December  2011, 31(4): 1151-1195. doi: 10.3934/dcds.2011.31.1151

An existence and uniqueness result for flux limited diffusion equations

1. 

DTIC, Universitat Pompeu Fabra, C/Roc Boronat 138, 08018 Barcelona, Spain

Received  November 2009 Revised  January 2010 Published  September 2011

We prove existence and uniqueness of entropy solutions of the Cauchy problem for the quasilinear parabolic equation $u_t$ $= div$ $a$$(u,Du)$ with initial condition $u_0$ $\in BV(\mathbb{R}^N)$, $u_0$$\geq 0$, where $a(z,\xi)$ = $\nabla_\xi f(z,\xi)$ and $f$ is a convex function of $\xi$ with linear growth as $\Vert \xi\Vert \to\infty$, satisfying other additional assumptions that cover the case of the so-called relativistic heat equation and other flux limited diffusion equations used in the theory of radiation hydrodynamics.
Citation: Vicent Caselles. An existence and uniqueness result for flux limited diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1151-1195. doi: 10.3934/dcds.2011.31.1151
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

[2]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear equations: The elliptic case, Annali della Scuola Norm. Sup. di Pisa. Cl. Sci. (5), 3 (2004), 555-587.

[3]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear equation: The parabolic case, Arch. Rat. Mech. Anal., 176 (2005), 415-453. doi: 10.1007/s00205-005-0358-5.

[4]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear elliptic equation, Nonlinear Analysis, 61 (2005), 637-669. doi: 10.1016/j.na.2004.11.020.

[5]

F. Andreu, V. Caselles and J. M. Mazón, The Cauchy problem for a strongly degenerate quasilinear equation, Journal European Math. Society (JEMS), 7 (2005), 361-393. doi: 10.4171/JEMS/32.

[6]

F. Andreu, V. Caselles, J. M. Mazón and S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal., 182 (2006), 269-297. doi: 10.1007/s00205-006-0428-3.

[7]

F. Andreu, V. Caselles and J. M. Mazón, Some regularity results on the 'relativistic' heat equation, J. Diff. Equat., 245 (2008), 3639-3663. doi: 10.1016/j.jde.2008.06.024.

[8]

F. Andreu, V. Caselles, J. M. Mazón and S. Moll, The Dirichlet problem associated to the relativistic heat equation, Math. Ann., 347 (2010), 135-199. doi: 10.1007/s00208-009-0428-3.

[9]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. di Matematica Pura ed Appl. (4), 135 (1983), 293-318.

[10]

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Normale Superiore di Pisa Cl. Sci. (4), 22 (1995), 241-273.

[11]

Ph. Bénilan and M. G. Crandall, Completely accretive operators, in "Semigroup Theory and Evolution Equations" (eds. Ph. Clement, et al.) (Delft, 1989), Lecture Notes in Pure and Appl. Math., 135, Dekker, New York, (1991), 41-75.

[12]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators," in preparation.

[13]

M. Bertsch and R. Dal Passo, Hyperbolic phenomena in a strongly degenerate parabolic equation, Arch Rational Mech. Anal., 117 (1992), 349-387. doi: 10.1007/BF00376188.

[14]

M. Bertsch and R. Dal Passo, A parabolic equation with a mean-curvature type operator, in "Nonlinear Diffusion Equations and their Equilibrium States, 3" (Gregynog, 1989), Progr. Nonlinear Differential Equation Appl., 7, Birkhäuser Boston, Boston, MA, (1992), 89-97.

[15]

Ph. Blanc, On the regularity of the solutions of some degenerate parabolic equations, Comm. in Partial Diff. Equat., 18 (1993), 821-846.

[16]

Ph. Blanc, "Sur une Classe d'Equations Paraboliques Degeneréesa une Dimension d'Espace Possedant des Solutions Discontinues," Ph.D. Thesis, number 798, Ecole Polytechnique Federale de Lausanne, 1989.

[17]

Y. Brenier, Extended Monge-Kantorovich theory, in "Optimal Transportation and Applications" (eds., L. A. Caffarelli and S. Salsa) (Martina-Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, 2003, 91-121.

[18]

F. Browder, Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Nat. Acad. Sci. USA, 74 (1977), 2659-2661. doi: 10.1073/pnas.74.7.2659.

