].
| |
| ** GWP for small B^{1,1} data (with an additional angular derivative of regularity) in the Lorentz gauge is in [Stz-p2].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''MKG and Yang-Mills in R^n, n>4'''</center>
| |
| | |
| * Scaling is s_c = n/2 - 1.
| |
| * LWP is almost certainly true for MKG-CG for s > s_c by adapting the results in [Sb-p5]. The corresponding question for Yang-Mills is still open.
| |
| ** For the model equations one can probably achieve this by adapting the results in [[references.html#Tt1999 Tt1999]]
| |
| * For dimensions n>= 6, GWP for small H^{n/2} data in MKG-CG is in [RoTa-p]. The corresponding question for Yang-Mills is still open, but a Besov result follows (in the Lorentz gauge) from [Stz-p3].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Yang-Mills-Higgs on R<sup>3</sup>'''</center>
| |
| | |
| * Suppose the potential energy V(<font face="Symbol">f</font>) behaves like |<font face="Symbol">f</font><nowiki>|^{p+1} (i.e. defocussing p^th power non-linearity). When p<=3, the Higgs term is negligible, and the theory mimics that of the ordinary Yang-Mills equation. The most interesting case is p=5, since the Higgs component is then H^1-critical.</nowiki>
| |
| * There is no perfect scale-invariance to this equation (unless p=3); the critical regularity is s_c = max(1/2, 3/2 - 2/(p-1))
| |
| * In the sub-critical case p<5 one has GWP for smooth data [[references.html#EaMc1982 EaMc1982]], [[references.html#GiVl1982b GiVl1981]]. This can be pushed to H^1 by the results in [[references.html#Ke1997 Ke1997]]. The local theory might be pushed even further.
| |
| * In the critical case p=5 one has GWP for s >= 1 [[references.html#Ke1997 Ke1997]].
| |
| * In the supercritical case p>5 one probably has LWP for s >= s_c (because this is true for the Yang-Mills and NLW equations separately), but this has not been rigorously shown. No large data global results are known, but this is also true for the supposedly simpler supercritical NLW. It seems possible however that one could obtain small-data GWP results.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center><br />'''The Maxwell-Klein-Gordon equation'''</center>
| |
| | |
| The Maxwell-Klein-Gordon is the special case of the [#ymh Yang-Mills-Higgs equation] when the Lie group G is just the circle U(1), and there is no potential energy term V(<font face="Symbol">f</font>). Thus A is now purely imaginary, and <font face="Symbol">f</font> is complex.
| |
| | |
| Despite the name, the Maxwell-Klein-Gordon equation is not really related to the (massive) Klein-Gordon equation. Rather, it is the Maxwell equation coupled with a massless scalar equation (i.e. a free wave equation). If the scalar field <font face="Symbol">f</font> is set to 0, the equation collapses to the linear Maxwell equations, which are basically a vector-valued variant of the free wave equation.
| |
| | |
| As with Yang-Mills, the three standard gauges are Lorentz gauge, Coloumb gauge, and Temporal gauge. The Lorentz gauge is most natural from a co-ordinate free viewpoint, but is difficult to work with technically. In principle the temporal gauge is the easiest to work with, being local in space, but in practice the Coloumb gauge is preferred because the null form structure of Maxwell-Klein-Gordon is most apparent in this gauge.
| |
| | |
| In the Coulomb gauge MKG has the schematic form
| |
| | |
| <center><font face="Symbol">D</font> A_0 = <font face="Symbol">ff</font>_t + <font face="Symbol">F</font><sup>3</sup><br /> \Box A = <font face="Symbol">Ñ</font>^{-1} Q(<font face="Symbol">f</font>, <font face="Symbol">f</font>) <br /> \Box <font face="Symbol">f</font> = Q(<font face="Symbol">Ñ</font>^{-1} A, <font face="Symbol">f</font>) + (A_0)_t <font face="Symbol">f</font> + A_0 <font face="Symbol">f</font>_t + <font face="Symbol">F</font><sup>3</sup></center>
| |
| | |
| where <font face="Symbol">F</font><sup>3</sup> denotes terms that are cubic in (A_0, A, <font face="Symbol">f</font>). Unfortunately, the equation for the A_0 portion of the Coloumb gauge is elliptic, which generates some low frequency issues. However, if we ignore the A_0 terms and the cubic terms then we reduce to the model equation
| |
| | |
| <center>\Box A = <font face="Symbol">Ñ</font>^{-1} Q(<font face="Symbol">f</font>, <font face="Symbol">f</font>) <br /> \Box <font face="Symbol">f</font> = Q(<font face="Symbol">Ñ</font>^{-1} A, <font face="Symbol">f</font>)</center>
| |
| | |
| which is slightly better than the corresponding model for Yang-Mills.
| |
| | |
| MKG has the advantage over YM that the Coulomb gauge is easily constructed globally using Riesz transforms, so there are less technical issues involved with this gauge.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Maxwell-Klein-Gordon on R'''</center>
| |
| | |
| * Scaling is s_c = -1/2.
| |
| * LWP can be shown in the temporal gauge for s>1/2 by energy estimates. For s<1/2 one begins to have difficulty interpreting the solution even in the distributional sense, but this might be avoidable, perhaps by a good choice of gauge. (The Coulomb gauge seems to have some technical difficulties however).
| |
| * GWP is easy to show in the temporal gauge for s >= 1 by energy methods and Hamiltonian conservation. Presumably one can improve the s >= 1 constraint substantially.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Maxwell-Klein-Gordon on R<sup>2</sup>'''</center>
| |
| | |
| * Scaling is s_c = 0.
| |
| * Heuristically, one expects X^{s,\delta} methods to give LWP for s > 1/4, but we do not know if this has been done rigorously.
| |
| ** Strichartz estimates give s > 1/2 [[references.html#PoSi1993 PoSi1993]], while energy methods give s>1.
