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 theory. This is despite the failure of the key bilinear estimate NaTkTs2001
  • For s within a logarithm for s=-3/4 MurTao-p.
  • Was proven for s > -3/4 KnPoVe1996.
  • Was proven for s > -5/8 in KnPoVe1993b.
  • Was proven for s >= 0 in Bo1993b.
  • Was proven for s > 3/4 in KnPoVe1993.
  • Was proven for s > 3/2 in BnSmr1975, Ka1975, Ka1979, GiTs1989, Bub1980, ....
  • 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 KnPoVe2001.
    • For real KdV this has also been established in CtCoTa-p, by the Miura transform and the corresponding result for mKdV.
    • Below -3/4 the solution map was known to not be C^3 Bo1993b, Bo1997; this was refined to C^2 in Tz1999b.
  • When the initial data is a real, rapidly decreasing measure one has a global smooth solution for t > 0 Kp1993. Without the rapidly decreasing hypothesis one can still construct a global weak solution Ts1989
• GWP in H^s for s > -3/4 (if u is real) CoKeStTkTa2003.
  • Was proven for s > -3/10 in CoKeStTkTa2001
  • Was proven for s>= 0 in Bo1993b. Global weak solutions in L^2 were constructed in Ka1983, KrFa1983, and were shown to obey the expected local smoothing estimate. These weak solutions were shown to be unique in Zh1997b
  • Was proven for s>= 1 in KnPoVe1993.
  • Was proven for s>= 2 in BnSmr1975, Ka1975, 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 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 and asymptotic stability have been obtained in 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 GeaGw1984 to study strongly interacting pairs of weakly nonlinear long waves, and studied further in BnPoSauTm1992. In 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.