Comptes Rendus
Research article - Partial differential equations
Nonlinear asymptotic stability in L ∞ for Lipschitz solutions to scalar conservation laws
William Golding1
1 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 581-592.

In this note, we show nonlinear stability in L ∞ for Lipschitz solutions to genuinely nonlinear, multi-dimensional scalar conservation laws. As an application, we are able to compute algebraic decay rates of the L ∞ norm of perturbations of global-in-time Lipschitz solutions, including perturbations of planar rarefaction waves. Our analysis uses the De Giorgi method applied to the kinetic formulation and is an extension of the method introduced recently by Silvestre in [Comm. Pure Appl. Math, 72 (6): 1321-1348, 2019].

Dans cette note, nous montrons la stabilité non linéaire L ∞ pour les solutions lipschitziennes des lois de conservation scalaires, multidimensionnelles et vraiment non linéaires. En tant qu’application, nous sommes capables de calculer des taux de décroissance algébriques explicites de la norme L ∞ des perturbations des solutions lipschitziennes globales en temps, y compris les perturbations des ondes de raréfaction planes. Notre analyse utilise la méthode de De Giorgi appliquée à la formulation cinétique et est une extension de la méthode récemment introduite par Silvestre dans [Comm. Pure Appl. Math, 72 (6) : 1321-1348, 2019].

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DOI: 10.5802/crmath.553
Classification: 35B40, 35B35, 35B65, 35L65

William Golding 1

1 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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William Golding. Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 581-592. doi : 10.5802/crmath.553. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.553/

[1] G.-Q. Chen; H. Frid Decay of entropy solutions of nonlinear conservation laws, Arch. Ration. Mech. Anal., Volume 146 (1999) no. 2, pp. 95-127 | DOI | MR | Zbl

[2] G. Crippa; F. Otto; M. Westdickenberg Regularizing effect of nonlinearity in multidimensional scalar conservation laws, Transport equations and multi-D hyperbolic conservation laws (F. Ancona; S. Bianchini; R. M. Colombo; C. De Lellis; A. Marson; A. Montanari, eds.) (Lecture Notes of the Unione Matematica Italiana), Volume 5, Springer, 2008, pp. 77-128 | DOI | Zbl

[3] C. Dafermos Long time behavior of periodic solutions to scalar conservation laws in several space dimensions, SIAM J. Math. Anal., Volume 45 (2013) no. 4, pp. 2064-2070 | DOI | MR | Zbl

[4] C. Dafermos Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, 325, Springer, 2016 | DOI | MR | Zbl

[5] C. Dafermos The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., Volume 70 (1979) no. 2, pp. 167-179 | DOI | MR | Zbl

[6] C. Dafermos Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. R. Soc. Edinb., Sect. A, Math., Volume 99 (1985) no. 3-4, pp. 201-239 | DOI | MR | Zbl

[7] E. De Giorgi Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., Volume 3 (1957) no. 3, pp. 25-43 | MR | Zbl

[8] R. DiPerna Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., Volume 28 (1979) no. 1, pp. 137-188 | DOI | MR | Zbl

[9] C. De Lellis; F. Otto; M. Westdickenberg Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws, Arch. Ration. Mech. Anal., Volume 170 (2003) no. 2, pp. 137-184 | DOI | MR | Zbl

[10] A. Debussche; J. Vovelle Long time behavior in scalar conservation laws, Differ. Integral Equ., Volume 22 (2009) no. 3-4 | MR | Zbl

[11] B. Enquist; E. Weinan Large time behavior and homogenization of solutions of two-dimensional conservation laws, Commun. Pure Appl. Math., Volume 46 (1993) no. 1, pp. 1-26 | DOI | MR | Zbl

[12] W. Golding Unconditional regularity and trace results for the isentropic Euler equations with γ=3 (2022) | arXiv

[13] D. Hoff The sharp form of Oleinik’ s entropy condition in several space variables, Trans. Am. Math. Soc., Volume 276 (1983) no. 2, pp. 707-714 | DOI | MR | Zbl

[14] H. K. Jenssen; C. Sinestrari On the spreading of characteristics for non-convex conservation laws, Proc. R. Soc. Edinb., Sect. A, Math., Volume 131 (2001) no. 4, pp. 909-925 | DOI | MR | Zbl

[15] S. N. Kružkov First order quasilinear equations in several independent variables, Math. USSR, Sb., Volume 10 (1970) no. 2, pp. 217-243 | DOI | Zbl

[16] P. Lax Hyperbolic systems of conservation laws. II, Commun. Pure Appl. Math., Volume 10 (1957), pp. 537-566 | DOI | MR | Zbl

[17] P.-L. Lions; B. Perthame; E. Tadmor A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Am. Math. Soc., Volume 7 (1994) no. 1, pp. 169-191 | DOI | Zbl

[18] O. A. Oleinik Discontinuous solutions of nonlinear differential equations, Transl., Ser. 2, Am. Math. Soc., Volume 26 (1963), pp. 95-172 | DOI | MR | Zbl

[19] E. Y. Panov On decay of periodic entropy solutions to a scalar conservation law, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 6, pp. 997-1007 | DOI | Numdam | MR | Zbl

[20] D. Serre Long-time stability in systems of conservation laws, using relative entropy/energy, Arch. Ration. Mech. Anal., Volume 219 (2016), pp. 679-699 | DOI | MR | Zbl

[21] D. Serre Divergence-free positive symmetric tensors and fluid dynamics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 5, pp. 1209-1234 | DOI | Numdam | MR | Zbl

[22] D. Serre Compensated integrability. Applications to the Vlasov-Poisson equation and other models in mathematical physics, J. Math. Pures Appl., Volume 127 (2019), pp. 67-88 | DOI | MR | Zbl

[23] L. Silvestre Oscillation properties of scalar conservation laws, Commun. Pure Appl. Math., Volume 72 (2019) no. 6, pp. 1321-1348 | DOI | MR | Zbl

[24] D. Serre; L. Silvestre Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates, Arch. Ration. Mech. Anal., Volume 234 (2019) no. 3, pp. 1391-1411 | DOI | MR | Zbl

[25] L. Tartar Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot–Watt Symposium, Vol. IV (Pitman Research Notes in Mathematics Series), Volume 39, Pitman, 1979, pp. 136-212 | Zbl

[26] E. Tadmor; T. Tao Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs, Commun. Pure Appl. Math., Volume 60 (2007), pp. 1488-1521 | DOI | MR | Zbl

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