| |
| * GWP is known for smooth data in the temporal gauge [[references.html#Mc1980 Mc1980]].
| |
| ** This should extend to s >= 1 and probably below, but we do not know if this is in the literature.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Maxwell-Klein-Gordon on R<sup>3</sup>'''</center>
| |
| | |
| [Thanks to Jacob Sterbenz for corrections - Ed.]
| |
| | |
| * Scaling is s_c = 1/2.
| |
| * LWP for s>1/2 in the Coulomb Gauge [MaStz-p]
| |
| ** For the model equation LWP fails for s < 3/4 [MaStz-p]. Thus the MKG result exploits additional structure in the MKG equation which is not present in the model equation.
| |
| ** For s>3/4 this was proven in the Coloumb gauge in [[references.html#Cu1999 Cu1999]].
| |
| ** For s>=1 this was proven in the Coulomb and Temporal gauges in [[references.html#KlMa1994 KlMa1994]].
| |
| ** For s>1 this follows (in any of the three gauges) from Strichartz estimates [[references.html#PoSi1993 PoSi1993]]
| |
| ** For s>3/2 this follows (in any of the three gauges) from energy estimates.
| |
| ** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorentz gauge.
| |
| ** The endpoint s=1/2 looks extremely difficult, even for the model equation. Perhaps things would be easier if one only had to deal with the null form <font face="Symbol">Ñ</font>^{-1} Q(<font face="Symbol">f</font>, <font face="Symbol">f</font>), as this is slightly smoother than Q(<font face="Symbol">Ñ</font>^{-1}A, <font face="Symbol">f</font>).
| |
| * GWP for s>7/8 in the Coloumb gauge [[references.html#KeTa-p KeTa-p]].
| |
| ** For s>= 1 this was proven in [[references.html#KlMa1997 KlMa1997]].
| |
| ** For smooth data this was proven in [[references.html#EaMc1982 EaMc1982]].
| |
| * For physical applications it is of interest to study MKG when the scalar field <font face="Symbol">f</font> propagates with a strictly slower velocity than the electromagnetic field A. In this case one cannot exploit the null form estimates; nevertheless, the estimates are more favourable, mainly because the two light cones are now transverse. Indeed, one has GWP for s>=1 in all three standard gauges [[references.html#Tg2000 Tg2000]]. The local and global theory for this equation may well be improvable.
| |
| * In the nonrelativistic limit this equation converges to a Maxwell-Poisson system [[references.html#MasNa2003 MasNa2003]]
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''The Maxwell-Dirac equation'''</center>
| |
| | |
| [More info on this equation would be greatly appreciated. - Ed.]
| |
| | |
| This equation essentially reads
| |
| | |
| <center>''D_A'' <font face="Symbol">y </font><nowiki>=</nowiki><font face="Symbol"> - y</font><br /> \Box A + <font face="Symbol">Ñ</font> (\div_{x,t} A)= <u><font face="Symbol">y</font></u><font face="Symbol"> y</font></center>
| |
| | |
| where <font face="Symbol">y</font> is a spinor field (solving a coupled massive Dirac equation), and ''D'' is the Dirac operator with connection A. We put <font face="Symbol">y</font> in H^{s_1} and A in H^{s_2} x H^{s_2 - 1}.
| |
| | |
| * Scaling is (s_1, s_2) = (n/2-3/2, n/2-1).
| |
| * When n=1, there is GWP for small smooth data [[references.html#Chd1973 Chd1973]]
| |
| * When n=3 there is LWP for (s_1, s_2) = (1, 1) in the Coulomb gauge [[references.html#Bou1999 Bou1999]], and for (s_1, s_2) = (1/2+, 1+) in the Lorentz gauge [[references.html#Bou1996 Bou1996]]
| |
| ** For (s_1, s_2) = (1,2) in the Coulomb gauge this is in [[references.html#Bou1996 Bou1996]]
| |
| ** This has recently been improved by Selberg to (1/4+, 1). Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. A) is kept fixed.
| |
| ** LWP for smooth data was obtained in [[references.html#Grs1966 Grs1966]]
| |
| ** GWP for small smooth data was obtained in [[references.html#Ge1991 Ge1991]]
| |
| * When n=4, GWP for small smooth data is known (Psarelli?)
| |
| | |
| In the nonrelativistic limit this equation converges to a Maxwell-Poisson system for data in the energy space [BecMauSb-p2]; furthermore one has a local well-posedness result which grows logarithmically in the asymptotic parameter. Earlier work appears in [[references.html#MasNa2003 MasNa2003]].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Dirac-Klein-Gordon equation'''</center>
| |
| | |
| [More info on this equation would be greatly appreciated. - Ed.]
| |
| | |
| This equation essentially reads
| |
| | |
| <center>''D'' <font face="Symbol">y </font><nowiki>= </nowiki><font face="Symbol">f y - y</font><br /> Box <font face="Symbol">f </font><nowiki>= </nowiki><u><font face="Symbol">y</font></u><font face="Symbol"> y</font></center>
| |
| | |
| where <font face="Symbol">y</font> is a spinor field (solving a coupled massive Dirac equation), ''D'' is the Dirac operator and <font face="Symbol">f </font>is a scalar (real) field. We put <br /><font face="Symbol">y </font>in H^{s_1} and (<font face="Symbol">f</font>, <font face="Symbol">f</font>_t) in H^{s_2} x H^{s_2 - 1}.
| |
| | |
| The energy class is essentially (s_1,s_2) = (1/2,1), but the energy density is not positive. However, the L^2 norm of <font face="Symbol">y </font>is also positive and conserved..
| |
| | |
| * Scaling is (s_1, s_2) = (n/2-3/2, n/2-1).
| |
| * When n=1 there is GWP for (s_1,s_2) = (1,1) [[references.html#Chd1973 Chd1973]], [[references.html#Bou2000 Bou2000]] and LWP for (s_1, s_2) = (0, 1/2) [[references.html#Bou2000 Bou2000]].
