Existence of a renormalized solution of nonlinear parabolic equations with general measure data
Résumé
In this paper we prove the existence of a renormalized solution for nonlinear parabolic equations of the type:
$$\displaystyle{\partial b(x,u)
\over
\partial t} - {\rm div}\Big(a(x,t,\nabla u)\Big)=\mu\qquad \text{in}\ \Omega\times (0,T),$$ where the right hand
side is a general measure, $b(x,u)$ is an
unbounded function of $u$ and $- {\rm div}(a(x,t,\nabla u))$
is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in
$\nabla u$.
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Références
Akdim, Y. Bennouna, J. Mekkour, M. and Redwane, H., Strongly nonlinear parabolic inequality in Orlicz spaces via a sequence of penalized equations, Afr. Mat. 26, no. 7-8, 1669-1695, (2015). https://doi.org/10.1007/s13370-014-0309-0
Akdim, Y. Bennouna, J. Mekkour, M and Redwane, H., Parabolic equations with measure data and three unbounded nonlinearities in weighted Sobolev spaces, Nonlinear Dyn. Syst. Theory 15, no. 2, 107-126, (2015).
Benilan, P. Boccardo, L. Gallouet, T. Gariepy, R. Pierre, M. and Vazquez, J., An L1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, 22, 241-273, (1995).
Blanchard, D., Truncation and monotonicity methods for parabolic equations equations, Nonlinear Anal., 21, 725-743, (1993). https://doi.org/10.1016/0362-546X(93)90120-H
Blanchard, D. and Francfort, G., A few results on a class of degenerate parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18, 213-249, (1993).
Blanchard, D. and Murat, F., Renormalized solutions of nonlinear parabolic problems with L1 data, Existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect., A127, 1137-1152, (1997). https://doi.org/10.1017/S0308210500026986
Blanchard, D. Murat, F. and Redwane, H., Existence et unicite de la solution reormalisee d'un probleme parabolique assez general, C. R. Acad. Sci. Paris S'er., I329, 575-580, (1999). https://doi.org/10.1016/S0764-4442(00)80004-X
Blanchard, D. Murat, F. and Redwane, H., Existence and Uniqueness of a Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems, J. Differential Equations, 177, 331-374, (2001). https://doi.org/10.1006/jdeq.2000.4013
Blanchard, D. and Porretta, A., Nonlinear parabolic equations with natural growth terms and measure initial data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4, 583-622, (2001).
Blanchard, D. and Porretta, A., Stefan problems with nonlinear diffusion and convection, J. Diff. Equations, 210, 383-428, (2005). https://doi.org/10.1016/j.jde.2004.06.012
Blanchard, D. and Redwane, H., Renormalized solutions of nonlinear parabolic evolution problems, J. Math. Pure Appl., 77, 117-151, (1998). https://doi.org/10.1016/S0021-7824(98)80067-6
Blanchard, D. and Redwane, H. and Petitta, F., Renormalized solutions of nonlinear parabolic equations with diffuse measure data, Manuscripta Math., 141, 601-635, (2013). https://doi.org/10.1007/s00229-012-0585-7
Blanchard, D. and Redwane, H., Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, natural growth terms and L 1 data, Arab J Math Sci 20(2), 157-176, (2014). https://doi.org/10.1016/j.ajmsc.2013.06.002
Boccardo, L, Dall'Aglio, A. Gallouet, T. and Orsina L., Nonlinear parabolic equations with measure data, J. Funct. Anal., 87, 49-169, (1997).
Boccardo, L. Gallouet, T. Orsina, L., Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincar'e Anal. Non Lin'eaire, 13 , 539-551, (1996). https://doi.org/10.1016/S0294-1449(16)30113-5
L. Boccardo, T. Gallou¨et , Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. https://doi.org/10.1016/0022-1236(89)90005-0
Boccardo, L. Murat, F. Puel, J. P., Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl. 152, 183-196, (1988). https://doi.org/10.1007/BF01766148
Carrillo, J., Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147(4), 269-361, (1999). https://doi.org/10.1007/s002050050152
Carrillo, J. and Wittbold, P., Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Differential Equations, 156, 93-121, (1999). https://doi.org/10.1006/jdeq.1998.3597
Carrillo, J. and Wittbold, P., Renormalized entropy solution of a scalar conservation law with boundary condition, J. Differential Equations, 185(1), 137-160, (2002). https://doi.org/10.1006/jdeq.2002.4179
Dal Maso, G. Murat, F. Orsina, L. and Prignet, A., Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28, 741-808, (1999).
