DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

doi:10.1016/S0252-9602(15)60001-7

Acta mathematica scientia,Series B ›› 2015, Vol. 35 ›› Issue (2): 281-302.doi: 10.1016/S0252-9602(15)60001-7

• Articles • Next Articles

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

Yunguang LU¹, Christian KLINGENBERG², Ujjwal KOLEY², Xuezhou LU³

1. Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China;
2. Institut für Mathematik, Julius-Maximilians-Universität Würzburg, Germany;
3. Laboratoire Ondes and Milieux Complexes UMR 6294, CNRS-Université|du Havre, France

Received:2013-12-24 Revised:2014-01-08 Online:2015-03-20 Published:2015-03-20
Contact: Xuezhou LU Laboratoire Ondes and Milieux Complexes UMR 6294, CNRS-Université du Havre, France E-mail: xuezhou.lu@gmail.com E-mail:xuezhou.lu@gmail.com
Supported by:
This work was partially supported by the Natural Science Foundation of China (11271105), a grant from the China Scholarship Council and a Humboldt fellowship of Germany.

Abstract

Abstract:

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

Key words: Degenerate convection-diffusion equations, regularity, decay rate, Lax-Oleinik type inequality

CLC Number:

35B50

Yunguang LU, Christian KLINGENBERG, Ujjwal KOLEY, Xuezhou LU. DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS[J].Acta mathematica scientia,Series B, 2015, 35(2): 281-302.

Trendmd

References

[1] Aronson D G. The Porous Medium Equation//Lecture notes in Mathematics. Berlin/New York: Springer- Verlag, 1985, 1224

[2] Bustos M C, Concha F, Bürger R, Tory E M. Sedimentation and thickening, volume 8 of Mathematical Modelling: Theory and Applications. Kluwer Academic Publishers, Dordrecht, 1999. Phenomenological foundation and mathematical theory

[3] Oleinik O A, Kalashnikov A S, Chzou Yui-lin. The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv Ahad Nauk SSR Ser Math, 1958, 22: 667-704

[4] Escobedo M, Zuazua E. Large time behaviour for convection-diffusion equations in Rⁿ. J Funct Anal, 1991, 100: 119-161

[5] Escobedo M, Vazquez J L, Zuazua E. Asymptotic behaviour and source type solutions for a diffusionconvection equation. Arch Rational Mech Anal, 1993, 124: 43-65

[6] Laurencot P H. Large time behaviour for diffusion equations with fast convection. Annali di Matematica pura ed applicata, 1998, 175: 233-251

[7] Liu T P, Pierre M. Source solutions and asymptotic behaviour in conservation laws. J Differ Equas, 1984, 51: 419-441

[8] Laurencot P H, Simondon F. Large time behaviour for porous medium equations with convection. Proc Royal Soc Edinburgh, 1998, 128A: 315-336

[9] Aronson D G. Regularity properties of flows through porous media. SIAM J Appl Math, 1969, 17: 461-467

[10] Jäger W, Lu Y G. Global regularity of solutions for general degenerate parabolic equations in 1-D. J Differ Equas, 1997, 140: 365-377

[11] Jäger W, Lu Y G. Hölder estimates of solutions for degenerate parabolic equations. Preprint 96-18, SISSA, Trieste and Abstract of the Workshop Hyperbolic Systems of Conservation Laws, Oberwolffach, Germany, April 28- May 4, 1996

[12] Smoller J A. Shock Waves and Reaction-Diffusion Equations. New York/Heidelberg/Berlin: Springer- Verlag, 1982

[13] Caffarelli L A, Vazquez J L, Wolanski N I. Lipschitz continuity of solutions and interfaces of the Ndimensional porous medium equation. Indiana Univ Math J, 1987, 36: 373-401

[14] Jäger W, Lu Y G. On solutions to Nonlinear Reaction-Diffusion-Convection Equations with Degenerate Diffusion. J Differ Equas, 2001, 170: 1-21

[15] Friedman A. Partial Differential Equations of Parabolic type. Prentice-Hall, Englewood Cliffs, NJ, 1964

[16] Itaya N. On the temporally global problem of the generalized Burgers' equation. J Math Kyoto University, 1974, 14: 129-177

[17] Ladyzhenskaya O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type//Translation of Mathematical Monographs, Vol 23. Providence, RI: Amer Math Soc, 1968

[18] Nash J. Le probleme de Cauchy pour les equations differentielles d'un iuide general. Bull Soc Math France, 1962, 90: 487-497

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Zhigang WU, Fuyi XU. STRONG SOLUTION FOR BIPOLAR COMPRESSIBLE NAVIER-STOKES-POISSON SYSTEM WITH UNEQUAL VISCOSITIES IN $L^P$-FRAMEWORK [J]. Acta mathematica scientia,Series B, 2025, 45(3): 1019-1044.
[2]	Di WU. ${C^{1, 1}}$ REGULARITY FOR SOLUTIONS TO THE DEGENERATE DUAL ORLICZ-MINKOWSKI PROBLEM [J]. Acta mathematica scientia,Series B, 2025, 45(2): 327-337.
[3]	Houzhi TANG, Shuxing ZHANG, Weiyuan ZOU. OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS [J]. Acta mathematica scientia,Series B, 2025, 45(2): 659-683.
[4]	Lihua TAN, Yingzhe FAN. LOW-REGULARITY SOLUTIONS TO FOKKER-PLANCK-TYPE SYSTEMS IN THE WHOLE SPACE [J]. Acta mathematica scientia,Series B, 2024, 44(6): 2361-2390.
[5]	Jun Chen, Xuemei Deng. ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS^* [J]. Acta mathematica scientia,Series B, 2024, 44(5): 1903-1915.
[6]	Cui NING, Chenxi HAO, Yaohong WANG. A LOW-REGULARITY FOURIER INTEGRATOR FOR THE DAVEY-STEWARTSON II SYSTEM WITH ALMOST MASS CONSERVATION [J]. Acta mathematica scientia,Series B, 2024, 44(4): 1536-1549.
[7]	Cuntao xiao, Hua qiu, Zheng-an yao. THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS [J]. Acta mathematica scientia,Series B, 2024, 44(3): 973-983.
[8]	Han Wang, Yinghui Zhang. THE OPTIMAL LARGE TIME BEHAVIOR OF 3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING [J]. Acta mathematica scientia,Series B, 2024, 44(3): 1064-1095.
[9]	Lvqiao LIU, Juan ZENG. THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL [J]. Acta mathematica scientia,Series B, 2024, 44(2): 455-473.
[10]	Changlin XIANG, Gaofeng ZHENG. SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION [J]. Acta mathematica scientia,Series B, 2024, 44(2): 420-430.
[11]	Zhenhua Guo, Xueyao Zhang. INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM^* [J]. Acta mathematica scientia,Series B, 2024, 44(1): 247-274.
[12]	Yu DONG, Yaofang HUANG, Li LI, Qing LU. THE REGULARITY CRITERIA OF WEAK SOLUTIONS TO 3D AXISYMMETRIC INCOMPRESSIBLE BOUSSINESQ EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(6): 2387-2397.
[13]	Renhai WANG, Boling GUO, Daiwen HUANG. THEORETICAL RESULTS ON THE EXISTENCE, REGULARITY AND ASYMPTOTIC STABILITY OF ENHANCED PULLBACK ATTRACTORS: APPLICATIONS TO 3D PRIMITIVE EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(6): 2493-2518.
[14]	Haitao WANG, Xiongtao ZHANG. THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1675-1716.
[15]	Bo Chen, Zhengmao Chen, Junhui Xie. PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1161-1174.