| |
| * When n=2 there are some LWP results in [[references.html#Bou2001 Bou2001]]
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Nonlinear Dirac equation'''</center>
| |
| | |
| This equation essentially reads
| |
| | |
| <center>''D'' <font face="Symbol">y - </font>m <font face="Symbol">y </font><nowiki>= </nowiki><font face="Symbol">l(g y, y) y</font></center>
| |
| | |
| where <font face="Symbol">y</font> is a spinor field, m > 0 is the mass, <font face="Symbol">l</font> is a complex parameter, <font face="Symbol">g</font> is the zeroth Pauli matrix, and (,) is the spinor inner product.
| |
| | |
| * Scaling is s_c =1 (at least in the massless case m=0).
| |
| * In R^3, LWP is known for H^s when s > 1 [[references.html#EscVe1997 EscVe1997]]
| |
| ** This can be improved to LWP in H^1 (and GWP for small H^1 data) if an epsilon of additional regularity as assumed in the radial variable [MacNkrNaOz-p]; in particular one has GWP for radial H^1 data.
| |
| * In R^3, GWP is known for small H^s data when s > 1 [MacNaOz-p2].Some results on the nonrelativistic limit of this equation are also obtained in that paper.
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''DDNLW'''</center>
| |
| | |
| We use DDNLW to denote a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.
| |
| | |
| <center>\Box <font face="Symbol">f</font> = <font face="Symbol">G</font>(<font face="Symbol">f</font>) D<font face="Symbol">f</font> D<font face="Symbol">f</font>.</center>
| |
| | |
| A fairly trivial example of such equations arise by considering fields of the form <font face="Symbol">f</font> = f(u), where f is a given smooth function (e.g. f(x) = exp(ix)) and u is a scalar solution to the free wave equation. In this case <font face="Symbol">f</font> solves the equation
| |
| | |
| <center>\Box <font face="Symbol">f</font><nowiki> = f''(</nowiki><font face="Symbol">f</font>) Q_0(<font face="Symbol">f</font>,<font face="Symbol">f</font>)</center>
| |
| | |
| where Q_0 is the null form
| |
| | |
| <center>Q_0(<font face="Symbol">f</font>, <font face="Symbol">y</font>) := \partial_<font face="Symbol">af</font> \partial^<font face="Symbol">a</font> <font face="Symbol">y</font> = <font face="Symbol">Ñf</font> . <font face="Symbol">Ñy</font> - <font face="Symbol">f</font>_t <font face="Symbol">y</font>_t.</center>
| |
| | |
| The above equation can be viewed as the wave equation on the one-dimensional manifold f(R), with the induced metric from R. The higher-dimensional version of this equation is known as the ''[#wm wave map equation]''.
| |
| | |
| DDNLW behaves like DNLW but with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate <br /> DDNLW and view the resulting as an instance of DNLW for the system of fields (<font face="Symbol">f</font>, D<font face="Symbol">f</font>). The reader should compare the results below with the [#dnlw-2 corresponding results for quadratic DNLW].
| |
| | |
| The critical regularity is s_c = d/2. For subcritical regularities s > s_c, <font face="Symbol">f</font> has some Holder continuity, and so one heuristically expects the <font face="Symbol">G</font>(<font face="Symbol">f</font>) terms <br /> to be negligible. However, this term must play a crucial role in the critical case s=s_c. For instance, Nirenberg [ref?] observed that the real scalar equation
| |
| | |
| <center>\Box <font face="Symbol">f</font> = -<font face="Symbol">f</font> Q_0(<font face="Symbol">f</font>, <font face="Symbol">f</font>)</center>
| |
| | |
| is globally well-posed in H^{d/2}, but the equation
| |
| | |
| <center>\Box <font face="Symbol">f</font> = <font face="Symbol">f</font> Q_0(<font face="Symbol">f</font>, <font face="Symbol">f</font>)</center>
| |
| | |
| is ill-posed in H^{d/2}; this is basically because the non-linear operator f -> exp(if) is continuous on (real-valued) H^{d/2}, while f -> exp(f) is not.
| |
| | |
| Energy estimates show that the general DDNLW equation is locally well-posed in H^s for s > s_c + 1. Using Strichartz estimates this can be improved to s > s_c + 3/4 in two dimensions and s > s_c + 1/2 in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put <font face="Symbol">f</font> into L<sup>2</sup>_t L^<font face="Symbol">¥</font>_x (or L^4_t L^<font face="Symbol">¥</font>_x in two dimensions), so that one can then use the energy method.
| |
| | |
| Using X^{s,b} estimates [[references.html#FcKl2000 FcKl2000]] instead of Strichartz estimates, one can improve this further to s > s_c + 1/4 in four dimensions and to the near-optimal s > s_c in five and higher dimensions [[references.html#Tt1999 Tt1999]].
| |
| | |
| Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in H<sup>2</sup> [[references.html#Lb1993 Lb1993]], although one can recover well-posedness in the Besov space B<sup>2</sup>_{2,1} [[references.html#Na1999 Na1999]], or with an epsilon of radial regularity [MacNkrNaOz-p]. It would be interesting to determine what the situation is in the other low dimensions.
| |
| | |
| If the quadratic non-linearity (<font face="Symbol">Ñf</font>)<sup>2</sup> is of the form Q_0(<font face="Symbol">f</font>,<font face="Symbol">f</font>) (or of a null form of similar strength) then the LWP theory can be pushed to s > s_c in all dimensions (see [KlMa1997], [KlMa1997b] for d >= 4, [[references.html#KlSb1997 KlSb1997]] for d >= 2, and [[references.html#KeTa1998b KeTa1998b]] for d=1).
| |
| | |
| If <font face="Symbol">G</font>(<font face="Symbol">f</font>) is constant and d is at least 4, then one has GWP outside of convex obstacles [Met-p2]
| |
| | |
| * For d >= 6 this is in [[references.html#ShbTs1986 ShbTs1986]]; for d >= 4 and the case of a ball this is in [[references.html#Ha1995 Ha1995]].
