Young measure solutions for a class of forward-backward convection-diffusion equations
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
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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
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- 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
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 (91c:49021), 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 (2006m:35145)
- Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890 (90e:49001)
- William Alan Day, The thermodynamics of simple materials with fading memory, Springer-Verlag, New York, 1972. Springer Tracts in Natural Philosophy, Vol. 22. MR 0366234 (51 \#2482)
- 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 (97a:35096), 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 (86g:35121), 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 (87j:35183), 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 (2003i:94005), 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, 1990. MR 1034481 (91a:35009)
- 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 (84m:35062), DOI https://doi.org/10.2307/1999317
- 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, 1983, pp. 409–422. MR 725537 (84k:35082)
- B. Kawohl, From Mumford-Shah to Perona-Malik in image processing, Math. Methods Appl. Sci. 27 (2004), no. 15, 1803–1814. MR 2087298 (2005d:68138), 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 (95f:49059), 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 (92m:49076), 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 (86j:35091), DOI https://doi.org/10.2307/1999911
- 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 (92m:35124), 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 (85e:35079)
- 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 (86a:35097), 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 (2011h:35144), 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 (2003g:47125), 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 (2005c:35165), DOI https://doi.org/10.1016/S0022-247X%2803%2900054-4
- L. C. Young, Lectures on the calculus of variations and optimal control theory, Foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia, 1969. MR 0259704 (41 \#4337)
- 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 (2006g:35133), 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 (2007f:35153), DOI https://doi.org/10.1007/s00526-005-0363-4
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Additional Information
Chunpeng Wang
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email:
wangcp@jlu.edu.cn
Yuanyuan Nie
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email:
nieyuanyuan@live.cn
Jingxue Yin
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, People’s Republic of China
Email:
yjx@scnu.edu.cn
Keywords:
Convection-diffusion equation,
forward-backward,
Young measure solution
Received by editor(s):
April 28, 2012
Published electronically:
January 8, 2014
Additional Notes:
Supported by the National Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China
Article copyright:
© Copyright 2014
Brown University