Asymptotic L1-decay of solutions of the porous medium equation to self-similarity

We consider the flow of gas in an N-dimensional porous medium with initial density v0NxO 0. The density vNx;tO then satisfies the nonlinear degenerate parabolic equa- tion vt E —v m where m> 1 is a physical constant. Assum- ing that R N1 Cj x j 2 Ov0NxOdx < 1, we prove that vNx;tO be- haves asymptotically, as t !1 , like the Barenblatt-Pattle solu- tion VNjxj;tO. We prove that the L 1 -distance decays at a rate t 1=NNNC2Om NO .M oreover, if N E1, we obtain an explicit time decay for the L 1 -distance at a suboptimal rate. The method we use is based on recent results we obtained for the Fokker-Planck equation (2), (3).

[1]  J. Vázquez,et al.  Eventual C∞-regularity and concavity for flows in one-dimensional porous media , 1987 .

[2]  Giuseppe Toscani,et al.  Sur l'inégalité logarithmique de Sobolev , 1997 .

[3]  J. Vázquez,et al.  Fundamental Solutions and Asymptotic Behaviour for the p-Laplacian Equation , 1988 .

[4]  J. Troutman.  Variational Calculus with Elementary Convexity , 1983 .

[5]  Avner Friedman,et al.  The asymptotic behavior of gas in an -dimensional porous medium , 1980 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  S. Kullback,et al.  A lower bound for discrimination information in terms of variation (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[8]  S. Kamenomostskaya.  Similar solutions and the asymptotics of filtration equations , 1976 .

[9]  Solomon Kullback,et al.  Correction to A Lower Bound for Discrimination Information in Terms of Variation , 1970, IEEE Trans. Inf. Theory.

[10]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[11]  S. Kamenomostskaya,et al.  The asymptotic behaviour of the solution of the filtration equation , 1973 .

[12]  William I. Newman,et al.  A Lyapunov functional for the evolution of solutions to the porous medium equation to self‐similarity. I , 1984 .

[13]  Giuseppe Toscani,et al.  Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation , 1999 .

[14]  James Ralston,et al.  A Lyapunov functional for the evolution of solutions to the porous medium equation to self‐similarity. II , 1984 .

[15]  Giuseppe Toscani,et al.  Exponential convergence toward equilibrium for homogeneous Fokker–Planck‐type equations , 1998 .

[16]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[17]  Juan Luis Vázquez,et al.  Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium , 1983 .