# American Institute of Mathematical Sciences

June  2016, 11(2): 203-222. doi: 10.3934/nhm.2016.11.203

## Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

 1 Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex 2 Université Abdou Moumouni de Niamey, École Normale Supérieure, Departement de Mathématiques, BP: 10963 Niamey, Niger

Received  May 2015 Revised  October 2015 Published  March 2016

We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.
Citation: Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203
##### References:
 [1] J. Aleksic and D. Mitrovic, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations, J. Hyperb. Differ. Equ., 10 (2013), 659-676. doi: 10.1142/S0219891613500239.  Google Scholar [2] K. Ammar, P. Wittbold and J. Carrillo, Scalar conservation laws with general boundary condition and continuous flux function, J. Differ. Equ., 228 (2006), 111-139. doi: 10.1016/j.jde.2006.05.002.  Google Scholar [3] B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, J. Hyperb. Differ. Equ., 7 (2010), 1-67. doi: 10.1142/S0219891610002062.  Google Scholar [4] B. Andreianov and F. Bouhsiss, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition, J. Evol. Equ., 4 (2004), 273-295. doi: 10.1007/s00028-004-0143-1.  Google Scholar [5] B. Andreianov and M. Karimou Gazibo, Entropy formulation of degenerate parabolic equation with zero-flux boundary condition, Z. Angew. Math. Phys., 64 (2013), 1471-1491. doi: 10.1007/s00033-012-0297-6.  Google Scholar [6] B. Andreianov and M. Karimou Gazibo, Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition， Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, Springer Proc. Math. Stat., 77 (2014), 303-311. doi: 10.1007/978-3-319-05684-5_29.  Google Scholar [7] B. Andreianov and K. Shibi, Scalar conservation laws with nonlinear boundary conditions, C. R. Acad. Paris Ser. I Math., 345 (2007), 431-434. doi: 10.1016/j.crma.2007.09.008.  Google Scholar [8] B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws, Trans. AMS, 367 (2015), 3763-3806. doi: 10.1090/S0002-9947-2015-05988-1.  Google Scholar [9] C. Bardos, A.-Y. LeRoux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. PDE, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar [10] Ph. Bénilan, Equations D'évolution dans un Espace de Banach Quelconque et Applications, Thèse d'état, Orsay, 1972. Google Scholar [11] Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Preprint book., ().   Google Scholar [12] Ph. Bénilan, J. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313-327.  Google Scholar [13] R. Bürger, H. Frid and K.H. Karlsen, On the well-posedness of entropy solution to conservation laws with a zero-flux boundary condition, J. Math. Anal. Appl., 326 (2007), 108-120. doi: 10.1016/j.jmaa.2006.02.072.  Google Scholar [14] R. Bürger, H. Frid and K. H. Karlsen, On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of presssure filtration, SIAM J. Math. Anal., 34 (2003) 611-635. Google Scholar [15] C. Cancès, Th. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media, Interfaces Free Bound, 11 (2009), 239-258. doi: 10.4171/IFB/210.  Google Scholar [16] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.  Google Scholar [17] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.  Google Scholar [18] R. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains, Acta Mathematica Sci., 35 (2015), 906-944. doi: 10.1016/S0252-9602(15)30028-X.  Google Scholar [19] F. Dubois and Ph. LeFloch, Boundary condition for nonlinear hyperbolic conservation laws, J. Differ. Equ., 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.  