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By '''quadratic-derivative nonlinear wave equation''' (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e. | By '''quadratic-derivative nonlinear wave equation''' (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e. | ||
<center>\Box f = G ( f ) D f D f .</center> | <center><math>\Box f = G ( f ) D f D f</math> .</center> | ||
A fairly trivial example of such equations arise by considering fields of the form f = 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 f solves the equation | A fairly trivial example of such equations arise by considering fields of the form f = 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 f solves the equation | ||
<center>\Box f | <center><math>\Box f = f''( f ) Q_0( f , f )</math></center> | ||
where Q_0 is the [[null form]] | where <math>Q_0</math> is the [[null form]] | ||
<center><math>Q_0( \phi , \psi ) := \partial_\alpha \phi \partial^\alpha \psi = \nabla \phi \cdot \nabla \psi - \phi_t \psi_t.</math></center> | <center><math>Q_0( \phi , \psi ) := \partial_\alpha \phi \partial^\alpha \psi = \nabla \phi \cdot \nabla \psi - \phi_t \psi_t.</math></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 [[wave maps|wave map equation]]. | The above equation can be viewed as the wave equation on the one-dimensional manifold <math>f(R)</math>, with the induced metric from <math>R</math>. The higher-dimensional version of this equation is known as the [[wave maps|wave map equation]]. | ||
Quadratic-derivative nonlinear wave equations behave like their [[linear-derivative nonlinear wave equations|linear-derivative counterparts]], with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate the DDNLW equation and view the resulting as an instance of a linear-derivative NLW for the system of fields ( f , D f ). The reader should compare the results below with the [[linear-derivative nonlinear wave equations|linear-derivative counterparts]]. | Quadratic-derivative nonlinear wave equations behave like their [[linear-derivative nonlinear wave equations|linear-derivative counterparts]], with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate the DDNLW equation and view the resulting as an instance of a linear-derivative NLW for the system of fields ( <math>f</math> , <math>D f</math> ). The reader should compare the results below with the [[linear-derivative nonlinear wave equations|linear-derivative counterparts]]. | ||
The critical regularity is s_c = d/2. For subcritical regularities s > s_c, f has some Holder continuity, and so one heuristically expects the G ( f ) 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 | The critical regularity is <math>s_c = d/2</math>. For subcritical regularities <math>s > s_c</math>, f has some Holder continuity, and so one heuristically expects the <math>G ( f )</math> terms <br /> to be negligible. However, this term must play a crucial role in the critical case <math>s=s_c</math>. For instance, Nirenberg [ref?] observed that the real scalar equation | ||
<center>\Box f = - f Q_0( f , f )</center> | <center><math>\Box f = - f Q_0( f , f )</math></center> | ||
is globally well-posed in H^{d/2}, but the equation | is globally well-posed in <math>H^{d/2}</math>, but the equation | ||
<center>\Box f = f Q_0( f , f )</center> | <center><math>\Box f = f Q_0( f , f )</math></center> | ||
is ill-posed in H^{d/2}; this is basically because the non-linear operator f | is ill-posed in <math>H^{d/2}</math>; this is basically because the non-linear operator <math>f \rightarrow exp(if)</math> is continuous on (real-valued) <math>H^{d/2}</math>, while <math>f \rightarrow exp(f)</math> 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 f into <math>L^2_t L^\infty_x</math> (or <math>L^4_t L^\infty_x</math> in two dimensions), so that one can then use the energy method. | Energy estimates show that the general DDNLW equation is locally well-posed in <math>H^s</math> for <math>s > s_c + 1</math>. Using Strichartz estimates this can be improved to <math>s > s_c + 3/4</math> in two dimensions and <math>s > s_c + 1/2</math> in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put <math>f</math> into <math>L^2_t L^\infty_x</math> (or <math>L^4_t L^\infty_x</math> in two dimensions), so that one can then use the energy method. | ||
Using X^{s,b} estimates ([[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 ([[Tt1999]]). | Using <math>X^{s,b}</math> estimates ([[FcKl2000]]) instead of Strichartz estimates, one can improve this further to <math>s > s_c + 1/4</math> in four dimensions and to the near-optimal <math>s > s_c</math> in five and higher dimensions ([[Tt1999]]). | ||
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in | Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in <math>H^2</math> ([[Lb1993]]), although one can recover well-posedness in the Besov space <math>B^2_{2,1}</math> ([[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 is of the form Q_0( f , f ) (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, [[KlSb1997]] for d \geq 2, and [[KeTa1998b]] for d=1). | If the quadratic non-linearity is of the form <math>Q_0( f , f )</math> (or of a null form of similar strength) then the LWP theory can be pushed to <math>s > s_c</math> in all dimensions (see [[KlMa1997], [[KlMa1997b]] for <math>d >= 4</math>, [[KlSb1997]] for <math>d \geq 2</math>, and [[KeTa1998b]] for <math>d=1</math>). | ||
