Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case

Molinet, Luc; Vento, Stéphane

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 531-560.

We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in $H^{-1}\left(ℝ\right)$ with a solution-map that is analytic from $H^{-1}\left(ℝ\right)$ to $C([0,T];H^{-1}\left(ℝ\right))$ whereas it is ill-posed in $H^s\left(ℝ\right)$, as soon as $s<-1$, in the sense that the flow-map $u_0\mapsto u\left(t\right)$ cannot be continuous from $H^s\left(ℝ\right)$ to even ${𝒟}^{\text{'}}\left(ℝ\right)$ at any fixed $t>0$ small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be useful to prove similar results for other dispersive-dissipative models.

Classification : 35E15, 35M11, 35Q53, 35Q60

[1] D. Bekiranov, The initial-value problem for the generalized Burgers’ equation, Differential Integral Equations 9 (6) (1996), 1253–1265. | MR | Zbl

[2] I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal. 233 (2006), 228–259. | MR | Zbl

[3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations. II. The KdV equation, Geom. Funct. Anal. 3 (1993), 209–262. | EuDML | MR | Zbl

[4] J. Bourgain, Periodic Korteveg de Vries equation with measures as initial data, Selecta Math. 3 (1993), 115–159. | MR | Zbl

[5] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), 1235–1293. | MR | Zbl

[6] D. B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burger’s equation, SIAM J. Math. Anal. 27 (1996), 708–724. | MR | Zbl

[7] P. Gérard, Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem, In: “International Congress of Mathematicians”, Vol. III, Eur. Math. Soc., Zürich, 2006, 157–182. | MR | Zbl

[8] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), In: “Séminaire Bourbaki 796”, Astérique 237, 1995, 163–187. | EuDML | Numdam | MR | Zbl

[9] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 133 (1997), 50–68. | MR | Zbl

[10] Z. Guo, Global Well-posedness of Korteweg-de Vries equation in $H^{-3/4}\left(ℝ\right)$, J. Math. Pures Appl. 91 (2009), 583–597. | MR | Zbl

[11] Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations 246 (2009), 3864–3901. | MR | Zbl

[12] T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(𝕋,ℝ)$, Duke Math. J. 135 (2006), 327–360. | MR | Zbl

[13] C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573–603. | MR | Zbl

[14] N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations 22 (2009), 447–464. | MR | Zbl

[15] L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case, Trans. Amer. Math. Soc., to appear. | MR | Zbl

[16] L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not. 37 (2002), 1979–2005. | MR | Zbl

[17] E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids 13 (1970), 1432–1434.

[18] T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), 839–908. | MR | Zbl

[19] T. Tao, Scattering for the quartic generalised Korteweg-de Vries equations, J. Differential Equations 232 (2007), 623–651. | MR | Zbl

[20] D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), 37–77. | MR | Zbl