一类限流扩散方程及其 $BV$ 理论进展——献给陈化教授 70 寿辰

Abstract

Abstract:

We briefly review the $BV$ entropy theory for a class of flux-limited diffusion equations, including the notion and well-posedness of entropy solutions, the hyperbolic features and entropy conditions, and the finite speed of propagation. The purpose is to serve as a basic theoretical framework for subsequent investigations of nonlinear parabolic equations exhibiting large-gradient degeneracy.

Key words: flux-limited diffusion, $BV$ entropy solution, hyperbolic features, finite propagation

CLC Number:

O175.29

Yong Luo, Chunpeng Wang, Jingxue Yin. Flux-Limited Diffusion Equations and Their $BV$ Theory[J].Acta mathematica scientia,Series A, 2026, 46(2): 604-615.

Trendmd

References 34

[1]	Ambrosio L, Fusco N, Pallara D. Functions of Bounded Variation and Free Discontinuity Problems. Oxford: Oxford Science Publications, 2000
[2]	Andreu F, Caselles V, Mazón J M. A strongly degenerate quasilinear elliptic equation. Nonlinear Analysis: Theory, Methods & Applications, 2005, 61(4): 637-669
[3]	Andreu F, Caselles V, Mazón J M. A Fisher-Kolmogorov equation with finite speed of propagation. Journal of Differential Equations, 2010, 248(10): 2528-2561
[4]	Andreu F, Caselles V, Mazón J M, Salvador Moll. Finite propagation speed for limited flux diffusion equations. Archive for Rational Mechanics and Analysis, 2006, 182: 269-297
[5]	Andreu F, Caselles V, Mazón J M, Moll S. The Dirichlet problem associated to the relativistic heat equation. Mathematische Annalen, 2010, 347(1): 135-199
[6]	Bertsch M, Dal Passo R. Hyperbolic phenomena in a strongly degenerate parabolic equation. Archive for Rational Mechanics and Analysis, 1992, 117(4): 349-387
[7]	Brenier Y. Extended monge-kantorovich theory//Optimal Transportation and Applications: Lectures given at the CIME Summer School, held in Martina Franca, Italy, September 2-8, 2001. Berlin: Springer, 2004, 91-121
[8]	Calvo J. Analysis of a class of degenerate parabolic equations with saturation mechanisms. SIAM Journal on Mathematical Analysis, 2015, 47(4): 2917-2951
[9]	Calvo J, Campos J, Caselles V, et al. Pattern formation in a flux limited reaction-diffusion equation of porous media type. Inventiones Mathematicae, 2016, 206(1): 57-108
[10]	Calvo J, Campos J, Caselles V, et al. Qualitative behaviour for flux-saturated mechanisms: travelling waves, waiting time and smoothing effects. Journal of the European Mathematical Society, 2017, 19(2): 441-472
[11]	Carrillo J A, Caselles V, Moll S. On the relativistic heat equation in one space dimension. Proceedings of the London Mathematical Society, 2013, 107(6): 1395-1423
[12]	Caselles V. An existence and uniqueness result for flux limited diffusion equations. Discrete and Continuous Dynamical Systems, 2011, 31(4): 1151-1195
[13]	Caselles V. Flux limited generalized porous media diffusion equations. Publicacions Matemàtiques, 2013, 57(1): 144-217
[14]	Caselles V. Convergence of flux limited porous media diffusion equations to its classical counterpart. Ann Scuola Norm Sup Pisa Cl Sci, 2015, XIV(5): 481-505
[15]	Caselles V. On the entropy conditions for some flux limited diffusion equations. Journal of Differential Equations, 2011, 250(8): 3311-3348
[16]	Andreu F, Caselles V, Mazón J M. The Cauchy problem for a strongly degenerate quasilinear equation. Journal of the European Mathematical Society, 2005, 7(3): 361-393
[17]	Chen G Q, Ziemer W P, Torres M. Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Communications on Pure and Applied Mathematics, 2009, 62(2): 242-304
[18]	Chen G Q, Irving C, Torres M. Extended divergence-measure fields, the Gauss-Green formula and Cauchy fluxes. Archive for Rational Mechanics and Analysis, 2025, 249(6): Art 79
[19]	Chertock A, Kurganov A, Rosenau P. Formation of discontinuities in flux-saturated degenerate parabolic equations. Nonlinearity, 2003, 16(6): Art 1875
[20]	Corli A, Malaguti L, Sovrano E. Wavefront solutions to reaction-convection equations with Perona-Malik diffusion. Journal of Differential Equations, 2022, 308: 474-506
[21]	Dal Passo R. Uniqueness of the entropy solution of a strongly degenerate parabolic equation. Communications in Partial Differential Equations, 1993, 18(1/2): 265-279
[22]	Elenius T. Characterizations and properties of solutions to parabolic problems of linear growth.Mathematische Annalen, arXiv:2025.16747
[23]	Giacomelli L. Finite speed of propagation and waiting time phenomena for degenerate parabolic equations with linear growth Lagrangian. SIAM Journal on Mathematical Analysis, 2015, 47(3): 2426-2441
[24]	Giacomelli L, Moll S, Petitta F. Optimal waiting time bounds for some flux-saturated diffusion equations. Communications in Partial Differential Equations, 2017, 42(4): 556-578
[25]	Kružkov S N. First order quasilinear equations with several independent variables (Russian). Mat Sb (NS), 1970, 81(123): 228-255
[26]	Levermore C D. Chapman-Enskog Approach to Flux-limited Diffusion Theory. Livermore: Lawrence Livermore National Lab (LLNL), 1979
[27]	Levermore C D, Pomraning G C. A flux-limited diffusion theory. Astrophysical Journal, 1981, 248: 321-334
[28]	Li X W, Li J Y, Mei M, Nave J C. Nonlinear stability of shock profiles to Burgers equation with critical fast diffusion and singularity. SIAM Journal on Mathematical Analysis, 2025, 57(6): 6403-6431
[29]	Mihalas D, Mihalas B W. Foundations of Radiation Hydrodynamics. New York: Oxford University Press, 1984
[30]	Pomraning G C. Flux-limited diffusion with relativistic corrections. Astrophysical Journal, 1983, 266: 841-847
[31]	Rosenau P. Tempered diffusion: A transport process with propagating fronts and inertial delay. Phys Rev A, 1992, 46(12): 7371-7374
[32]	Wang C P, Yin J X, Wang Z J. Similar entropy solutions of a singular diffusion equation. Computers & Mathematics with Applications, 2005, 49(7): 1059-1068
[33]	Wang C P, Yin J X, Wang Z J. Self-similar entropy solutions of a singular diffusion equation in arbitrary spatial dimension. Nonlinear Analysis: Theory, Methods & Applications, 2008, 69(7): 2192-2208
[34]	Yin J X, Li H L, Pang P Y H. BV solutions of a singular diffusion equation. Mathematische Nachrichten, 2003, 253(1): 92-106