| |
| * Without an obstacle, one can use the [#gwp_qnlw general theory of quasilinear NLW].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Two-speed DDNLW'''</center>
| |
| | |
| * One can consider two-speed variants of DDNLW ([#two-speed see Overview]), when both F and G have the form <font face="Symbol">G</font>(U) DU DU.
| |
| * The Strichartz and energy estimates carry over without difficulty to this setting. The results obtained by X^{s,\theta} estimates change, however. The null forms are no longer as useful, however the estimates are usually more favourable because of the transversality of the two light cones. Of course, if F contains DuDu or G contains DvDv then one cannot do any better than the one-speed case.
| |
| * For d=2 one can obtain LWP for the near-optimal range s>3/2 when F does not contain DuDu and G does not contain DvDv [Tg-p].
| |
| * For d=1 one can obtain LWP for the near-optimal range s>1 when F does not contain DuDu and G does not contain DvDv [Tg-p].
| |
| * For d=3 one can obtain GWP for small compactly supported data for quasilinear equations with multiple speeds, as long as the nonlinearity has no explicit dependence on U [KeSmhSo-p3]
| |
| | |
| A special case of two-speed DDNLS arises in elasticity (more on this to be added in later).
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Wave maps'''</center>
| |
| | |
| Wave maps are maps <font face="Symbol">f</font> from R^{d+1} to a Riemannian manifold M which are critical points of the Lagrangian
| |
| | |
| <center>\int <font face="Symbol">f</font>_<font face="Symbol">a</font> . <font face="Symbol">f</font>^<font face="Symbol">a</font> dx dt.</center>
| |
| | |
| When M is flat, wave maps just obey the wave equation (if viewed in flat co-ordinates). More generally, they obey the equation
| |
| | |
| <center>Box <font face="Symbol">f</font> = <font face="Symbol">G</font>(<font face="Symbol">f</font>) Q_0(<font face="Symbol">f</font>, <font face="Symbol">f</font>)</center>
| |
| | |
| where <font face="Symbol">G</font>(<font face="Symbol">f</font>) is the second fundamental form and Q_0 is the null form [#ddnlw mentioned earlier]. When the target manifold is a unit sphere, this simplifies to
| |
| | |
| <center>Box <font face="Symbol">f</font> = -<font face="Symbol">f</font> Q_0(<font face="Symbol">f</font>,<font face="Symbol">f</font>)</center>
| |
| | |
| where <font face="Symbol">f</font> is viewed in Cartesian co-ordinates (and must therefore obey |<font face="Symbol">f</font><nowiki>|=1 at all positions and times in order to stay on the sphere). The sphere case has special algebraic structure (beyond that of other symmetric spaces) while also staying compact, and so the sphere is usually considered the easiest case to study. Some additional simplifications arise if the target is a Riemann surface (because the connection group becomes U(1), which is abelian); thus S</nowiki><sup>2</sup> is a particularly simple case.
| |
| | |
| This equation is highly geometrical, and can be rewritten in many different ways. It is also related to the Einstein equations (if one assumes various symmetry assumptions on the metric; see e.g. [[references.html#BgCcMc1995 BgCcMc1995]]).
| |
| | |
| The critical regularity is s_c = d/2. Thus the two-dimensional case is especially interesting, as the equation is then energy-critical. The sub-critical theory s > d/2 is fairly well understood, but the s_c = d/2 theory is quite delicate. A big problem is that H^{d/2} does not control L^<font face="Symbol">¥</font>, so one cannot localize to a small co-ordinate patch (or perform algebraic operations properly).
| |
| | |
| The positive and negative curvature cases are suspected to behave differently, especially at the critical regularity. Intuitively, the negative curvature space spreads the solution out more, thus giving a better chance for LWP and GWP.More recently, distinctions have arisen between the boundedly parallelizable case (where the exists an orthonormal frame whose structure constants and derivatives are bounded), and the isometrically embeddable case.For instance, hyperbolic space is in the former category but not in the latter; smooth compact manifolds such as the sphere are in both.
| |
| | |
| The general LWP/GWP theory (except for the special [#wm_on_R one-dimensional] and [#wm_on_R^2 two-dimensional] cases, which are covered in more detail below) is as follows.
| |
| | |
| * For d>=2 one has LWP in H^{n/2}, and GWP and regularity for small data, if the manifold can be isometrically embedded in Euclidean space [Tt-p2]
| |
| ** Earlier global regularity results in H^{n/2} are as follows.
| |
| *** For a sphere in d>=5, see [Ta-p5]; for a sphere in d >= 2, see [Ta-p6]..
| |
| *** The d >= 5 has been generalized to arbitrary manifolds which are boundedly parallelizable [KlRo-p].
| |
| *** This has been extended to d=4 by [[references.html#SaSw2001 SaSw2001]] and [[references.html#NdStvUh2003b NdStvUh2003b]]. In the constant curvature case one also has global well-posedness for small data in H^{n/2} [[references.html#NdStvUh2003b NdStvUh2003b]]. This can be extended to manifolds with bounded second fundamental form [[references.html#SaSw2001 SaSw2001]].
| |
| *** This has been extended to d=3 when the target is a Riemann surface [[references.html#Kri2003 Kri2003]], and to d=2 for hyperbolic space [Kri-p]
| |
| ** For the critical Besov space B^{d/2,1}_2 this is in [[references.html#Tt1998 Tt1998]] when d >= 4 and [[references.html#Tt2001b Tt2001b]] when d>=2. (See also [[references.html#Na1999 Na1999]] in the case when the wave map lies on a geodesic). For small data one also has GWP and scattering.
| |
| ** In the sub-critical spaces H^s, s > d/2 this was shown in [[references.html#KlMa1995b KlMa1995b]] for the d>=4 case and in [[references.html#KlSb1997 KlSb1997]] for d>=2.