[19]

J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Diff. Equat., 156 (1999), 93-121. doi: 10.1006/jdeq.1998.3597.

[20]

V. Caselles, On the entropy conditions for some flux limited diffusion equations, J. Diff. Equat., 250 (2011), 3311-3348. doi: 10.1016/j.jde.2011.01.027.

[21]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.

[22]

M. G. Crandall, Nonlinear semigroups and evolution equations governed by accretive operators, in "Nonlinear Functional Analysis and its Applications, Part 1" (Berkeley, Calif., 1983), Proc. of Symp. in Pure Mat., 45, Part I, AMS, Providence, RI, (1986), 305-337.

[23]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.

[24]

G. Dal Maso, Integral representation on $BV(\Omega)$ of $\Gamma$-limits of variational integrals, Manuscripta Math., 30 (1978/80), 387-416. doi: 10.1007/BF01301259.

[25]

R. Dal Passo, Uniqueness of the entropy solution of a strongly degenerate parabolic equation, Comm. in Partial Diff. Equat., 18 (1993), 265-279.

[26]

A. Chertock, A. Kurganov and P. Rosenau, Formation of discontinuities in flux-saturated degenerate parabolic equations, Nonlinearity, 16 (2003), 1875-1898. doi: 10.1088/0951-7715/16/6/301.

[27]

V. De Cicco, N. Fusco and A. Verde, On $L^1$-lower semicontinuity in $BV$, J. Convex Anal., 12 (2005), 173-185.

[28]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, (Italian) [New functionals in the calculus of variations], Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat Fis. Natur. (8), 82 (1988), 199-210.

[29]

J. J. Duderstadt and G. A. Moses, "Inertial Confinement Fusion," John Wiley and Sons, 1982.

[30]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Math., CRC Press, Boca Raton, FL, 1992.

[31]

S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.

[32]

R. Mc Cann and M. Puel, Construting a relativistic heat flow by transport time steps, Ann. de Inst. Henri Poincaré Anal. Non Linéaire, 26 (2009), 2539-2580.

[33]

D. Mihalas and B. Mihalas, "Foundations of Radiation Hydrodynamics," Oxford University Press, New York, 1984.

[34]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.

[35]

P. Rosenau, Free energy functionals at the high gradient limit, Phys. Review A, 41 (1990), 2227-2230. doi: 10.1103/PhysRevA.41.2227.

[36]

P. Rosenau, Tempered diffusion: A transport process with propagating front and inertial delay, Phys. Review A, 46 (1992), 7371-7374. doi: 10.1103/PhysRevA.46.R7371.

[37]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," GTM, 120, Springer-Verlag, New York, 1989.

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.

[2]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear equations: The elliptic case, Annali della Scuola Norm. Sup. di Pisa. Cl. Sci. (5), 3 (2004), 555-587.

[3]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear equation: The parabolic case, Arch. Rat. Mech. Anal., 176 (2005), 415-453. doi: 10.1007/s00205-005-0358-5.

[4]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear elliptic equation, Nonlinear Analysis, 61 (2005), 637-669. doi: 10.1016/j.na.2004.11.020.

[5]

F. Andreu, V. Caselles and J. M. Mazón, The Cauchy problem for a strongly degenerate quasilinear equation, Journal European Math. Society (JEMS), 7 (2005), 361-393. doi: 10.4171/JEMS/32.

[6]

F. Andreu, V. Caselles, J. M. Mazón and S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal., 182 (2006), 269-297. doi: 10.1007/s00205-006-0428-3.

[7]

F. Andreu, V. Caselles and J. M. Mazón, Some regularity results on the 'relativistic' heat equation, J. Diff. Equat., 245 (2008), 3639-3663. doi: 10.1016/j.jde.2008.06.024.

[8]

F. Andreu, V. Caselles, J. M. Mazón and S. Moll, The Dirichlet problem associated to the relativistic heat equation, Math. Ann., 347 (2010), 135-199. doi: 10.1007/s00208-009-0428-3.

[9]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. di Matematica Pura ed Appl. (4), 135 (1983), 293-318.

[10]

Ph. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Normale Superiore di Pisa Cl. Sci. (4), 22 (1995), 241-273.

[11]

Ph. Bénilan and M. G. Crandall, Completely accretive operators, in "Semigroup Theory and Evolution Equations" (eds. Ph. Clement, et al.) (Delft, 1989), Lecture Notes in Pure and Appl. Math., 135, Dekker, New York, (1991), 41-75.

