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 . 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 -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.
@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] Entropy formulation for fractal conservation laws, Journal of Evolution Equations 7 no. 1 (2007), 145-175 | MR 2305729 | Zbl 1116.35013
,[2] 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] 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] Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46 | MR 1637513 | Zbl 0911.35100
, , ,[6] Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231-252 | MR 1708995 | Zbl 0931.35015
, , ,[7] Asymptotics for conservation laws involving Lévy diffusion generators, Studia Math. 148 (2001), 171-192 | MR 1881259 | Zbl 0990.35023
, , ,[8] 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] 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] 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] Instabilities and nonlinear patterns of overdriven detonations in gases, , (ed.), Nonlinear PDE's in Condensed Matter and Reactive Flows, Kluwer (2002), 49-97 | Zbl 1271.76102
,[14] The Theory of Max Min, Springer, Berlin (1967)
,[15] 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] A numerical method for fractal conservation laws, Math. Comp. 79 (2010), 71-94 | MR 2552219
,[17] 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] Fractal first order partial differential equations, Arch. Ration. Mech. Anal. 182 no. 2 (2006), 299-331 | MR 2259335 | Zbl 1111.35144
, ,[19] 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] 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] Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli 11 (2005), 689-714 | MR 2158256 | Zbl 1122.60063
, , ,[22] 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] 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] First order quasilinear equations with several independent variables, Math. Sb. (N.S.) 81 no. 123 (1970), 228-255 | MR 267257 | Zbl 0215.16203
,[26] Well-posedness of the Cauchy problem for fractional power dissipative equations, Nonlinear Anal. 68 (2008), 461-484 | MR 2372358 | Zbl 1132.35047
, , ,[27] 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] 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] Lévy processes in the physical sciences, Lévy Processes, Birkhäuser Boston, Boston, MA (2001), 241-266 | MR 1833700 | Zbl 0982.60043
,