| |
| *** For the model wave map equation this was shown for d>=3 in [[references.html#KlMa1997b KlMa1997b]].
| |
| ** If one replaces the critical Besov space by H^{n/2} then one has failure of analytic or C^2 local well-posedness for d>=3 [DanGe-p], and one has failure of continuous local well-posedness for d=1 [[references.html#Na1999 Na1999]], [[references.html#Ta2000 Ta2000]]
| |
| ** GWP is also known for smooth data close to a geodesic [[references.html#Si1989 Si1989]]. For smooth data close to a point this was in [[references.html#Cq1987 Cq1987]].
| |
| * For d >= 3 singularities can form from large data, even when the data is smooth and rotationally symmetric [[references.html#CaSaTv1998 CaSaTv1998]]
| |
| ** For d=3 this was proven in [[references.html#Sa1988 Sa1988]]
| |
| ** For d>=7 one can have singularities even when the target has negative curvature [[references.html#CaSaTv1998 CaSaTv1998]]
| |
| ** For d=3, numerics suggest that there is a transition between global existence for small data and blowup for large data, with the self-similar blowup solution being an intermediate attractor [Lie-p]
| |
| | |
| For further references see [[references.html#Sw1997 Sw1997]], [[references.html#SaSw1998 SaSw1998]], [KlSb-p].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Wave maps on R'''</center>
| |
| | |
| * Scaling is s_c = 1/2.
| |
| * LWP in H^s for s > 1/2 [[references.html#KeTa1998b KeTa1998b]]
| |
| ** Proven for s >= 1 in [Zh-p]
| |
| ** Proven for s > 3/2 by energy methods
| |
| ** One also has LWP in the space L^1_1 [[references.html#KeTa1998b KeTa1998b]]. Interpolants of this with the H^s results are probably possible.
| |
| ** One has ill-posedness for H^{1/2}, and similarly for Besov spaces near H^{1/2} such as B^{1/2,1}_2 [[references.html#Na1999 Na1999]], [[references.html#Ta2000 Ta2000]]. However, the ill-posedness is not an instance of blowup, only of a discontinuous solution map, and perhaps a weaker notion of solution still exists and is unique.
| |
| * GWP in H^s for s>3/4 [[references.html#KeTa1998b KeTa1998b]] when the target manifold is a sphere
| |
| ** Was proven for s >= 1 in [[references.html#Zh1999 Zh1999]] for general manifolds
| |
| ** Was proven for s >= 2 for general manifolds in [[references.html#Gu1980 Gu1980]], [[references.html#LaSh1981 LaSh1981]], [[references.html#GiVl1982 GiVl1982]], [[references.html#Sa1988 Sa1988]]
| |
| ** One also has GWP and scattering in L^1_1. [[references.html#KeTa1998b KeTa1998b]] One probably also has asymptotic completeness.
| |
| ** Scattering fails when the initial velocity is not conditionally integrable [[references.html#KeTa1998b KeTa1998b]].
| |
| ** It should be possible to improve the s>3/4 result by correction term methods, and perhaps to obtain interpolants with the L^{1,1} result. One should also be able to extend to general manifolds.
| |
| * ''Remark''<nowiki>: The non-linear term has absolutely no smoothing properties, because of the double derivative in the non-linearity and the lack of dispersion in the one-dimensional case.</nowiki>
| |
| * ''Remark''<nowiki>: The equation is completely integrable [</nowiki>[references.html#Pm1976 Pm1976]], but not in the same way as KdV, mKdV or 1D NLS. (The additional conserved quantities do not control H^s norms, but rather the pointwise distribution of the energy. Indeed, the energy density itself obeys the free wave equation!).When the target is a symmetric space, homoclinic periodic multisoliton solutions were constructed in [TeUh-p2].
| |
| * Remark: When the target manifold is S<sup>2</sup>, the wave map equation is related to the [#Sine-Gordon sine-Gordon equation] [[references.html#Pm1976 Pm1976]].Homoclinic periodic breather solutions were constructed in [[references.html#SaSr1996 SaSr1996]].
| |
| * When the target is a Lorentzian manifold, local existence for smooth solutions was established in [Cq-p2].A criterion on the target manifold to guarantee global existence of smooth solutions is in [Woo-p]; however if the target manifold is the Lorentz sphere S^{1,n-1} then there is a large class of data which blows up [Woo-p].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Wave maps on R<sup>2</sup>'''</center>
| |
| | |
| * Scaling is s_c = 1 (energy-critical).
| |
| * LWP in H^1 [Tt-p2]
| |
| ** For B^{1,1}_2 this is in [[references.html#Tt2001b Tt2001b]].
| |
| ** LWP in H^s, s>1 was shown in [[references.html#KlSb1997 KlSb1997]].
| |
| ** For s>7/4 this can be shown by Strichartz methods.
| |
| ** For s>2 this can be shown by energy estimates.
| |
| * GWP and regularity is known for small energy when the target manifold is boundedly parallelizable [Tt-p2]
| |
| ** When the target manifold is H^2, regularity was obtained by Krieger [Kri-p]
| |
| ** When the target manifold is a sphere, regularity was obtained in [Ta-p6]
| |
| ** For small B^{1,1}_2 data GWP is in [[references.html#Tt2001b Tt2001b]]
| |
| ** GWP and regularity for small H^1 data was known for corotational wave maps, and can be extended to large H^1 data when the target is geodesically convex [[references#SaSw1993 SaSw1993]], [Sw-p2]; see also [[references.html#SaTv1992 SaTv1992]], [Gl-p]. In the later papers the result is obtained for quite general rotationally symmetric manifolds, such as non-compact manifolds, although one generically expects blow-up for certain manifolds such as the sphere (see e.g. [Sw-p2], or the numerics in [[references.html#BizCjTb2001 BizCjTb2001]], [IbLie-p]). The question of large H^1 GWP and regularity is equivalent to the non-existence of non-constant harmonic maps on the target [Sw-p2]. The corotational results have been extended to wave maps with torsion in [[references.html#AcIb2000 AcIb2000]].
