Communications on Pure and Applied Analysis (CPAA)

On the solvability conditions for the diffusion equation with convection terms

Pages: 365 - 373, Volume 11, Issue 1, January 2012      doi:10.3934/cpaa.2012.11.365

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Vitali Vougalter - University of Cape Town, Department of Mathematics, Rondebosch, 7701, South Africa (email)
Vitaly Volpert - Institute of Mathematics, University Lyon 1, 69622 Villeurbann, France (email)

Abstract: A linear second order elliptic equation describing heat or mass diffusion and convection on a given velocity field is considered in $R^3$. The corresponding operator $L$ may not satisfy the Fredholm property. In this case, solvability conditions for the equation $L u = f$ are not known. In this work, we derive solvability conditions in $H^2(R^3)$ for the non self-adjoint problem by relating it to a self-adjoint Schrödinger type operator, for which solvability conditions are obtained in our previous work [13].

Keywords:  Solvability conditions, porous medium, adjoint operator, continuous spectrum.
Mathematics Subject Classification:  Primary: 35J10, 35P10; Secondary: 35P25.

Received: January 2010;      Revised: August 2010;      Available Online: September 2011.