Regularity of solutions for the critical N-dimensional Burgers' equation

Chan, Chi Hin; Czubak, Magdalena

Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, pp. 471-501.

We consider the fractional Burgers' equation on ${ℝ}^N$ with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur and show existence of smooth solutions given any initial datum in $L^2\left({ℝ}^N\right)$.

Nous considérons l'équation de Burgers avec diffusion fractionnelle dans ${ℝ}^N$. Nous montrons l'existence de solutions globales regulières pour toute donnée initiale dans $L^2\left({ℝ}^N\right)$, en utilisant une version parabolique de la méthode de De Giorgi introduite par Caffarelli et Vasseur.

[1] Nathael Alibaud, Boris Andreianov, Non-uniqueness of weak solutions for the fractal burgers equation, arXiv:0907.3695 (2009) | Numdam | MR | Zbl

[2] Nathaël Alibaud, Jérôme Droniou, Julien Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ. 4 no. 3 (2007), 479-499 | MR | Zbl

[3] Nathael Alibaud, Cyril Imbert, Grzegorz Karch, Asymptotic properties of entropy solutions to fractal burgers equation, arXiv:0903.3394 (2009) | MR | Zbl

[4] Luis Caffarelli, Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 no. 7-9 (2007), 1245-1260 | MR | Zbl

[5] Luis Caffarelli, Alexis Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., in press; arXiv:math/0608447 | MR

[6] Peter Constantin, Diego Cordoba, Jiahong Wu, On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J. 50 no. special issue (2001), 97-107 | MR | Zbl

[7] Peter Constantin, Jiahong Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 no. 5 (1999), 937-948 | MR | Zbl

[8] Antonio Córdoba, Diego Córdoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations, Proc. Natl. Acad. Sci. USA 100 no. 26 (2003), 15316-15317 | MR | Zbl

[9] Antonio Córdoba, Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 no. 3 (2004), 511-528 | MR | Zbl

[10] Hongjie Dong, Dapeng Du, Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space, Discrete Contin. Dyn. Syst. 21 no. 4 (2008), 1095-1101 | MR | Zbl

[11] Hongjie Dong, Dapeng Du, Dong Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J. 58 no. 2 (2009), 807-821 | MR | Zbl

[12] Hongjie Dong, Nataša Pavlović, A regularity criterion for the dissipative quasi-geostrophic equations, arXiv:0710.5201 (2007)

[13] J. Droniou, T. Gallouët, J. Vovelle, Global solution smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ. 3 no. 3 (2003), 499-521 | MR | Zbl

[14] Grzegorz Karch, Changxing Miao, Xiaojing Xu, On convergence of solutions of fractal Burgers equation toward rarefaction waves, SIAM J. Math. Anal. 39 no. 5 (2008), 1536-1549 | MR | Zbl

[15] A. Kiselev, F. Nazarov, A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math. 167 no. 3 (2007), 445-453 | MR | Zbl

[16] Alexander Kiselev, Fedor Nazarov, Roman Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 no. 3 (2008), 211-240 | MR | Zbl

[17] Changxing Miao, Gang Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, arXiv:0805.3465 | MR | Zbl

[18] Serge Resnick, Dynamical problems in nonlinear advective partial differential equations, PhD thesis, University of Chicago, 1995