带有分数扩散的多维Burgers方程的衰减估计

数学物理学报 ›› 2016, Vol. 36 ›› Issue (2): 340-352.

带有分数扩散的多维Burgers方程的衰减估计

余沛

电子科技大学数学科学学院成都 611731

收稿日期:2015-09-11 修回日期:2016-01-28 出版日期:2016-04-25 发布日期:2016-04-25
作者简介:余沛,yp9106@gmail.com
基金资助:
国家自然科学基金(11571063)资助

Decay of Weak Solutions to the Multi-Dimensional Burgers' Equation with Fractional Diffusion

Yu Pei

School of Mathematical Sciences, University Electronic Sciences & Technology, Chengdu 611731

Received:2015-09-11 Revised:2016-01-28 Online:2016-04-25 Published:2016-04-25
Supported by:
Supported by the NSFC (11571063)

摘要/Abstract

摘要：

研究了多维分数阶Burgers方程整体弱解的衰减性质.特别是,对u₀∈L²∩L^p (p≠2)或u₀∈L²∩L^n/2α-1分别建立了解的一致L²或L^p (p>n/2α-1)衰减估计;而对u₀仅仅属于L²,证明了解一致L²衰减的不存在性.

关键词: 分数阶Burgers方程, 一般初值, 最优衰减估计

Abstract:

In this paper, we investigate the time decay properties of the global weak solutions to the multi-dimensional Burgers' equation with fractional diffusion. We establish the optimal decay rates in L² or L^p norm to solutions with initial data u₀∈L²∩L^p for p≠2. If u₀∈L² only, we also show that it is impossible to obtain a uniform decay. Finally, for u₀∈L²∩L^n/2α-1, we obtain a uniform decay estimate of solutions in L^p norm for any p>n/2α-1.

Key words: Fractional Burgers' equation, Large initial data, Optimal decay rate

中图分类号:

O175.29

余沛. 带有分数扩散的多维Burgers方程的衰减估计[J]. 数学物理学报, 2016, 36(2): 340-352.

Yu Pei. Decay of Weak Solutions to the Multi-Dimensional Burgers' Equation with Fractional Diffusion[J]. Acta mathematica scientia,Series A, 2016, 36(2): 340-352.

参考文献

[1] Alibaud N. Entropy formulation for fractal conservation laws. J Evol Equ, 2007, 12:145-175
[2] Alibaud N, Andreianov B. Non-uniqueness of weak solutions for the fractal burgers equation. Ann I H Poincaré-AN, 2010, 27:997-1016
[3] Alibaud N, Droniou J, Vovelle J. Occurrence and non-appearance of shocks in fractal Burgers equations. J Hyperbolic Differ Equ, 2007, 4(3):479-499
[4] Alibaud N, Imbert C, Karch G. Asymptotic properties of entropy solutions to fractal burgers equation. SIAM J Math Anal, 2010, 42:354-376
[5] Biler P, Funaki T, Woyczynski W. Fractal Burgers equations. J Diff Eq, 1998, 148:9-46
[6] Biler P, Karch G, Woyczynski W. Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann I H Poincaré-AN, 2001, 18:613-637
[7] Chan C H, Czubak M. Regularity of solutions for the critical N-dimensional Burgers equation. Ann I H Poincaré-AN, 2010, 27:471-501
[8] Clavin P. Instabilities and nonlinear patterns of overdriven detonations in gases//Berestycki H, Pomeau Y. Nonlinear PDE's in Condensed Matter and Reactive Flows. Dordrecht:Kluwer, 2002:49-97
[9] Cordoba A, Cordoba D. A pointwise estimate for fractionary derivatives with applications to partial differential equations. PNAS, 2003, 100:15316-15317
[10] Cordoba A, Cordoba D. A maximum principle applied to quasi-geostrophic equations. Commun Math Phys, 2004, 249:511-528
[11] Dong H, Du D, Li D. Finite time singularities and global well-posedness for fractal Burgers equations. Indiana Univ Math J, 2009, 58:807-821
[12] Droniou J, Gallouet T, Vovelle J. Global solution and smoothing effect for a non-local regularization of a hyperbolic equation. J Evol Equ, 2003, 3:499-521
[13] Ju N. The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Commun Math Phys, 2005, 255:161-181
[14] Karch G, Miao C, Xu X. On convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM J Math Anal, 2008, 39:1536-1549
[15] Kato T. Strong L^p-solutions of the Navier-Stokes equation in R^m, with applications to weak solutions. Math Z, 1984, 187:471-480
[16] Kato T, Fujita H. On the non-stationary Navier-Stokes system. Rend Sem Mat Univ Padova, 1962, 32:243-260
[17] Kiselev A, Nazarov F, Shterenberg R. Blow up and regularity for fractal Burgers equation. Dynamics of PDE, 2008, 5:211-240
[18] Kiselev A, Ryzhik L. Biomixing by chemotaxis and enhancement of biological reactions. Comm Part Diff Eqs, 2012, 37:298-318
[19] Li F, Rong F. Decay of solutions to fractal parabolic conservation laws with large initial data. Comm Pure Appl Anal, 2013, 12:973-984
[20] Li F, Wang W. The pointwise estimate of solutions to the parabolic conservation law in multi-dimensions. Nonlinear Differ Equ Appl NoDEA, 2014, 21:87-103
[21] Miao C, Wu G. Global well-posedness of the critical Burgers equation in critical Besov spaces. J Diff Eq, 2009, 247:1673-1693
[22] Niche C J, Orive-Illera R. Decay of solutions to a porous media equation with fractional diffusion. Advances in Differential Equations, 2014, 19:133-160
[23] Niche C J, Schonbek M E. Decay of weak solutions to the 2D dissipative quasi-geostrophic equation. Comm Math Phys, 2007, 276:93-115
[24] Schonbek M E. Decay of solutions to parabolic conservation laws. Comm Part Diff Eqs, 1980, 5:449-473
[25] Schonbek M E. L² decay of weak solutions of the Navier-Stokes equations. Arch Ration Mech Anal, 1985, 8:209-222
[26] Schonbek M E. Uniform decay rates for parabolic conservation laws. Nonlinear Anal, 1986, 10:943-956
[27] Schonbek M E. Large time behaviour of solutions to the Navier-Stokes equations. Comm Part Diff Eqs. 1986, 11:733-763
[28] Wang L, Wang W. Large-time behavior of periodic solutions to fractal Burgers equation with large initial data. Chin Ann Math, 2012, 33B:405-418
[29] Woyczynski W. Lévy processes in the physical sciences//Lévy Processes. Boston, MA:Birkháuser Boston, 2001:241-266
[30] Wu J. Dissipative quasi-geostrophic equations with L^p data. Electronic Journal of Differential Equations, 2001, 2001:1-13