| |
| ** Regularity is also known for large smooth radial data [[references.html#CdTv1993 CdTv1993]] assuming a convexity condition on the target manifold. This convexity condition was relaxed in [[references.html#Sw2002 Sw2002]], and then removed completely in [[references.html#Sw2003 Sw2003]]. One also has a pointwise bound on the diameter of the range of the wave map in the radial case under similar conditions on the manifold [[references.html#CdTv1993b CdTv1993b]]
| |
| ** It is an important open problem whether one has regularity for all large smooth data, at least in the negative curvature case. A slightly harder problem would be to obtain GWP in the critical space H^1.
| |
| *** When the target manifold is a sphere, numerical evidence seems to suggest energy concentration and singularity formulation for large equivariant data [IbLie-p].In the equivariant case, examples of blowup in H^{1+eps} on domains |x|^alpha < t can be constructed if one adds a forcing term on the right-hand side [GeIv-p]
| |
| ** Global weak solutions are known for large energy data [[references.html#MuSw1996 MuSw1996]], [[references.html#FrMuSw1998 FrMuSw1998]], but as far as is known these solutions might develop singularities or become "ghost" solutions.
| |
| * When the domain and target are S<sup>2</sup>, stationary-rotating solutions exist and are stable with respect to corotational perturbations [[references.html#SaTv1997 SaTv1997]]
| |
| * BMO-type estimates on distance functions were obtained in [[references#Gl1998 Gl1998]]
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Quasilinear wave equations (QNLW)'''</center>
| |
| | |
| In a local co-ordinate chart, quasilinear wave equations (QNLW) take the form
| |
| | |
| <center>partial_<font face="Symbol">a</font> g^{<font face="Symbol">ab</font>}(u) partial_<font face="Symbol">b</font> u = F(u, Du).</center>
| |
| | |
| One could also consider equations where the metric depends on derivatives of u, but one can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case g^{00} = 1, g^{0i} = g^{i0} = 0 by a suitable change of variables. F is usually quadratic in the derivatives Du, as this formulation is then robust under many types of changes of variables.
| |
| | |
| Quasilinear NLWs appear frequently in general relativity. The most famous example is the [#Einstein Einstein equations], but there are others (coming from relativistic elasticity, hydrodynamics, [#Minimal_Surface_Equation minimal surfaces], etc. [Ed: anyone willing to contribute information on these other equations (even just their name and form) would be greatly appreciated.]). The most interesting dimension is of course the physical dimension d=3.
| |
| | |
| Classically one has LWP for H^s when s > d/2+1 [[references.html#HuKaMar1977 HuKaMar1977]], but the [#dnlw semilinear theory] suggests that we should be able to improve this to s > s_c = d/2 with a null condition, and to s > d/2 + max(1/2, (d-5)/4) without one (these results would be sharp even in the semilinear case). In principle Strichartz estimates should be able to push down to s > d/2 + 1/2, but only partial results of this type are known. Specifically:
| |
| | |
| * When d=2 one has LWP in the expected range s > d/2 + 3/4 without a null condition [SmTt-p]
| |
| ** For s > d/2 + 3/4 + 1/12 this is in [Tt-p5] (using the FBI transform).
| |
| ** For s > d/2 + 3/4 + 1/8 this is in [[references.html#BaCh1999 BaCh1999]] (using FIOs) and [[references.html#Tt2000 Tt2000]] (using the FBI transform).
| |
| * When d=3,4,5 one has LWP for s > d/2 + 1/2 [SmTt-p] (using parametrices and the equation for the metric); in the specific case of the [Einstein%20equations Einstein equations] see [KlRo-p3], [KlRo-p4], [KlRo-p5] (using vector fields and the equation for the metric)
| |
| ** For s > d/2 + 1/2 + 1/7 (approx) and d=3 this is in [KlRo-p2] (vector fields and the equation for the metric)
| |
| ** For s > d/2 + 1/2 + 1/6 and d=3 this is in [Tt-p5] (using the FBI transform).
| |
| ** For s > d/2 + 1/2 + 1/5 (approx) and d=3 this is in [Kl-p2] (vector fields methods).
| |
| ** For s > d/2 + 1/2 + 1/4 and d>= 3 this is in [[references.html#BaCh1999 BaCh1999]] (using FIOs) and [[references.html#Tt2000 Tt2000]] (using the FBI transform). See also [[references.html#BaCh1999b BaCh1999b]].
| |
| | |
| A special type of QNLW is the cubic equations, where g itself obeys an elliptic equaton of the form Delta g = |Du|^2, and the non-linearity is of the form Dg Du. For such equations, we have LPW for s > d/2 + 1/6 when d >= 4 [BaCh-p], [[references.html#BaCh2002 BaCh2002]]. This equation has some similarity with the differentiated wave map equation in the Coulomb gauge.
| |
| | |
| For small smooth compactly supported data of size <font face="Symbol">e</font> and smooth non-linearities, the GWP theory for QNLW is as follows.
| |
| | |
| * If the non-linearity is a null form, then one has GWP for d>=3; in fact one can take the data in a weighted Sobolev space H^{4,3} x H^{3,4} [[references.html#Cd1986 Cd1986]].
| |
| ** Without the null structure, one has almost GWP in d=3 [[references.html#Kl1985b Kl1985b]], and this is sharp [[references.html#Jo1981 Jo1981]], [[references.html#Si1983 Si1983]]
| |
| *** In the semi-linear case and when the nonlinearity is quadratic in the derivatives, this is also true outside of a compact non-trapping obstacle [KeSmhSo-p2]. This has been generalized to the quasi-linear case in [KeSmhSo-p3] (and non-linear Dirichlet wave equations are also treated there, as are multiple speeds).