Google Scholar [20] S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal., 37 (2000), 1838-1860. doi: 10.1137/S0036142998336138.  Google Scholar [21] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modeles Nonlinéaires de L'ingénierie Pétroliere, Math. et Appl., 22. Springer-Verlag, Berlin, 1996.  Google Scholar [22] M. Karimou Gazibo, Degenerate Convection-Diffusion Equation with a Robin boundary condition, In F. Ancona et al., eds. Hyperbolic Problems : Theory, Numerics, Applications, Proceedings of 14th HYP conference in Padua, AIMS series in Appl. Math., 8 (2014), 583-590. Google Scholar [23] M. Karimou Gazibo, Degenerate parabolic equation with zero flux boundary condition and its approximations,, Preprint available at , ().   Google Scholar [24] M. Karimou Gazibo, Études Mathématiques et Numériques des Problèmes Paraboliques Avec Des Conditions Aux Limites, Thèse de Doctorat Besançon, 2013. Google Scholar [25] S. N. Kruzhkov, First order quasi-linear equations in several independent variables, Math. USSR Sb., 10 (1970), 217-243. Google Scholar [26] Y. S. Kwon, Strong traces for degenerate parabolic-hyperbolic equations, Discrete Contin. Dyn. Syst., 25 (2009), 1275-1286. doi: 10.3934/dcds.2009.25.1275.  Google Scholar [27] M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problems, J. Evol. Equ., 3 (2003), 603-622. doi: 10.1007/s00028-003-0105-z.  Google Scholar [28] F. Otto, Initial-boundary value problem for a scalar conservation laws, C. R. Acad. Sci. Paris Sér I Math., 322 (1996), 729-734.  Google Scholar [29] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation, Arch. Ration. Mech. Anal., 163 (2002), 87-124. doi: 10.1007/s002050200184.  Google Scholar [30] A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods, SIAM J. Numer. Anal., 41 (2003), 2262-2293. doi: 10.1137/S0036142902406612.  Google Scholar [31] E. Yu. Panov, On the theory of generalized entropy solutions of Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions, Iszvestiya Math., 66 (2002), 1171-1218 (in Russian). doi: 10.1070/IM2002v066n06ABEH000411.  Google Scholar [32] E. Yu. Panov, Existence of strong traces for quasi-solutions of multi-dimensional scalar conservation laws, J. Hyp. Diff. Equ., 4 (2009), 729-770. doi: 10.1142/S0219891607001343.  Google Scholar [33] E. Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux, J. Differ. Equ., 247 (2009), 2821-2870. doi: 10.1016/j.jde.2009.08.022.  Google Scholar [34] A. Porretta and J. Vovelle, $L^1$ solutions to first order hyperbolic equations in bounded domains, Comm. PDEs, 28 (2003), 381-408. doi: 10.1081/PDE-120019387.  Google Scholar [35] E. Rouvre and G. Gagneux, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, An. Fac. Sci. Toulouse, 10 (2001), 163-183. doi: 10.5802/afst.987.  Google Scholar [36] T. Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation, Math. Res. Lett., 3 (1996), 475-490. doi: 10.4310/MRL.1996.v3.n4.a6.  Google Scholar [37] G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advance in Math. Sci. Appl., 15 (2005), 423-450.  Google Scholar [38] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.  Google Scholar [39] A. I. Vol'pert and S. I. Hudjaev, Cauchy problem for degenerate second order quasilinear parabolic equations, Math. USSR Sbornik, 78 (1969), 374-396.  Google Scholar [40] J. Vovelle, Convergence of finite volume monotones schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596. doi: 10.1007/s002110100307.  Google Scholar

