Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Authors: Chao Zhang and Shulin Zhou
Journal: Proc. Amer. Math. Soc. 145 (2017), 2577-2589
MSC (2010): Primary 35D05; Secondary 35D10
DOI: https://doi.org/10.1090/proc/13406
Published electronically: November 30, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] 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
  • [3] Lucio Boccardo and Luigi Orsina, Leray-Lions operators with logarithmic growth, J. Math. Anal. Appl. 423 (2015), no. 1, 608-622. MR 3273197, https://doi.org/10.1016/j.jmaa.2014.09.065
  • [4] D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations 210 (2005), no. 2, 383-428. MR 2119989, https://doi.org/10.1016/j.jde.2004.06.012
  • [5] 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, https://doi.org/10.1007/s00205-013-0646-4
  • [6] 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, https://doi.org/10.1016/j.jfa.2009.08.007
  • [7] Y. Chen, Parabolic Partial Differential Equations of Second Order, Peking University Press, Beijing, 2003 (in Chinese).
  • [8] Maria Colombo and Giuseppe Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), no. 2, 443-496. MR 3294408, https://doi.org/10.1007/s00205-014-0785-2
  • [9] Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
  • [10] D. M. Duc and J. Eells, Regularity of exponentially harmonic functions, Internat. J. Math. 2 (1991), no. 4, 395-408. MR 1113568, https://doi.org/10.1142/S0129167X91000223
  • [11] Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384
  • [12] Emmanuele DiBenedetto, Real analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1897317
  • [13] 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, https://doi.org/10.2307/1971423
  • [14] 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, https://doi.org/10.1007/s00030-007-5018-z
  • [15] Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
  • [16] 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, https://doi.org/10.1007/s002291020227
  • [17] 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
  • [18] P.-L. Lions and F. Murat, Solutions renormalisées d'équations elliptiques non linéaires, unpublished.
  • [19] Hisashi Naito, On a local Hölder continuity for a minimizer of the exponential energy functional, Nagoya Math. J. 129 (1993), 97-113. MR 1210004
  • [20] 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, https://doi.org/10.1007/s00028-011-0115-1
  • [21] 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
  • [22] 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, https://doi.org/10.1016/S0362-546X(96)00030-2
  • [23] 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
  • [24] 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
  • [25] Chao Zhang and Shulin Zhou, Entropy solutions for a non-uniformly parabolic equation, Manuscripta Math. 131 (2010), no. 3-4, 335-354. MR 2592084, https://doi.org/10.1007/s00229-009-0321-0
  • [26] 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, https://doi.org/10.5186/aasfm.2012.3709

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35D05, 35D10

Retrieve articles in all journals with MSC (2010): 35D05, 35D10


Additional Information

Chao Zhang
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email: czhangmath@hit.edu.cn

Shulin Zhou
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: szhou@math.pku.edu.cn

DOI: https://doi.org/10.1090/proc/13406
Keywords: Renormalized solutions, existence, uniqueness, parabolic
Received by editor(s): February 6, 2016
Received by editor(s) in revised form: July 27, 2016
Published electronically: November 30, 2016
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society