Young measure solutions for a class of forward-backward convection-diffusion equations

Wang, Chunpeng; Nie, Yuanyuan; Yin, Jingxue

doi:10.1090/S0033-569X-2014-01338-8

Authors: Chunpeng Wang, Yuanyuan Nie and Jingxue Yin
Journal: Quart. Appl. Math. 72 (2014), 177-192
MSC (2010): Primary 35R35, 35K55
DOI: https://doi.org/10.1090/S0033-569X-2014-01338-8
Published electronically: January 8, 2014
MathSciNet review: 3185137
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the first initial boundary value problems of a class of forward-backward convection-diffusion equations. The existence theorem and the continuous dependence theorem of Young measure solutions are established.

References

J. M. Ball, A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988) Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 207–215. MR 1036070, DOI https://doi.org/10.1007/BFb0024945
Yan Chen and Kewei Zhang, Young measure solutions of the two-dimensional Perona-Malik equation in image processing, Commun. Pure Appl. Anal. 5 (2006), no. 3, 615–635. MR 2217594, DOI https://doi.org/10.3934/cpaa.2006.5.617
Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
William Alan Day, The thermodynamics of simple materials with fading memory, Springer-Verlag, New York-Heidelberg, 1972. Springer Tracts in Natural Philosophy, Vol. 22. MR 0366234
Sophia Demoulini, Young measure solutions for a nonlinear parabolic equation of forward-backward type, SIAM J. Math. Anal. 27 (1996), no. 2, 376–403. MR 1377480, DOI https://doi.org/10.1137/S0036141094261847
Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI https://doi.org/10.1007/BF00752112
Charles M. Elliott, The Stefan problem with a nonmonotone constitutive relation, IMA J. Appl. Math. 35 (1985), no. 2, 257–264. Special issue: IMA conference on crystal growth (Oxford, 1985). MR 839202, DOI https://doi.org/10.1093/imamat/35.2.257
Selim Esedoḡlu, An analysis of the Perona-Malik scheme, Comm. Pure Appl. Math. 54 (2001), no. 12, 1442–1487. MR 1852979, DOI https://doi.org/10.1002/cpa.3008
Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1034481
Klaus Höllig, Existence of infinitely many solutions for a forward backward heat equation, Trans. Amer. Math. Soc. 278 (1983), no. 1, 299–316. MR 697076, DOI https://doi.org/10.1090/S0002-9947-1983-0697076-8
Klaus Höllig and John A. Nohel, A diffusion equation with a nonmonotone constitutive function, Systems of nonlinear partial differential equations (Oxford, 1982), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 111, Reidel, Dordrecht-Boston, Mass., 1983, pp. 409–422. MR 725537
B. Kawohl, From Mumford-Shah to Perona-Malik in image processing, Math. Methods Appl. Sci. 27 (2004), no. 15, 1803–1814. MR 2087298, DOI https://doi.org/10.1002/mma.564
David Kinderlehrer and Pablo Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal. 4 (1994), no. 1, 59–90. MR 1274138, DOI https://doi.org/10.1007/BF02921593
David Kinderlehrer and Pablo Pedregal, Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal. 23 (1992), no. 1, 1–19. MR 1145159, DOI https://doi.org/10.1137/0523001
Alan V. Lair, Uniqueness for a forward backward diffusion equation, Trans. Amer. Math. Soc. 291 (1985), no. 1, 311–317. MR 797062, DOI https://doi.org/10.1090/S0002-9947-1985-0797062-5

P. Perona and J. Malik, Scale-space edge detection using anisotropic diffusion, IEEE Trans. on Pattern Analysis and Machine Intelligence, 12 (7)(1990), 629–639.

M. Slemrod, Dynamics of measure valued solutions to a backward-forward heat equation, J. Dynam. Differential Equations 3 (1991), no. 1, 1–28. MR 1094722, DOI https://doi.org/10.1007/BF01049487

Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR 725524

L. Tartar, Étude des oscillations dans les équations aux dérivées partielles non linéaires, Trends and applications of pure mathematics to mechanics (Palaiseau, 1983) Lecture Notes in Phys., vol. 195, Springer, Berlin, 1984, pp. 384–412 (French). MR 755737, DOI https://doi.org/10.1007/3-540-12916-2_68

Chunpeng Wang, Jingxue Yin, and Bibo Lu, Anti-shifting phenomenon of a convective nonlinear diffusion equation, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 3, 1211–1236. MR 2670192, DOI https://doi.org/10.3934/dcdsb.2010.14.1211

Augusto Visintin, Forward-backward parabolic equations and hysteresis, Calc. Var. Partial Differential Equations 15 (2002), no. 1, 115–132. MR 1920717, DOI https://doi.org/10.1007/s005260100120

Jingxue Yin and Chunpeng Wang, Young measure solutions of a class of forward-backward diffusion equations, J. Math. Anal. Appl. 279 (2003), no. 2, 659–683. MR 1974053, DOI https://doi.org/10.1016/S0022-247X%2803%2900054-4

L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704

Kewei Zhang, On existence of weak solutions for one-dimensional forward-backward diffusion equations, J. Differential Equations 220 (2006), no. 2, 322–353. MR 2183375, DOI https://doi.org/10.1016/j.jde.2005.01.011

Kewei Zhang, Existence of infinitely many solutions for the one-dimensional Perona-Malik model, Calc. Var. Partial Differential Equations 26 (2006), no. 2, 171–199. MR 2222243, DOI https://doi.org/10.1007/s00526-005-0363-4