The well-posedness of renormalized solutions for a non-uniformly parabolic equation

Zhang, Chao; Zhou, Shulin

doi:10.1090/proc/13406

The well-posedness of renormalized solutions for a non-uniformly parabolic equation
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Proc. Amer. Math. Soc. 145 (2017), 2577-2589 Request permission

Abstract:

In this paper we present a unified approach to establish the existence of renormalized solutions and a comparison result for a class of non-uniformly parabolic initial-boundary value problems. As a consequence, the uniqueness of renormalized solutions and the equivalence between entropy and renormalized solutions for such equations are obtained. The results extend the well-posedness results for the classical $p$-Laplacian type equations to a larger class of non-linear elliptic and parabolic PDEs including the nearly linear growth operators.

References

Philippe Bénilan, Lucio Boccardo, Thierry Gallouët, Ron Gariepy, Michel Pierre, and Juan Luis Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 241–273. MR 1354907
Philippe Bénilan, Jose Carrillo, and Petra Wittbold, Renormalized entropy solutions of scalar conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 2, 313–327. MR 1784177
Lucio Boccardo and Luigi Orsina, Leray-Lions operators with logarithmic growth, J. Math. Anal. Appl. 423 (2015), no. 1, 608–622. MR 3273197, DOI 10.1016/j.jmaa.2014.09.065
D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations 210 (2005), no. 2, 383–428. MR 2119989, DOI 10.1016/j.jde.2004.06.012
Verena Bögelein, Frank Duzaar, and Paolo Marcellini, Parabolic systems with $p,q$-growth: a variational approach, Arch. Ration. Mech. Anal. 210 (2013), no. 1, 219–267. MR 3073153, DOI 10.1007/s00205-013-0646-4
Yongyong Cai and Shulin Zhou, Existence and uniqueness of weak solutions for a non-uniformly parabolic equation, J. Funct. Anal. 257 (2009), no. 10, 3021–3042. MR 2568684, DOI 10.1016/j.jfa.2009.08.007
Y. Chen, Parabolic Partial Differential Equations of Second Order, Peking University Press, Beijing, 2003 (in Chinese).
Maria Colombo and Giuseppe Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443–496. MR 3294408, DOI 10.1007/s00205-014-0785-2
Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890, DOI 10.1007/978-3-642-51440-1
D. M. Duc and J. Eells, Regularity of exponentially harmonic functions, Internat. J. Math. 2 (1991), no. 4, 395–408. MR 1113568, DOI 10.1142/S0129167X91000223
Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
Emmanuele DiBenedetto, Real analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1897317, DOI 10.1007/978-1-4612-0117-5
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), no. 2, 321–366. MR 1014927, DOI 10.2307/1971423
Jérôme Droniou and Alain Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 1-2, 181–205. MR 2346459, DOI 10.1007/s00030-007-5018-z
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Martin Fuchs and Giuseppe Mingione, Full $C^{1,\alpha }$-regularity for free and constrained local minimizers of elliptic variational integrals with nearly linear growth, Manuscripta Math. 102 (2000), no. 2, 227–250. MR 1771942, DOI 10.1007/s002291020227
Gary M. Lieberman, On the regularity of the minimizer of a functional with exponential growth, Comment. Math. Univ. Carolin. 33 (1992), no. 1, 45–49. MR 1173745
P.-L. Lions and F. Murat, Solutions renormalisées d’équations elliptiques non linéaires, unpublished.
Hisashi Naito, On a local Hölder continuity for a minimizer of the exponential energy functional, Nagoya Math. J. 129 (1993), 97–113. MR 1210004, DOI 10.1017/S0027763000004335
Francesco Petitta, Augusto C. Ponce, and Alessio Porretta, Diffuse measures and nonlinear parabolic equations, J. Evol. Equ. 11 (2011), no. 4, 861–905. MR 2861310, DOI 10.1007/s00028-011-0115-1
Francesco Petitta and Alessio Porretta, On the notion of renormalized solution to nonlinear parabolic equations with general measure data, J. Elliptic Parabol. Equ. 1 (2015), 201–214. MR 3403419, DOI 10.1007/BF03377376
Alain Prignet, Existence and uniqueness of “entropy” solutions of parabolic problems with $L^1$ data, Nonlinear Anal. 28 (1997), no. 12, 1943–1954. MR 1436364, DOI 10.1016/S0362-546X(96)00030-2
Mohammed Saadoune and Michel Valadier, Extraction of a “good” subsequence from a bounded sequence of integrable functions, J. Convex Anal. 2 (1995), no. 1-2, 345–357. MR 1363378
Lihe Wang and Shulin Zhou, Existence and uniqueness of weak solutions for a nonlinear parabolic equation related to image analysis, J. Partial Differential Equations 19 (2006), no. 2, 97–112. MR 2227688
Chao Zhang and Shulin Zhou, Entropy solutions for a non-uniformly parabolic equation, Manuscripta Math. 131 (2010), no. 3-4, 335–354. MR 2592084, DOI 10.1007/s00229-009-0321-0
Chao Zhang and Shulin Zhou, Renormalized solutions for a non-uniformly parabolic equation, Ann. Acad. Sci. Fenn. Math. 37 (2012), no. 1, 175–189. MR 2920432, DOI 10.5186/aasfm.2012.3709