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March  2013, 12(2): 973-984. doi: 10.3934/cpaa.2013.12.973

Decay of solutions to fractal parabolic conservation laws with large initial data

Fengbai Li 1, and  Feng Rong 1,

1. 

Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China, China

Received  October 2011 Revised  February 2012 Published  September 2012

In this paper, we study the time-asymptotic behavior of solutions to the Cauchy problem for multi-dimensional parabolic conservation laws with fractional dissipation. For arbitrarily large initial data, we obtain the optimal decay rates in $L^2$ and homogeneous Sobolev spaces for solutions to the equation with the power of Laplacian $\frac{1}{2} < \alpha \le 1$ by using the time-frequency decomposition method and the energy method. The argument is based on a maximum principle.
Keywords: optimal decay rate., parabolic conservation laws, large initial data, Time-frequency cut-off operator.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35R1.
Citation: Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973
References:
[1]

P. Biler, T. Funaki and W. Woyczyński, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.  Google Scholar

[2]

P. Biler, G. Karch and W. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré, 18 (2001), 613-637. doi: 10.1016/S0294-1449(01)00080-4.  Google Scholar

[3]

T. Cazenave, "Semilinear Schrödinger Equation," Courant Lecture Notes in Math. 10, Courant Ins. Math. Sci. and Amer. Math. Soc., 2003.  Google Scholar

[4]

C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Math. 34, Springer-Verlag, New York, 2006.  Google Scholar

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P. Constantin, D. Córdoba and J. Wu, On the critical dissipative quasi-geostrophic equations, Indiana Univ. Math. J., 50 (2001), 97-107. doi: 10.1512/iumj.2001.50.2153.  Google Scholar

[6]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333.  Google Scholar

[7]

A. Cródoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.  Google Scholar

[8]

J. Droniou, T. Gallouët and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499-521. doi: 10.1007/s00028-003-0503-1.  Google Scholar

[9]

J. Han and W. Wang, Decay estimates and the application in the stability for the solutions to the subcritical dissipative quasi-geostrophic equations, preprint, 2009. Google Scholar

[10]

E. Hopf, The partial differential equation $u_t+uu_x=\mu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302.  Google Scholar

[11]

N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[12]

D. B. Kotlow, Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl., 35 (1971), 563-576. doi: 10.1016/0022-247X(71)90204-6.  Google Scholar

[13]

T.-T. Li and Y.-M. Chen, "Nonlinear Evolution Equations," in Chinese, Science Press, 1989. Google Scholar

[14]

S. Resnick, "Dynamical Problems in Non-linear Advective Partial Differential Equations," Ph.D. thesis, University of Chicago, 1995.  Google Scholar

[15]

M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473. doi: 10.1080/0360530800882145.  Google Scholar

[16]

M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal., 10 (1986), 943-956. doi: 10.1016/0362-546X(86)90080-5.  Google Scholar

[17]

M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375. doi: 10.1137/S0036141002409362.  Google Scholar

[18]

W. Wang and L. Yu, Decay of solutions to parabolic conservation laws with large initial data, preprint, 2009. Google Scholar

show all references

References:
[1]

P. Biler, T. Funaki and W. Woyczyński, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.  Google Scholar

[2]

P. Biler, G. Karch and W. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré, 18 (2001), 613-637. doi: 10.1016/S0294-1449(01)00080-4.  Google Scholar

[3]

T. Cazenave, "Semilinear Schrödinger Equation," Courant Lecture Notes in Math. 10, Courant Ins. Math. Sci. and Amer. Math. Soc., 2003.  Google Scholar

[4]

C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Math. 34, Springer-Verlag, New York, 2006.  Google Scholar

[5]

P. Constantin, D. Córdoba and J. Wu, On the critical dissipative quasi-geostrophic equations, Indiana Univ. Math. J., 50 (2001), 97-107. doi: 10.1512/iumj.2001.50.2153.  Google Scholar

[6]

P. Constantin and J. Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333.  Google Scholar

[7]

A. Cródoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1.  Google Scholar

[8]

J. Droniou, T. Gallouët and J. Vovelle, Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3 (2003), 499-521. doi: 10.1007/s00028-003-0503-1.  Google Scholar

[9]

J. Han and W. Wang, Decay estimates and the application in the stability for the solutions to the subcritical dissipative quasi-geostrophic equations, preprint, 2009. Google Scholar

[10]

E. Hopf, The partial differential equation $u_t+uu_x=\mu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302.  Google Scholar

[11]

N. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z.  Google Scholar

[12]

D. B. Kotlow, Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl., 35 (1971), 563-576. doi: 10.1016/0022-247X(71)90204-6.  Google Scholar

[13]

T.-T. Li and Y.-M. Chen, "Nonlinear Evolution Equations," in Chinese, Science Press, 1989. Google Scholar

[14]

S. Resnick, "Dynamical Problems in Non-linear Advective Partial Differential Equations," Ph.D. thesis, University of Chicago, 1995.  Google Scholar

[15]

M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473. doi: 10.1080/0360530800882145.  Google Scholar

[16]

M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal., 10 (1986), 943-956. doi: 10.1016/0362-546X(86)90080-5.  Google Scholar

[17]

M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375. doi: 10.1137/S0036141002409362.  Google Scholar

[18]

W. Wang and L. Yu, Decay of solutions to parabolic conservation laws with large initial data, preprint, 2009. Google Scholar

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