September  2014, 34(9): 3575-3589. doi: 10.3934/dcds.2014.34.3575

Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics

1. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania

2. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

Received  June 2013 Revised  November 2013 Published  March 2014

In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions. Such scaling argument is very common for the study of long time behavior of solutions to evolutionary problems where a scaling invariance of the main part of the operator is present.
Citation: Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matematica Espaola, Madrid, 2010.

[2]

J. Bourgain, H. Brezis and P. Mironescu, Optimal control and partial differential equations, in Proceedings of the conference in honour of Professor Alain Bensoussan's 60th birthday, Paris, France, 2000 (eds. J. L. Menaldi et al), IOS Press, (2001), 439-455.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[4]

M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $R^N$, J. Funct. Anal., 100 (1991), 119-161. doi: 10.1016/0022-1236(91)90105-E.

[5]

M. Escobedo, J. L. Vázquez and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65. doi: 10.1007/BF00392203.

[6]

K. Hammer, Non-linear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168. doi: 10.1093/qjmam/24.2.155.

[7]

L. I. Ignat, T. I. Ignat and D. Stancu-Dumitru, A compactness tool for the analysis of nonlocal evolution equations, preprint, arXiv:1301.6019.

[8]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437. doi: 10.1016/j.jfa.2007.07.013.

[9]

G. Karch and K. Suzuki, Spikes and diffusion waves in a one-dimensional model of chemotaxis, Nonlinearity, 23 (2010), 3119-3137. doi: 10.1088/0951-7715/23/12/007.

[10]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439-465. doi: 10.1016/S0022-0396(02)00158-4.

[11]

P. Laurençot, Asymptotic self-similarity for a simplified model for radiating gases, Asymptot. Anal., 42 (2005), 251-262.

[12]

S. Schochet and E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws, Arch. Rational Mech. Anal., 119 (1992), 95-107. doi: 10.1007/BF00375117.

[13]

M. Schonbek, The Fourier splitting method, Advances in Geometric Analysis and Continuum Mechanics (Stanford, CA, 1993), Int. Press, Cambridge, MA, (1995), 269-274.

[14]

D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws, Evolutionary Equations, Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 473-553.

[15]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[16]

J. Terra and N. Wolanski, Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data, Discrete Contin. Dyn. Syst., 31 (2011), 581-605. doi: 10.3934/dcds.2011.31.581.

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI; Real Sociedad Matematica Espaola, Madrid, 2010.

[2]

J. Bourgain, H. Brezis and P. Mironescu, Optimal control and partial differential equations, in Proceedings of the conference in honour of Professor Alain Bensoussan's 60th birthday, Paris, France, 2000 (eds. J. L. Menaldi et al), IOS Press, (2001), 439-455.

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[4]

M. Escobedo and E. Zuazua, Large time behavior for convection-diffusion equations in $R^N$, J. Funct. Anal., 100 (1991), 119-161. doi: 10.1016/0022-1236(91)90105-E.

[5]

M. Escobedo, J. L. Vázquez and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., 124 (1993), 43-65. doi: 10.1007/BF00392203.

[6]

K. Hammer, Non-linear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168. doi: 10.1093/qjmam/24.2.155.

[7]

L. I. Ignat, T. I. Ignat and D. Stancu-Dumitru, A compactness tool for the analysis of nonlocal evolution equations, preprint, arXiv:1301.6019.

[8]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437. doi: 10.1016/j.jfa.2007.07.013.

[9]

G. Karch and K. Suzuki, Spikes and diffusion waves in a one-dimensional model of chemotaxis, Nonlinearity, 23 (2010), 3119-3137. doi: 10.1088/0951-7715/23/12/007.

[10]

C. Lattanzio and P. Marcati, Global well-posedness and relaxation limits of a model for radiating gas, J. Differential Equations, 190 (2003), 439-465. doi: 10.1016/S0022-0396(02)00158-4.

[11]

P. Laurençot, Asymptotic self-similarity for a simplified model for radiating gases, Asymptot. Anal., 42 (2005), 251-262.

[12]

S. Schochet and E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws, Arch. Rational Mech. Anal., 119 (1992), 95-107. doi: 10.1007/BF00375117.

[13]

M. Schonbek, The Fourier splitting method, Advances in Geometric Analysis and Continuum Mechanics (Stanford, CA, 1993), Int. Press, Cambridge, MA, (1995), 269-274.

[14]

D. Serre, $L^1$-stability of nonlinear waves in scalar conservation laws, Evolutionary Equations, Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 473-553.

[15]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[16]

J. Terra and N. Wolanski, Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data, Discrete Contin. Dyn. Syst., 31 (2011), 581-605. doi: 10.3934/dcds.2011.31.581.

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