Flux-Limited Diffusion Equations and Their $BV$ Theory

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 34

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Lingjun Wang. Hydroelastic Small-Amplitude Solitary and Generalized Solitary Waves with Constant Vorticity [J]. Acta mathematica scientia,Series A, 2026, 46(2): 628-645.
[2]	Hao Liu, Yun Wang, Chunjing Xie. Existence of Discretely Self-Similar Solutions to Four-Dimensional Steady Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2026, 46(2): 709-723.
[3]	Jiaqing Bao, Nanbo Chen, Xiaochun Liu, Sainan Liang. On the Schrödinger-Bopp-Podolsky-Proca System with Singular Nonlinearity on Closed Manifolds [J]. Acta mathematica scientia,Series A, 2026, 46(2): 751-769.
[4]	Tianyi Wang, Zhengyang Yu. Rigidity of Steady Inhomogeneous Incompressible Euler Equations in Two-Dimensional Annular Domains [J]. Acta mathematica scientia,Series A, 2026, 46(2): 819-839.
[5]	Pingping Zeng, Pingan Zeng, Lu Liu. Optical Soliton Resonances and Soliton Molecules for the Hiorta Equation [J]. Acta mathematica scientia,Series A, 2026, 46(1): 238-248.
[6]	Yulu Tian, Dengshan Wang, Liang Zhao. Normalized Solutions to a $p$-Laplacian Equation with an $L^2$ Constraint: The Mass Supercritical Case [J]. Acta mathematica scientia,Series A, 2026, 46(1): 1-30.
[7]	Chao Zhu, Qingming Hao, Zhigang Pan, Yanhua Wang. Steady-State Bifurcation to a Class of Extended Fisher-Kolmogorov System with Cooperative and Self-Limiting Effects [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1432-1443.
[8]	Weibin Wang, Zhiqiang Shao. Delta Shocks and Vacuum States in Vanishing Pressure and Magnetic Field Limits of Solutions to the Isentropic Magnetogasdynamics with the Logarithmic Equation of State Under the Body Force [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1477-1491.
[9]	Xiaolu Li, Yuanze Wu. The Existence and Multiplicity of Spiked Solutions for Nonlinear Schrödinger Equation with Variable Exponents [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1535-1552.
[10]	Zhu Weipeng, Li Jinlu, Wu Xing. Global Smooth Solutions of the Damped Boussinesq Equations with a Class of Large Initial Data [J]. Acta mathematica scientia,Series A, 2025, 45(4): 1077-1085.
[11]	Chen Xuejiao, Li Dandan, Shi Jincheng, Zeng Peng. Continuous Dependence of Harmonic Equations on Base Perturbation in Prismatic Cylinder [J]. Acta mathematica scientia,Series A, 2025, 45(1): 101-109.
[12]	Li Changhao, Yuan Baoquan. Global Regularity for the 2D Micropolar Rayleigh-Bénard Convection System with Velocity Zero Dissipation and Temperature Fractional Diffusion [J]. Acta mathematica scientia,Series A, 2024, 44(4): 914-924.
[13]	Li Yaohong, Tian Shoufu. Persistence Property and Propagation Speed for the mCH-CH Equation [J]. Acta mathematica scientia,Series A, 2024, 44(3): 609-620.
[14]	Duan Xueliang, Wu Xiaofan, Wei Gongming, Yang Haitao. Multiple Positive Solutions of Kirchhoff Type Linearly Coupled System with Critical Exponent [J]. Acta mathematica scientia,Series A, 2024, 44(3): 699-716.
[15]	He Fen, Wang Zhen. Riemann Problem for a Class of Mixed-type Chemotaxis Models [J]. Acta mathematica scientia,Series A, 2024, 44(2): 354-360.