Korteweg-de Vries equation on R

The local and global well-posedness theory for the Korteweg-de Vries equation on the real line is as follows.

• Scaling is s_c = -3/2.
• LWP in H^s for s >= -3/4 [CtCoTa-p], using a modified Miura transform and the [#mKdV_on_R mKdV theory]. This is despite the failure of the key bilinear estimate references.html#NaTkTs-p NaTkTs2001
  • For s within a logarithm for s=-3/4 [MurTao-p].
  • Was proven for s > -3/4 references.html#KnPoVe1996 KnPoVe1996.
  • Was proven for s > -5/8 in references.html#KnPoVe1993b KnPoVe1993b.
  • Was proven for s >= 0 in references.html#Bo1993b Bo1993b.
  • Was proven for s > 3/4 in references.html#KnPoVe1993 KnPoVe1993.
  • Was proven for s > 3/2 in references.html#BnSm1975 BnSm1975, references.html#Ka1975 Ka1975, references.html#Ka1979 Ka1979, references.html#GiTs1989 GiTs1989, references.html#Bu1980 Bu1980, ....
  • One has local ill-posedness(in the sense that the map is not uniformly continuous) for s < -3/4 (in the complex setting) by soliton examples [[references.html#KnPoVe-p KnPoVe-p]].
    • For real KdV this has also been established in [CtCoTa-p], by the Miura transform and the [#mKdV_on_R corresponding result for mKdV].
    • Below -3/4 the solution map was known to not be C^3 references.html#Bo1993b Bo1993b, references.html#Bo1997 Bo1997; this was refined to C^2 in references.html#Tz1999b Tz1999b.
  • When the initial data is a real, rapidly decreasing measure one has a global smooth solution for t > 0 referencs.html#Kp1993 Kp1993. Without the rapidly decreasing hypothesis one can still construct a global weak solution references.html#Ts1989 Ts1989
• GWP in H^s for s > -3/4 (if u is real) references.html#CoKeStTaTk2003 CoKeStTkTa2003.
  • Was proven for s > -3/10 in references.html#CoKeStTkTa2001 CoKeStTkTa2001
  • Was proven for s>= 0 in references.html#Bo1993b Bo1993b. Global weak solutions in L^2 were constructed in references.html#Ka1983 Ka1983, references.html#KrFa1983 KrFa1983, and were shown to obey the expected local smoothing estimate. These weak solutions were shown to be unique in references.html#Zh1997b Zh1997b
  • Was proven for s>= 1 in references.html#KnPoVe1993 KnPoVe1993.
  • Was proven for s>= 2 in references.html#BnSm1975 BnSm1975, references.html#Ka1975 Ka1975, references.html#Ka1979 Ka1979, ....
  • Remark: In the complex setting GWP fails for large data with Fourier support on the half-line [Bona/Winther?], [Birnir], ????. This result extends to a wide class of dispersive PDE.
• By use of the inverse scattering transform one can show that smooth solutions eventually resolve into solitons, that two colliding solitons emerge as (slightly phase shifted) solitons, etc.
• Solitons are orbitally H^1 stable references.html#Bj1972 Bj1972
  • In H^s, 0 <= s < 1, the orbital stability of solitons is at most polynomial (the distance to the ground state manifold in H^s norm grows like at most O(t^{1-s+}) in time) [RaySt-p]
  • In L^2, orbital stability has been obtained in references.html#MeVe2003 MeVe2003.

KdV-like systems

The KdV equation can also be generalized to a 2x2 system

$\partial _{t}u+\partial _{x}^{3}u+a_{3}\partial _{x}^{3}v+u\partial _{x}u+a_{1}v\partial _{x}v+a_{2}\partial _{x}(uv)=0$
$b_{1}\partial _{t}v+\partial _{x}^{3}v+b_{2}a_{3}\partial _{x}^{3}u+v\partial _{x}v+b_{2}a_{2}u\partial _{x}u+b_{2}a_{1}\partial _{x}(uv)+r\partial _{x}v$

where b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in references.html#GeaGr1984 GeaGr1984 to study strongly interacting pairs of weakly nonlinear long waves, and studied further in references.html#BnPoSauTm1992 BnPoSauTm1992. In references.html#AsCoeWgg1996 AsCoeWgg1996 it was shown that this system was also globally well-posed on L^2.
It is an interesting question as to whether these results can be pushed further to match the KdV theory; the apparent lack of complete integrability in this system (for generic choices of parameters b_i, a_i, r) suggests a possible difficulty.