# American Institute of Mathematical Sciences

June  2016, 11(2): 251-262. doi: 10.3934/nhm.2016.11.251

## Non-critical fractional conservation laws in domains with boundary

 1 Laboratoire de Mathématiques de Besancon U.F.R. S.T, 16 route de Gray, 25030 BESANCON, France

Received  May 2015 Revised  July 2015 Published  March 2016

We study bounded solutions for a multidimensional conservation law coupled with a power $s\in (0,1)$ of the Dirichlet laplacian acting in a domain. If $s \leq 1/2$ then the study centers on the concept of entropy solutions for which existence and uniqueness are proved to hold. If $s >1/2$ then the focus is rather on the $C^\infty$-regularity of weak solutions. This kind of results is known in $\mathbb{R}^N$ but perhaps not so much in domains. The extension given here relies on an abstract spectral approach, which would also allow many other types of nonlocal operators.
Citation: Matthieu Brassart. Non-critical fractional conservation laws in domains with boundary. Networks & Heterogeneous Media, 2016, 11 (2) : 251-262. doi: 10.3934/nhm.2016.11.251
##### References:
 [1] N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z.  Google Scholar [2] N. Alibaud and B. Andreianov, Non-uniqueness of weak solutions for fractal Burgers equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 997-1016. doi: 10.1016/j.anihpc.2010.01.008.  Google Scholar [3] N. Alibaud and M. Brassart, Entropy solutions to fractional conservation laws in domains, In preparation, 2015. Google Scholar [4] N. Alibaud and M. Brassart, Parabolicity for fractional conservation laws in domains, In preparation, 2015. Google Scholar [5] N. Alibaud, J. Droniou and J. Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ., 4 (2007), 479-499. doi: 10.1142/S0219891607001227.  Google Scholar [6] P.L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer-Verlag, 1967. Google Scholar [7] C. Bardos, A. Leroux and J. C. Nedelec, First-order quasilinear equations with boundary conditions, Comm. in Part. Diff. Eq., 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar [8] C. H. Chan and M. Czubak, Regularity of solutions for the critical N-dimensional Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008.  Google Scholar [9] C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization for the slightly supercritical quasi-geostrophic equation, Discrete and Continuous Dynamical Systems, 27 (2010), 847-861. doi: 10.3934/dcds.2010.27.847.  Google Scholar [10] S. Cifani and E. R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413-441. doi: 10.1016/j.anihpc.2011.02.006.  Google Scholar [11] H. Dong and D. Du, Finite time singularities and global well-posedness for fractal Burgers' equations, Indiana Univ. Math, 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505.  Google Scholar [12] J. Droniou, Th. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499-521. doi: 10.1007/s00028-003-0503-1.  Google Scholar [13] J. Endal and E. R. Jakobsen, $L^1$ contraction for bounded (nonintegrable) solutions of degenerate parabolic equations, SIAM J. Math. Anal., 46 (2014), 3957-3982. doi: 10.1137/140966599.  Google Scholar [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, 1981.  Google Scholar [15] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Mathematicae, 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar [16] S. N. Kruzhkov, First order quasilinear equations with several independent variables, Math. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar [17] C. Miao and G. Wu, Global well-posedness for the critical Burgers equation in critical Besov spaces, J. Diff. Eq., 247 (2009), 1673-1693. doi: 10.1016/j.jde.2009.03.028.  Google Scholar [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [19] K. Yosida, Functional Analysis, 4th edition, Springer-Verlag, 1974.  Google Scholar

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##### References:
 [1] N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z.  Google Scholar [2] N. Alibaud and B. Andreianov, Non-uniqueness of weak solutions for fractal Burgers equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 997-1016. doi: 10.1016/j.anihpc.2010.01.008.  Google Scholar [3] N. Alibaud and M. Brassart, Entropy solutions to fractional conservation laws in domains, In preparation, 2015. Google Scholar [4] N. Alibaud and M. Brassart, Parabolicity for fractional conservation laws in domains, In preparation, 2015. Google Scholar [5] N. Alibaud, J. Droniou and J. Vovelle, Occurrence and non-appearance of shocks in fractal Burgers equations, J. Hyperbolic Differ. Equ., 4 (2007), 479-499. doi: 10.1142/S0219891607001227.  Google Scholar [6] P.L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer-Verlag, 1967. Google Scholar [7] C. Bardos, A. Leroux and J. C. Nedelec, First-order quasilinear equations with boundary conditions, Comm. in Part. Diff. Eq., 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar [8] C. H. Chan and M. Czubak, Regularity of solutions for the critical N-dimensional Burgers' equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008.  Google Scholar [9] C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization for the slightly supercritical quasi-geostrophic equation, Discrete and Continuous Dynamical Systems, 27 (2010), 847-861. doi: 10.3934/dcds.2010.27.847.  Google Scholar [10] S. Cifani and E. R. Jakobsen, Entropy solution theory for fractional degenerate convection-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 413-441. doi: 10.1016/j.anihpc.2011.02.006.  Google Scholar [11] H. Dong and D. Du, Finite time singularities and global well-posedness for fractal Burgers' equations, Indiana Univ. Math, 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505.  Google Scholar [12] J. Droniou, Th. Gallouet and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499-521. doi: 10.1007/s00028-003-0503-1.  Google Scholar [13] J. Endal and E. R. Jakobsen, $L^1$ contraction for bounded (nonintegrable) solutions of degenerate parabolic equations, SIAM J. Math. Anal., 46 (2014), 3957-3982. doi: 10.1137/140966599.  Google Scholar [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, 1981.  Google Scholar [15] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Mathematicae, 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.  Google Scholar [16] S. N. Kruzhkov, First order quasilinear equations with several independent variables, Math. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar [17] C. Miao and G. Wu, Global well-posedness for the critical Burgers equation in critical Besov spaces, J. Diff. Eq., 247 (2009), 1673-1693. doi: 10.1016/j.jde.2009.03.028.  Google Scholar [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [19] K. Yosida, Functional Analysis, 4th edition, Springer-Verlag, 1974.  Google Scholar
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