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January  2012, 11(1): 365-373. doi: 10.3934/cpaa.2012.11.365

## On the solvability conditions for the diffusion equation with convection terms

 1 University of Cape Town, Department of Mathematics, Rondebosch, 7701, South Africa 2 Institute of Mathematics, University Lyon 1, 69622 Villeurbann

Received  January 2010 Revised  August 2010 Published  September 2011

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].
Citation: Vitali Vougalter, Vitaly Volpert. On the solvability conditions for the diffusion equation with convection terms. Communications on Pure & Applied Analysis, 2012, 11 (1) : 365-373. doi: 10.3934/cpaa.2012.11.365
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