Ill-posedness of nonlocal Burgers equations |
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Authors: | Sylvie Benzoni-Gavage Jean-François Coulombel Nikolay Tzvetkov |
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Institution: | aUniversité de Lyon, Université Lyon 1, INSA de Lyon, Ecole Centrale de Lyon, CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France;bCNRS, Université Lille 1, Laboratoire Paul Painlevé (UMR CNRS 8524) and EPI SIMPAF of INRIA Lille Nord Europe, Cité scientifique, Bâtiment M2, 59655 Villeneuve d?Ascq Cedex, France;cDépartement de Mathématiques (UMR CNRS 8088), Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France |
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Abstract: | Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter J.K. Hunter, Nonlinear surface waves, in: Current Progress in Hyberbolic Systems: Riemann Problems and Computations, Brunswick, ME, 1988, in: Contemp. Math., vol. 100, Amer. Math. Soc., 1989, pp. 185–202], and more recently by Benzoni-Gavage and Rosini S. Benzoni-Gavage, M. Rosini, Weakly nonlinear surface waves and subsonic phase boundaries, Comput. Math. Appl. 57 (3–4) (2009) 1463–1484], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage S. Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (3–4) (2009) 303–320] under an appropriate stability condition originally pointed out by Hunter. In this article, it is shown that the latter condition is not only sufficient for well-posedness in Sobolev spaces but also necessary. The main point of the analysis is to show that when the stability condition is violated, nonlocal Burgers equations reduce to second order elliptic PDEs. The resulting ill-posedness result encompasses various cases previously studied in the literature. |
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Keywords: | MSC: 35A10 35J15 35L65 35Q35 35S10 |
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