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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 and Applied Analysis, 2012, 11 (1) : 365-373. doi: 10.3934/cpaa.2012.11.365
##### References:
 [1] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, "Schrödinger Operators with Application to Quantum Mechanics and Global Geometry," Springer-Verlag, Berlin-Heidelberg-New York, 1987. [2] A. Ducrot, M. Marion and V. Volpert, "Reaction-diffusion Waves (with the Lewis number different from 1)," Publibook, Paris, 2009. [3] F. Hamel, H. Berestycki and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: 10.1007/S0022000412019. [4] F. Hamel, N. Nadirashvili and E. Russ, An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift, C.R. Math. Acad. Sci. Paris, 340 (2005), 347-352. doi: 10.1016/j.crma.2005.01.012. [5] B. L. G. Jonsson, M. Merkli, I. M. Sigal and F. Ting, "Applied Analysis," 349 pages (in preparation). [6] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279. doi: 10.1007/BF01360915. [7] E. Lieb and M. Loss, "Analysis," Graduate studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997. [8] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III. Scattering Theory," Academic Press, 1979. [9] I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/S0022200303254. [10] R. Texier-Picard and V. Volpert, Reaction-diffusion-convection problems in unbounded cylinders, Revista Matematica Complutense, 16 (2003), 233-276. [11] V. Volpert and A. Volpert, Convective instability of reaction fronts. Linear stability analysis, Eur. J. Appl. Math., 9 (1998), 507-525. doi: 10.1017/S095679259800357X. [12] A. Volpert and V. Volpert, Fredholm property of elliptic operators in unbounded domains, Trans. Moscow Math. Soc., 67 (2006), 127-197. doi: 10.1090/S0077155406001592. [13] V. Vougalter and V. Volpert, Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc., 54 (2011), 249-271. doi: 10.1017/S0013091509000236. [14] V. Vougalter and V.Volpert, On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60 (2010), 169-191.

show all references

##### References:
 [1] H. L. Cycon, R. G. Froese, W. Kirsch and B. Simon, "Schrödinger Operators with Application to Quantum Mechanics and Global Geometry," Springer-Verlag, Berlin-Heidelberg-New York, 1987. [2] A. Ducrot, M. Marion and V. Volpert, "Reaction-diffusion Waves (with the Lewis number different from 1)," Publibook, Paris, 2009. [3] F. Hamel, H. Berestycki and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: 10.1007/S0022000412019. [4] F. Hamel, N. Nadirashvili and E. Russ, An isoperimetric inequality for the principal eigenvalue of the Laplacian with drift, C.R. Math. Acad. Sci. Paris, 340 (2005), 347-352. doi: 10.1016/j.crma.2005.01.012. [5] B. L. G. Jonsson, M. Merkli, I. M. Sigal and F. Ting, "Applied Analysis," 349 pages (in preparation). [6] T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279. doi: 10.1007/BF01360915. [7] E. Lieb and M. Loss, "Analysis," Graduate studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997. [8] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III. Scattering Theory," Academic Press, 1979. [9] I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513. doi: 10.1007/S0022200303254. [10] R. Texier-Picard and V. Volpert, Reaction-diffusion-convection problems in unbounded cylinders, Revista Matematica Complutense, 16 (2003), 233-276. [11] V. Volpert and A. Volpert, Convective instability of reaction fronts. Linear stability analysis, Eur. J. Appl. Math., 9 (1998), 507-525. doi: 10.1017/S095679259800357X. [12] A. Volpert and V. Volpert, Fredholm property of elliptic operators in unbounded domains, Trans. Moscow Math. Soc., 67 (2006), 127-197. doi: 10.1090/S0077155406001592. [13] V. Vougalter and V. Volpert, Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc., 54 (2011), 249-271. doi: 10.1017/S0013091509000236. [14] V. Vougalter and V.Volpert, On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60 (2010), 169-191.
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