[12]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators," in preparation.

[13]

M. Bertsch and R. Dal Passo, Hyperbolic phenomena in a strongly degenerate parabolic equation, Arch Rational Mech. Anal., 117 (1992), 349-387. doi: 10.1007/BF00376188.

[14]

M. Bertsch and R. Dal Passo, A parabolic equation with a mean-curvature type operator, in "Nonlinear Diffusion Equations and their Equilibrium States, 3" (Gregynog, 1989), Progr. Nonlinear Differential Equation Appl., 7, Birkhäuser Boston, Boston, MA, (1992), 89-97.

[15]

Ph. Blanc, On the regularity of the solutions of some degenerate parabolic equations, Comm. in Partial Diff. Equat., 18 (1993), 821-846.

[16]

Ph. Blanc, "Sur une Classe d'Equations Paraboliques Degeneréesa une Dimension d'Espace Possedant des Solutions Discontinues," Ph.D. Thesis, number 798, Ecole Polytechnique Federale de Lausanne, 1989.

[17]

Y. Brenier, Extended Monge-Kantorovich theory, in "Optimal Transportation and Applications" (eds., L. A. Caffarelli and S. Salsa) (Martina-Franca, 2001), Lecture Notes in Math., 1813, Springer, Berlin, 2003, 91-121.

[18]

F. Browder, Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Nat. Acad. Sci. USA, 74 (1977), 2659-2661. doi: 10.1073/pnas.74.7.2659.

[19]

J. Carrillo and P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Diff. Equat., 156 (1999), 93-121. doi: 10.1006/jdeq.1998.3597.

[20]

V. Caselles, On the entropy conditions for some flux limited diffusion equations, J. Diff. Equat., 250 (2011), 3311-3348. doi: 10.1016/j.jde.2011.01.027.

[21]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.

[22]

M. G. Crandall, Nonlinear semigroups and evolution equations governed by accretive operators, in "Nonlinear Functional Analysis and its Applications, Part 1" (Berkeley, Calif., 1983), Proc. of Symp. in Pure Mat., 45, Part I, AMS, Providence, RI, (1986), 305-337.

[23]

M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. doi: 10.2307/2373376.

[24]

G. Dal Maso, Integral representation on $BV(\Omega)$ of $\Gamma$-limits of variational integrals, Manuscripta Math., 30 (1978/80), 387-416. doi: 10.1007/BF01301259.

[25]

R. Dal Passo, Uniqueness of the entropy solution of a strongly degenerate parabolic equation, Comm. in Partial Diff. Equat., 18 (1993), 265-279.

[26]

A. Chertock, A. Kurganov and P. Rosenau, Formation of discontinuities in flux-saturated degenerate parabolic equations, Nonlinearity, 16 (2003), 1875-1898. doi: 10.1088/0951-7715/16/6/301.

[27]

V. De Cicco, N. Fusco and A. Verde, On $L^1$-lower semicontinuity in $BV$, J. Convex Anal., 12 (2005), 173-185.

[28]

E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, (Italian) [New functionals in the calculus of variations], Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat Fis. Natur. (8), 82 (1988), 199-210.

[29]

J. J. Duderstadt and G. A. Moses, "Inertial Confinement Fusion," John Wiley and Sons, 1982.

[30]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Math., CRC Press, Boca Raton, FL, 1992.

[31]

S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.

[32]

R. Mc Cann and M. Puel, Construting a relativistic heat flow by transport time steps, Ann. de Inst. Henri Poincaré Anal. Non Linéaire, 26 (2009), 2539-2580.

[33]

D. Mihalas and B. Mihalas, "Foundations of Radiation Hydrodynamics," Oxford University Press, New York, 1984.

[34]

M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.

[35]

P. Rosenau, Free energy functionals at the high gradient limit, Phys. Review A, 41 (1990), 2227-2230. doi: 10.1103/PhysRevA.41.2227.

[36]

P. Rosenau, Tempered diffusion: A transport process with propagating front and inertial delay, Phys. Review A, 46 (1992), 7371-7374. doi: 10.1103/PhysRevA.46.R7371.

[37]

W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation," GTM, 120, Springer-Verlag, New York, 1989.

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