| |
| ** With a null structure and outside a star-shaped obstacle with Dirichlet conditions and d=3, one has GWP for small data in H^{9,8} x H^{8,9} which are compatible with the boundary [KeSmhSo-p]. Earlier work in this direction is in [[references.html#Dt1990 Dt1990]].
| |
| *** For radial data and obstacle this was obtained in [[references.html#Go1995 Go1995]]; see also [[references.html#Ha1995 Ha1995]], [[references.html#Ha2000 Ha2000]].
| |
| *** In the semilinear case, the non-trapping condition was removed in [MetSo-p], even in the multiple speed case, provided one has an exponential decay result near the obstacle (this is true, for instance, if the obstacle is a union of a finite number of sufficiently separated strictly convex bodies).
| |
| ** For d>3 or for cubic nonlinearities one has GWP regardless of the null structure [[refernces.html#KlPo1983 KlPo1983]], [[references.html#Sa1982 Sa1982]], [[references.html#Kl1985b Kl1985b]].
| |
| *** In three dimensions with a null structure, for systems with multiple wave speeds, one has GWP [[references.html#So2001 So2001]]
| |
| *** In the exterior of an nontrapping obstacle with Dirichlet conditions, with multiple speeds, one has GWP for sufficiently smooth and decaying data obeying the usual compatibility conditions at the boundary [MetSo-p2], if the quasilinear terms obey some symmetry conditions and the semilinear terms are quadratic in Du
| |
| ***# When the obstacle is a ball this is in [[references.html#Ha1995 Ha1995]].
| |
| ***# For d >= 6 outside of a starshaped obstacle this is in [[references.html#ShbTs1984 ShbTs1984]], [[references.html#ShbTs1986 ShbTs1986]].
| |
| | |
| <div class="MsoNormal" style="text-align: center"><center>
| |
| ----
| |
| </center></div>
| |
| | |
| <center>'''Einstein equations'''</center>
| |
| | |
| [Note: This is an immense topic, and we do not even begin to do it justice with this very brief selection of results. For more detail, we recommend the very nice [http://relativity.livingreviews.org/Articles/lrr-2002-6/index.html survey on existence and global dynamics of the Einstein equations by Alan Rendall]. Further references are, of course, always appreciated. We thank Uwe Brauer, Daniel Pollack, and some anonymous contributors to this section.]
| |
|
| |
|
| The (vacuum) Einstein equations take the form
| | All relativistic field equations in (classical) physics are variants of the [[free wave equation]] or [[Klein-Gordon equation]] on [[Minkowski space]]. |
|
| |
|
| <center>R_{<font face="Symbol">a</font> <font face="Symbol">b</font>} = C R g_{<font face="Symbol">a</font> <font face="Symbol">b</font>}</center>
| | There are several ways to perturb this equation. There are ''linear perturbations'', which include the addition of potential terms, connection terms, and drag terms, as well as the replacement of the flat Minkowski metric with a more general curved metric, or by placing obstacles or otherwise changing the topology of the domain manifold <math>R^{1+d}</math>. |
|
| |
|
| where g is the metric for a 3+1-dimensional manifold, R is the Ricci curvature tensor, and C is an absolute constant. The Cauchy data for this problem is thus a three-dimensional Riemannian manifold together with the second fundamental form of this manifold (roughly speaking, this is like the initial position and initial velocity for the metric g). However, these two quantities are not completely independent; they must obey certain ''constraint equations''. These equations are now known to be well behaved for all s > 3/2 [Max-p], [Max2005] (see also earlier work in higher regularities in [[references.html#RenFri2000 RenFri2000]], [[references.html#Ren2002 Ren2002]]).
| | Here we shall focus more on purely ''non-linear'' perturbations, which collapse to a constant-coefficient wave equation in the [[small amplitude limit]]. |
| | In the fullest generality, this would mean studying equations of the form |
|
| |
|
| Because of the diffeomorphism invariance of the Einstein equations, these equations are not hyperbolic as stated. However, this can be remedied by choosing an appropriate choice of co-ordinate system (a gauge, if you will). One popular choice is ''harmonic co-ordinates'' or ''wave co-ordinates'', where the co-ordinate functions x<sup><font face="Symbol">a</font></sup> are assumed to obey the wave equation Box<sub>g</sub> x<sup><font face="Symbol">a</font></sup> = 0 with respect to the metric g. In this case the Einstein equations take a form which (in gross caricature) looks something like
| | <center><math>F(f, Df, D^2_{}f) = 0</math></center> |
|
| |
|
| <center>Box<sub>g</sub> g = <font face="Symbol">G</font>(g) Q(dg, dg) + lower order terms</center> | | where <math>D</math> denotes differentiation in space or time and the Taylor expansion of <math>F</math> to first order is the free wave or Klein-Gordon equation. Such fully non-linear equations, though, are very difficult to study, and have only really been analyzed in the one-dimensional case (in which case it can be subsumed into the general theory of 1+1-dimensional hyperbolic systems). In higher dimensions the only known tool to analyze this case is to differentiate the equation, turning it into a quasi-linear system. As such we do not discuss fully non-linear wave equations here. Instead, we consider three less general types of equations, which in increasing order of complexity are the [[NLW|semi-linear]], [[DNLW|semi-linear with derivatives]], and [[QNLW|quasi-linear]] equations. |
|
| |
|
| where Q is some quadratic form of the first two derivatives. In other words, it becomes a [#Quasilinear quasilinear wave equation]. One would then specify initial <br /> data on the initial surface x = 0; the co-ordinate x plays the role of time, locally at least.
| | Non-linear wave equations are often the [[Euler-Lagrange equation]] for some [[variational problem]]. This usually generates the conserved [[stress-energy tensor]], which is of fundamental importance in the analysis of such equations, especially for the global-in-time theory. |
|
| |
|
| * Scaling is s_c = 3/2. Thus energy is super-critical, which seems to make a large data global theory extremely difficult.
