• Previous Article
    Spatiotemporal dynamics of cooperation and spite behavior by conformist transmission
  • CPAA Home
  • This Issue
  • Next Article
    Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis
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

[1]

Edoardo Mainini. On the signed porous medium flow. Networks & Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525

[2]

María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012

[3]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[4]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337

[5]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393

[6]

Wen Deng. Resolvent estimates for a two-dimensional non-self-adjoint operator. Communications on Pure & Applied Analysis, 2013, 12 (1) : 547-596. doi: 10.3934/cpaa.2013.12.547

[7]

Nguyen Dinh Cong, Roberta Fabbri. On the spectrum of the one-dimensional Schrödinger operator. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 541-554. doi: 10.3934/dcdsb.2008.9.541

[8]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013

[9]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

[10]

Ansgar Jüngel, Ingrid Violet. Mixed entropy estimates for the porous-medium equation with convection. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 783-796. doi: 10.3934/dcdsb.2009.12.783

[11]

Jing Li, Yifu Wang, Jingxue Yin. Non-sharp travelling waves for a dual porous medium equation. Communications on Pure & Applied Analysis, 2016, 15 (2) : 623-636. doi: 10.3934/cpaa.2016.15.623

[12]

Goro Akagi. Energy solutions of the Cauchy-Neumann problem for porous medium equations. Conference Publications, 2009, 2009 (Special) : 1-10. doi: 10.3934/proc.2009.2009.1

[13]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23

[14]

Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761

[15]

Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123

[16]

Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927

[17]

Panagiota Daskalopoulos, Eunjai Rhee. Free-boundary regularity for generalized porous medium equations. Communications on Pure & Applied Analysis, 2003, 2 (4) : 481-494. doi: 10.3934/cpaa.2003.2.481

[18]

Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725

[19]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

[20]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]