Dall'Aglio, A. and Orsina, L., Nonlinear parabolic equations with natural growth conditions and L1 data, Nonlinear Anal., 27, 59-73, (1996). https://doi.org/10.1016/0362-546X(94)00363-M
Di Perna, R. J. and Lions, P. L., On the Cauchy problem for Boltzmann equations : Global existence and weak stability, Ann. Math., 130, pp. 321-366, (1989). https://doi.org/10.2307/1971423
Droniou, J. Prignet, A., Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA Nonlinear Diff. Eq. Appl., 14, 181-205, (2007) https://doi.org/10.1007/s00030-007-5018-z
Maitre, E. Sur une classe d'equations a double non linearite : application a la simulation numerique dun ecoulement visqueux compressible, These Universite Joseph Fourier - Grenoble 1, (1997).
Dall'Aglio, A. Orsina, L., Existence results for some nonlinear parabolic equations with nonregular data, Differential Integral Equations, 5, 1335-1354, (1992).
Fukushima, M. Sato, K. Taniguchi, S., On the closable part of pre-Dirichlet forms and the fine supports of underlying measures, Osaka J. Math., 28, 517-535, (1991).
Lions, J. L., Quelques methodes de resolution des problemes aux limites non lineaire, Dunod et Gauthier-Villars, Paris, (1969).
Landes, R., On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect., A89, 217-237, (1981). https://doi.org/10.1017/S0308210500020242
Murat, F., Soluciones renormalizadas de EDP elipticas non lineales, Technical Report R93023, Laboratoire d'Analyse Numerique, Paris VI, Cours a l'Universite de Seville, (1993).
Murat, F., Equations elliptiques non lin'eaires avec second membre L1 ou mesure, in Compte Rendus du 26 eme Congres d'Analyse Numerique, les Karellis, A12-A24, (1994).
Petitta, F., Asymptotic behavior of solutions for linear parabolic equations with general measure data, C.R. Math. Acad. Sci. Paris 344 , no.9, 571-576, (2007). https://doi.org/10.1016/j.crma.2007.03.021
Petitta, F., Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura ed Appl., 187 (4) , 563-604, (2008). https://doi.org/10.1007/s10231-007-0057-y
Porretta, A., Existence results for nonlinear parabolic equations via strong convergence of trauncations, Ann. Mat. Pura ed Applicata, 177,1 43-172, (1999). https://doi.org/10.1007/BF02505907
Prignet, A., Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures, Rend. Mat. Appl., (15), 321-337, (1995),
Pierre, M., Parabolic Capacity and Sobolev Spaces, SIAM J. Math Anal. 14, 522-533, (1983). https://doi.org/10.1137/0514044
Prignet, A., Existence and uniqueness of entropy solutions of parabolic problems with L1 data, Nonlinear Anal., 28, 1943-1954, (1997). https://doi.org/10.1016/S0362-546X(96)00030-2
Marah, A. and Redwane, H., Nonlinear parabolic equation with diffuse measure data, J. Nonl. Evol. Equ. Appl., 3, 27-48, (2017).
Droniou, J. Porretta, A. and Prignet, A., Parabolic capacity and soft measure for nonlinear equations, Potential Anal., 19, no.2, 99-161, (2003).
Redwane, H., Existence of a solution for a class of parabolic equations with three unbounded nonlinearities, Adv. Dyn. Syst. Appl., 2 , 241-264, (2007).
Serrin, J., Pathological solution of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18, 385-387, (1964).
Simon, J., Compact sets in L p (0, T; B), Ann. Mat. Pura Appl., 146, 65-96, (1987). https://doi.org/10.1007/BF01762360
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