| | The principle of relativity asserts that the equations of physics are covariant with respect to the underlying geometry of spacetime. This can be exploited in a number of ways. One is via [[stress-energy tensor]] mentioned previously. Another is via [[conformal transformation]] of spacetime. A third is via [[finite speed of propagation]]. The covariance also generates some important [[null structure]]s in the nonlinear components of the equation. |
| * LWP is known in H^s for s > 5/2 by energy estimates (see [[references.html#HuKaMar1977 HuKaMar1977]], [AnMc-p]; for smooth data s > 4 this is in [[references.html#Cq1952 Cq1952]]) - given that the initial data obeys the constraint equations, of course.
| |
| ** This result can be improved to s>2 by the [#Quasilinear recent quasilinear theory] (see in particular [KlRo-p3], [KlRo-p4], [KlRo-p5]).
| |
| ** This result has now been improved further to s=2 [KlRo-p6], [KlRo-p7], [KlRo-p8]
| |
| ** For smooth data, one has a (possibly geodesically incomplete) maximal Cauchy development [[references.html#CqGc1969 CqGc1969]].
| |
| * GWP for small smooth asymptotically flat data was shown in [[references.html#CdKl1993 CdKl1993]] (see also [[references.html#CdKl1990 CdKl1990]]). In other words, Minkowski space is stable.
| |
| ** Another proof using the double null foliation is in [[references.html#KlNi2003 KlNi2003]], [KlNi-p]
| |
| ** Another proof of this fact (using the Lorenz gauge, and assuming Schwarzschild metric outside of a compact set) is in [LbRo-p] (see also [[references.html#LbRo2003 LbRo2003]] for a treatment of the asymptotic dynamics)
| |
| ** Singularities must form if there is a trapped surface [[references.html#Pn1965 Pn1965]].
| |
| * Many special solutions (Schwarzschild space, Kerr space, etc.) The stability of these spaces is a very interesting (and difficult) question.
| |
| * The equations can simplify under additional symmetry assumptions. The U(1)-symmetric case reduces to a system of equations which closely resembles the [#wm_on_R^2 two-dimensional wave maps equation] (with the target manifold being hyperbolic space H^2).
| |
| * Another important question is the ''Cosmic Censorship Hypothesis''. Informally, this asserts that singularities are always (or at least generically) concealed by black holes. Another (slightly different) version of the conjecture asserts that the maximal Cauchy development is always inextendable as a (suitably regular) Lorentzian manifold. This question is already interesting in the U(1)-symmetric case (perhaps with a matter coupling).
| |
|
| |
|
| <div class="MsoNormal" style="text-align: center"><center>
| | The perturbative theory for nonlinear wave equations rests on various linear, bilinear, and nonlinear [[wave estimates|estimates for the linear wave equation]]. |
| ----
| |
| </center></div>
| |
|
| |
|
| <center>'''Minimal surface equation'''</center>
| | ===Dependence on dimension=== |
|
| |
|
| This quasilinear equation takes the form
| | The one-dimensional case <math>d=1</math> is special for several reasons. Firstly, there is the very convenient null co-ordinate system <math>u = t+x, v = t-x</math> which can be used to factorize <math>\Box</math>. Also, the stress-energy tensor often becomes trace-free, which leads to better conformal invariance properties. There are a vastly larger number of conformal transformations, indeed anything of the form <math>(u,v) \rightarrow ( F (u), Y (v))</math> is conformal. Also, the one-dimensional wave equation has no decay, local smoothing, or dispersion properties, and its solutions are essentially travelling waves. Finally, there are a much larger range of spaces beyond Sobolev spaces which are available for well-posedness theory, because the free wave evolution operator preserves all translation-invariant spaces. (In two and higher dimensions only <math>L^2</math>-based spaces such as Sobolev spaces H^s are preserved, because waves can focus at a point (or defocus from a point)). |
|
| |
|
| <center>partial<sub><font face="Symbol">a</font></sub> [ (1 + <font face="Symbol">f<sub>b</sub>f<sup>b</sup></font>)^{-1/2} <font face="Symbol">f<sup>a</sup></font> ] = 0</center> | | The higher-dimensional case <math>d>1</math> is usually quite different from the one-dimensional case, although in spherically symmetric situations one can obtain similar behaviour, especially when viewed in the null co-ordinates <math>u = t+r, v = t-r</math>. Indeed one can think of spherically symmetric wave equations as one-dimensional wave equations with a singular drag term <math>(n-1) f _r / r</math>. |
|
| |
|
| where <font face="Symbol">f</font> is a scalar function on R^{n-1}xR (the graph of a surface in R^n x R<font face="Symbol">).</font>This is the Minkowski analogue of the minimal surface equation in Euclidean space, see [[references.html#Hp1994 Hp1994]].
| | ===Specific wave equations=== |
|
| |
|
| * This is a [#Quasilinear quasilinear wave equation], and so LWP in H^s for s > n/2 + 1 follows from energy methods, with various improvements via Strichartz possible. However, it is likely that the special structure of this equation allows us to do better. | | * [[semilinear NLW|Semilinear wave equations]] ([[sine-Gordon]], etc.) |
| * GWP for small smooth compactly supported data is in [Lb-p]. | | * [[DNLW|NLW with derivatives]] |
| | ** [[linear-derivative nonlinear wave equations]] ([[YM|Yang-Mills]], [[YMH|Yang-Mills-Higgs]], [[MKG|Maxwell-Klein-Gordon]]) |
| | ** [[DDNLW|quadratic-derivative nonlinear wave equations]] ([[wave maps]]) |
| | * [[Dirac equations|Dirac-type equations]] |
| | * [[QNLW|Quasilinear wave equations]] ([[Einstein]], [[minimal surface equation|minimal surface]], etc.) |
| | * [[GMPDE|Generalized Microstructure PDE]] |
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| </div>
| | [[Category:Wave]] |
| | [[Category:Equations]] |
