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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
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, (1987).   Google Scholar

[2]

A. Ducrot, M. Marion and V. Volpert, "Reaction-diffusion Waves (with the Lewis number different from 1),", Publibook, (2009).   Google Scholar

[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.  doi: 10.1007/S0022000412019.  Google Scholar

[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.  doi: 10.1016/j.crma.2005.01.012.  Google Scholar

[5]

B. L. G. Jonsson, M. Merkli, I. M. Sigal and F. Ting, "Applied Analysis,", 349 pages (in preparation)., ().   Google Scholar

[6]

T. Kato, Wave operators and similarity for some non-selfadjoint operators,, Math. Ann., 162 (): 258.  doi: 10.1007/BF01360915.  Google Scholar

[7]

E. Lieb and M. Loss, "Analysis,", Graduate studies in Mathematics, 14 (1997).   Google Scholar

[8]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III. Scattering Theory,", Academic Press, (1979).   Google Scholar

[9]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials,, Invent. Math., 155 (2004), 451.  doi: 10.1007/S0022200303254.  Google Scholar

[10]

R. Texier-Picard and V. Volpert, Reaction-diffusion-convection problems in unbounded cylinders,, Revista Matematica Complutense, 16 (2003), 233.   Google Scholar

[11]

V. Volpert and A. Volpert, Convective instability of reaction fronts. Linear stability analysis,, Eur. J. Appl. Math., 9 (1998), 507.  doi: 10.1017/S095679259800357X.  Google Scholar

[12]

A. Volpert and V. Volpert, Fredholm property of elliptic operators in unbounded domains,, Trans. Moscow Math. Soc., 67 (2006), 127.  doi: 10.1090/S0077155406001592.  Google Scholar

[13]

V. Vougalter and V. Volpert, Solvability conditions for some non Fredholm operators,, Proc. Edinb. Math. Soc., 54 (2011), 249.  doi: 10.1017/S0013091509000236.  Google Scholar

[14]

V. Vougalter and V.Volpert, On the solvability conditions for some non Fredholm operators,, Int. J. Pure Appl. Math., 60 (2010), 169.   Google Scholar

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, (1987).   Google Scholar

[2]

A. Ducrot, M. Marion and V. Volpert, "Reaction-diffusion Waves (with the Lewis number different from 1),", Publibook, (2009).   Google Scholar

[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.  doi: 10.1007/S0022000412019.  Google Scholar

[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.  doi: 10.1016/j.crma.2005.01.012.  Google Scholar

[5]

B. L. G. Jonsson, M. Merkli, I. M. Sigal and F. Ting, "Applied Analysis,", 349 pages (in preparation)., ().   Google Scholar

[6]

T. Kato, Wave operators and similarity for some non-selfadjoint operators,, Math. Ann., 162 (): 258.  doi: 10.1007/BF01360915.  Google Scholar

[7]

E. Lieb and M. Loss, "Analysis,", Graduate studies in Mathematics, 14 (1997).   Google Scholar

[8]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, III. Scattering Theory,", Academic Press, (1979).   Google Scholar

[9]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials,, Invent. Math., 155 (2004), 451.  doi: 10.1007/S0022200303254.  Google Scholar

[10]

R. Texier-Picard and V. Volpert, Reaction-diffusion-convection problems in unbounded cylinders,, Revista Matematica Complutense, 16 (2003), 233.   Google Scholar

[11]

V. Volpert and A. Volpert, Convective instability of reaction fronts. Linear stability analysis,, Eur. J. Appl. Math., 9 (1998), 507.  doi: 10.1017/S095679259800357X.  Google Scholar

[12]

A. Volpert and V. Volpert, Fredholm property of elliptic operators in unbounded domains,, Trans. Moscow Math. Soc., 67 (2006), 127.  doi: 10.1090/S0077155406001592.  Google Scholar

[13]

V. Vougalter and V. Volpert, Solvability conditions for some non Fredholm operators,, Proc. Edinb. Math. Soc., 54 (2011), 249.  doi: 10.1017/S0013091509000236.  Google Scholar

[14]

V. Vougalter and V.Volpert, On the solvability conditions for some non Fredholm operators,, Int. J. Pure Appl. Math., 60 (2010), 169.   Google Scholar

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