Non-uniqueness of weak solutions for the fractal Burgers equation
Alibaud, Nathaël ; Andreianov, Boris
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 997-1016 / Harvested from Numdam

La notion de solution entropique de Kruzhkov a été étendue par Alibaud en 2007 aux lois de conservation avec un terme diffusif fractionnaire ; ceci a permis de démontrer que le prolème de Cauchy est bien posé dans le cadre L . Dans cet article, on montre que si l'ordre de l'opérateur de diffusion est strictement plus petit que un, alors il peut exister plusieurs solutions faibles ; on apporte ainsi une motivation supplémentaire à l'utilisation des solutions entropiques.

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional Laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the L -framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.

Publié le : 2010-01-01
DOI : https://doi.org/10.1016/j.anihpc.2010.01.008
Classification:  35L65,  35L67,  35L82,  35S10,  35S30
@article{AIHPC_2010__27_4_997_0,
     author = {Alibaud, Natha\"el and Andreianov, Boris},
     title = {Non-uniqueness of weak solutions for the fractal Burgers equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {997-1016},
     doi = {10.1016/j.anihpc.2010.01.008},
     mrnumber = {2659155},
     zbl = {1201.35006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_4_997_0}
}
Alibaud, Nathaël; Andreianov, Boris. Non-uniqueness of weak solutions for the fractal Burgers equation. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 997-1016. doi : 10.1016/j.anihpc.2010.01.008. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_4_997_0/

[1] N. Alibaud, Entropy formulation for fractal conservation laws, Journal of Evolution Equations 7 no. 1 (2007), 145-175 | MR 2305729 | Zbl 1116.35013

[2] N. Alibaud, J. Droniou, J. Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equation, Journal of Hyperbolic Differential Equations 4 no. 3 (2007), 479-499 | MR 2339805 | Zbl 1144.35038

[3] N. Alibaud, C. Imbert, Fractional semi-linear parabolic equations with unbounded data, Trans. Amer. Math. Soc. 361 (2009), 2527-2566 | MR 2471928 | Zbl 1173.35525

[4] N. Alibaud, C. Imbert, G. Karch, Asymptotic properties of entropy solutions to fractal Burgers equation, SIAM Journal on Mathematical Analysis, in press | MR 2607346

[5] P. Biler, T. Funaki, W. Woyczyński, Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46 | MR 1637513 | Zbl 0911.35100

[6] P. Biler, G. Karch, W. Woyczyński, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231-252 | MR 1708995 | Zbl 0931.35015

[7] P. Biler, G. Karch, W. Woyczyński, Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math. 148 (2001), 171-192 | MR 1881259 | Zbl 0990.35023

[8] P. Biler, G. Karch, W. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 613-637 | Numdam | MR 1849690 | Zbl 0991.35009

[9] J.-M. Bony, P. Courège, P. Priouret, Semi-groupe de Feller sur une variété à bord compacte et prblèmes aux limites intégro-différentiels du second-ordre donnant lieu au principe du maximum, Ann. Inst. Fourier 18 no. 2 (1968), 396-521 | Numdam | MR 245085 | Zbl 0181.11704

[10] S. Cifani, E.R. Jakobsen, K.H. Karlsen, The discontinuous Galerkin method for fractal conservation laws, 2009, submitted for publication | MR 2832791

[11] C.H. Chan, M. Czubak, Regularity of solutions for the critical N-dimensional Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 2 (2010), 471-501 | Numdam | MR 2595188 | Zbl 1189.35354

[12] C.H. Chan, M. Czubak, L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation, 2009, submitted for publication | MR 2600693

[13] P. Clavin, Instabilities and nonlinear patterns of overdriven detonations in gases, H. Berestycki, Y. Pomeau (ed.), Nonlinear PDE's in Condensed Matter and Reactive Flows, Kluwer (2002), 49-97 | Zbl 1271.76102

[14] J.-M. Danskyn, The Theory of Max Min, Springer, Berlin (1967)

[15] H. Dong, D. Du, D. Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J. 58 no. 2 (2009), 807-822 | MR 2514389 | Zbl 1166.35030

[16] J. Droniou, A numerical method for fractal conservation laws, Math. Comp. 79 (2010), 71-94 | MR 2552219

[17] J. Droniou, T. Gallouët, J. Vovelle, Global solution and smoothing effect for a non-local regularization of an hyperbolic equation, Journal of Evolution Equations 3 no. 3 (2003), 499-521 | MR 2019032 | Zbl 1036.35123

[18] J. Droniou, C. Imbert, Fractal first order partial differential equations, Arch. Ration. Mech. Anal. 182 no. 2 (2006), 299-331 | MR 2259335 | Zbl 1111.35144

[19] W. Hoh, Pseudo differential operators with negative definite symbols and the martingale problem, Stochastics 55 no. 3–4 (1995), 225-252 | MR 1378858 | Zbl 0880.47029

[20] B. Jourdain, S. Méléard, W. Woyczyński, A probabilistic approach for nonlinear equations involving the fractional Laplacian and singular operator, Potential Analysis 23 (2005), 55-81 | MR 2136209 | Zbl 1069.60056

[21] B. Jourdain, S. Méléard, W. Woyczyński, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli 11 (2005), 689-714 | MR 2158256 | Zbl 1122.60063

[22] G. Karch, C. Miao, X. Xu, On convergence of solutions of fractal Burgers equation toward rarefaction waves, SIAM J. Math. Anal. 39 (2008), 1536-1549 | MR 2377288 | Zbl 1154.35080

[23] K.H. Karlsen, S. Ulusoy, Stability of entropy solutions for Lévy mixed hyperbolic parabolic equations, 2009, submitted for publication | MR 2836797

[24] A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dynamics of Partial Differential Equations 5 no. 3 (2008), 211-240 | MR 2455893 | Zbl 1186.35020

[25] S.N. Kruzhkov, First order quasilinear equations with several independent variables, Math. Sb. (N.S.) 81 no. 123 (1970), 228-255 | MR 267257 | Zbl 0215.16203

[26] C. Miao, B. Yuan, B. Zhang, Well-posedness of the Cauchy problem for fractional power dissipative equations, Nonlinear Anal. 68 (2008), 461-484 | MR 2372358 | Zbl 1132.35047

[27] C. Miao, G. Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, J. Differential Equations 247 no. 6 (2009), 1673-1693 | MR 2553854 | Zbl 1184.35003

[28] O.A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Uspekhi Mat. Nauk 12 no. 3 (1957), Russian Mathematical Surveys 3 (1957) | MR 94541 | Zbl 0079.31402

[29] W. Woyczyński, Lévy processes in the physical sciences, Lévy Processes, Birkhäuser Boston, Boston, MA (2001), 241-266 | MR 1833700 | Zbl 0982.60043