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

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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

Abstract
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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 and 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.
[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.
[3]	T. Cazenave, "Semilinear Schrödinger Equation," Courant Lecture Notes in Math. 10, Courant Ins. Math. Sci. and Amer. Math. Soc., 2003.
[4]	C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Math. 34, Springer-Verlag, New York, 2006.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	T.-T. Li and Y.-M. Chen, "Nonlinear Evolution Equations," in Chinese, Science Press, 1989.
[14]	S. Resnick, "Dynamical Problems in Non-linear Advective Partial Differential Equations," Ph.D. thesis, University of Chicago, 1995.
[15]	M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473. doi: 10.1080/0360530800882145.
[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.
[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.
[18]	W. Wang and L. Yu, Decay of solutions to parabolic conservation laws with large initial data, preprint, 2009.

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.
[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.
[3]	T. Cazenave, "Semilinear Schrödinger Equation," Courant Lecture Notes in Math. 10, Courant Ins. Math. Sci. and Amer. Math. Soc., 2003.
[4]	C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Math. 34, Springer-Verlag, New York, 2006.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	T.-T. Li and Y.-M. Chen, "Nonlinear Evolution Equations," in Chinese, Science Press, 1989.
[14]	S. Resnick, "Dynamical Problems in Non-linear Advective Partial Differential Equations," Ph.D. thesis, University of Chicago, 1995.
[15]	M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473. doi: 10.1080/0360530800882145.
[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.
[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.
[18]	W. Wang and L. Yu, Decay of solutions to parabolic conservation laws with large initial data, preprint, 2009.

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