Advanced Search
Article Contents
Article Contents

On the solvability conditions for the diffusion equation with convection terms

Abstract Related Papers Cited by
  • 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].
    Mathematics Subject Classification: Primary: 35J10, 35P10; Secondary: 35P25.


    \begin{equation} \\ \end{equation}
  • [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.


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


    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.


    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.


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


    T. KatoWave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279. doi: 10.1007/BF01360915.


    E. Lieb and M. Loss, "Analysis," Graduate studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.


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


    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.


    R. Texier-Picard and V. Volpert, Reaction-diffusion-convection problems in unbounded cylinders, Revista Matematica Complutense, 16 (2003), 233-276.


    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.


    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.


    V. Vougalter and V. Volpert, Solvability conditions for some non Fredholm operators, Proc. Edinb. Math. Soc., 54 (2011), 249-271.doi: 10.1017/S0013091509000236.


    V. Vougalter and V.Volpert, On the solvability conditions for some non Fredholm operators, Int. J. Pure Appl. Math., 60 (2010), 169-191.

  • 加载中

Article Metrics

HTML views() PDF downloads(215) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint