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Non-critical fractional conservation laws in domains with boundary
1. | Laboratoire de Mathématiques de Besancon U.F.R. S.T, 16 route de Gray, 25030 BESANCON, France |
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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