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##### References:
 [1] J. Aleksic and D. Mitrovic, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations, J. Hyperb. Differ. Equ., 10 (2013), 659-676. doi: 10.1142/S0219891613500239.  Google Scholar [2] K. Ammar, P. Wittbold and J. Carrillo, Scalar conservation laws with general boundary condition and continuous flux function, J. Differ. Equ., 228 (2006), 111-139. doi: 10.1016/j.jde.2006.05.002.  Google Scholar [3] B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, J. Hyperb. Differ. Equ., 7 (2010), 1-67. doi: 10.1142/S0219891610002062.  Google Scholar [4] B. Andreianov and F. Bouhsiss, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition, J. Evol. Equ., 4 (2004), 273-295. doi: 10.1007/s00028-004-0143-1.  Google Scholar [5] B. Andreianov and M. Karimou Gazibo, Entropy formulation of degenerate parabolic equation with zero-flux boundary condition, Z. Angew. Math. Phys., 64 (2013), 1471-1491. doi: 10.1007/s00033-012-0297-6.  Google Scholar [6] B. Andreianov and M. Karimou Gazibo, Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition， Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, Springer Proc. Math. Stat., 77 (2014), 303-311. doi: 10.1007/978-3-319-05684-5_29.  Google Scholar [7] B. Andreianov and K. Shibi, Scalar conservation laws with nonlinear boundary conditions, C. R. Acad. Paris Ser. I Math., 345 (2007), 431-434. doi: 10.1016/j.crma.2007.09.008.  Google Scholar [8] B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws, Trans. AMS, 367 (2015), 3763-3806. doi: 10.1090/S0002-9947-2015-05988-1.  Google Scholar [9] C. Bardos, A.-Y. LeRoux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. PDE, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.  Google Scholar [10] Ph. Bénilan, Equations D'évolution dans un Espace de Banach Quelconque et Applications, Thèse d'état, Orsay, 1972. Google Scholar [11] Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Preprint book., ().   Google Scholar [12] Ph. Bénilan, J. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313-327.  Google Scholar [13] R. Bürger, H. Frid and K.H. Karlsen, On the well-posedness of entropy solution to conservation laws with a zero-flux boundary condition, J. Math. Anal. Appl., 326 (2007), 108-120. doi: 10.1016/j.jmaa.2006.02.072.  Google Scholar [14] R. Bürger, H. Frid and K. H. Karlsen, On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of presssure filtration, SIAM J. Math. Anal., 34 (2003) 611-635. Google Scholar [15] C. Cancès, Th. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media, Interfaces Free Bound, 11 (2009), 239-258. doi: 10.4171/IFB/210.  Google Scholar [16] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147 (1999), 269-361. doi: 10.1007/s002050050152.  Google Scholar [17] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.  Google Scholar [18] R. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains, Acta Mathematica Sci., 35 (2015), 906-944. doi: 10.1016/S0252-9602(15)30028-X.  Google Scholar [19] F. Dubois and Ph. LeFloch, Boundary condition for nonlinear hyperbolic conservation laws, J. Differ. Equ., 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.  Google Scholar [20] S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal., 37 (2000), 1838-1860. doi: 10.1137/S0036142998336138.  Google Scholar [21] G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modeles Nonlinéaires de L'ingénierie Pétroliere, Math. et Appl., 22. Springer-Verlag, Berlin, 1996.  Google Scholar [22] M. Karimou Gazibo, Degenerate Convection-Diffusion Equation with a Robin boundary condition, In F. Ancona et al., eds. Hyperbolic Problems : Theory, Numerics, Applications, Proceedings of 14th HYP conference in Padua, AIMS series in Appl. Math., 8 (2014), 583-590. Google Scholar [23] M. Karimou Gazibo, Degenerate parabolic equation with zero flux boundary condition and its approximations,, Preprint available at , ().   Google Scholar [24] M. Karimou Gazibo, Études Mathématiques et Numériques des Problèmes Paraboliques Avec Des Conditions Aux Limites, Thèse de Doctorat Besançon, 2013. Google Scholar [25] S. N. Kruzhkov, First order quasi-linear equations in several independent variables, Math. USSR Sb., 10 (1970), 217-243. Google Scholar [26] Y. S. Kwon, Strong traces for degenerate parabolic-hyperbolic equations, Discrete Contin. Dyn. Syst., 25 (2009), 1275-1286. doi: 10.3934/dcds.2009.25.1275.  Google Scholar [27] M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problems, J. Evol. Equ., 3 (2003), 603-622. doi: 10.1007/s00028-003-0105-z.  Google Scholar [28] F. Otto, Initial-boundary value problem for a scalar conservation laws, C. R. Acad. Sci. Paris Sér I Math., 322 (1996), 729-734.  Google Scholar [29] C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation, Arch. Ration. Mech. Anal., 163 (2002), 87-124. doi: 10.1007/s002050200184.  Google Scholar [30] A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods, SIAM J. Numer. Anal., 41 (2003), 2262-2293. doi: 10.1137/S0036142902406612.  Google Scholar [31] E. Yu. Panov, On the theory of generalized entropy solutions of Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions, Iszvestiya Math., 66 (2002), 1171-1218 (in Russian). doi: 10.1070/IM2002v066n06ABEH000411.  Google Scholar [32] E. Yu. Panov, Existence of strong traces for quasi-solutions of multi-dimensional scalar conservation laws, J. Hyp. Diff. Equ., 4 (2009), 729-770. doi: 10.1142/S0219891607001343.  Google Scholar [33] E. Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux, J. Differ. Equ., 247 (2009), 2821-2870. doi: 10.1016/j.jde.2009.08.022.  Google Scholar [34] A. Porretta and J. Vovelle, $L^1$ solutions to first order hyperbolic equations in bounded domains, Comm. PDEs, 28 (2003), 381-408. doi: 10.1081/PDE-120019387.  Google Scholar [35] E. Rouvre and G. Gagneux, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées, An. Fac. Sci. Toulouse, 10 (2001), 163-183. doi: 10.5802/afst.987.  Google Scholar [36] T. Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation, Math. Res. Lett., 3 (1996), 475-490. doi: 10.4310/MRL.1996.v3.n4.a6.  Google Scholar [37] G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Advance in Math. Sci. Appl., 15 (2005), 423-450.  Google Scholar [38] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.  Google Scholar [39] A. I. Vol'pert and S. I. Hudjaev, Cauchy problem for degenerate second order quasilinear parabolic equations, Math. USSR Sbornik, 78 (1969), 374-396.  Google Scholar [40] J. Vovelle, Convergence of finite volume monotones schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596. doi: 10.1007/s002110100307.  Google Scholar
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