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

Non-critical fractional conservation laws in domains with boundary

Matthieu Brassart 1,

1. 

Laboratoire de Mathématiques de Besancon 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.
Keywords: nonlocal conservation law, $L^1$-contraction principle., Entropy formulation, Kato inequality.
Mathematics Subject Classification: 35B30, 35L65, 35L82, 35S10, 35S3.
Citation: Matthieu Brassart. Non-critical fractional conservation laws in domains with boundary. Networks and 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.

[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.

[3]

N. Alibaud and M. Brassart, Entropy solutions to fractional conservation laws in domains, In preparation, 2015.

[4]

N. Alibaud and M. Brassart, Parabolicity for fractional conservation laws in domains, In preparation, 2015.

[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.

[6]

P.L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer-Verlag, 1967.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, 1981.

[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.

[16]

S. N. Kruzhkov, First order quasilinear equations with several independent variables, Math. Sb. (N.S.), 81 (1970), 228-255.

[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.

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

K. Yosida, Functional Analysis, 4th edition, Springer-Verlag, 1974.

show all references

References:
[1]

N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., 7 (2007), 145-175. doi: 10.1007/s00028-006-0253-z.

[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.

[3]

N. Alibaud and M. Brassart, Entropy solutions to fractional conservation laws in domains, In preparation, 2015.

[4]

N. Alibaud and M. Brassart, Parabolicity for fractional conservation laws in domains, In preparation, 2015.

[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.

[6]

P.L. Butzer and H. Berens, Semigroups of Operators and Approximation, Springer-Verlag, 1967.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, 1981.

[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.

[16]

S. N. Kruzhkov, First order quasilinear equations with several independent variables, Math. Sb. (N.S.), 81 (1970), 228-255.

[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.

[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1.

[19]

K. Yosida, Functional Analysis, 4th edition, Springer-Verlag, 1974.

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