If G ( f ) is constant and d is at least 4, then one has GWP outside of convex obstacles [[Met-p2]] | If <math>G ( f )</math> is constant and <math>d</math> is at least 4, then one has GWP outside of convex obstacles [[Met-p2]] | ||
* For d \geq 6 this is in [[ShbTs1986]]; for d \geq 4 and the case of a ball this is in [[Ha1995]]. | * For <math>d \geq 6</math> this is in [[ShbTs1986]]; for <math>d \geq 4</math> and the case of a ball this is in [[Ha1995]]. | ||
* Without an obstacle, one can use the [[QNLW|general theory of quasilinear NLW]]. | * Without an obstacle, one can use the [[QNLW|general theory of quasilinear NLW]]. | ||
Latest revision as of 20:10, 3 August 2006
By quadratic-derivative nonlinear wave equation (DDNLW), we mean a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.
A fairly trivial example of such equations arise by considering fields of the form f = 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 f solves the equation
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q_0} is the null form
The above equation can be viewed as the wave equation on the one-dimensional manifold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(R)} , with the induced metric from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R} . The higher-dimensional version of this equation is known as the wave map equation.
Quadratic-derivative nonlinear wave equations behave like their linear-derivative counterparts, with all fields requiring one more derivative of regularity. One explicit way to make this connection is to differentiate the DDNLW equation and view the resulting as an instance of a linear-derivative NLW for the system of fields ( Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D f} ). The reader should compare the results below with the linear-derivative counterparts.
The critical regularity is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = d/2}
. For subcritical regularities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c}
, f has some Holder continuity, and so one heuristically expects the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G ( f )}
terms
to be negligible. However, this term must play a crucial role in the critical case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=s_c}
. For instance, Nirenberg [ref?] observed that the real scalar equation
is globally well-posed in , but the equation
is ill-posed in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2}} ; this is basically because the non-linear operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \rightarrow exp(if)} is continuous on (real-valued) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{d/2}} , while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f \rightarrow exp(f)} is not.
Energy estimates show that the general DDNLW equation is locally well-posed in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c + 1} . Using Strichartz estimates this can be improved to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c + 3/4} in two dimensions and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c + 1/2} in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f} into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2_t L^\infty_x} (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^4_t L^\infty_x} in two dimensions), so that one can then use the energy method.
Using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X^{s,b}} estimates (FcKl2000) instead of Strichartz estimates, one can improve this further to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c + 1/4} in four dimensions and to the near-optimal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c} in five and higher dimensions (Tt1999).
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^2} (Lb1993), although one can recover well-posedness in the Besov space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B^2_{2,1}} (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 is of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q_0( f , f )} (or of a null form of similar strength) then the LWP theory can be pushed to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > s_c} in all dimensions (see [[KlMa1997], KlMa1997b for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d >= 4} , KlSb1997 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \geq 2} , and KeTa1998b for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d=1} ).
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G ( f )} is constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d} is at least 4, then one has GWP outside of convex obstacles Met-p2
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \geq 6} this is in ShbTs1986; for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d \geq 4} and the case of a ball this is in Ha1995.
- Without an obstacle, one can use the general theory of quasilinear NLW.
The two-speed analogue of these equations has also been studied.
