June  2013, 33(6): 2299-2318. doi: 10.3934/dcds.2013.33.2299

Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas

1. 

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan, Taiwan

2. 

Mathematics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan

Received  October 2011 Revised  October 2012 Published  December 2012

Arising from one-particle distribution functions of stationary dissipative plasmas, we consider a coupled elliptic system with singular data in the plane. The existence and uniqueness of solutions to the Dirichlet boundary value problem are proved. In addition, the structure of other solutions, including blow-up solutions, is also clarified.
Citation: Jann-Long Chern, Zhi-You Chen, Yong-Li Tang. Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2299-2318. doi: 10.3934/dcds.2013.33.2299
References:
[1]

W. H. Bennet, Magnetically self-focusing streams,, Phys. Rev., 45 (1934), 890.   Google Scholar

[2]

J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles,, Comm. Math. Phys., 296 (2010), 323.  doi: 10.1007/s00220-010-1021-z.  Google Scholar

[3]

Z.-Y. Chen, J.-L. Chern and Y.-L. Tang, On the solutions to a Liouville-type system involving singularity,, Calc. Var. Partial Differential Equations, 43 (2012), 57.  doi: 10.1007/s00526-011-0403-1.  Google Scholar

[4]

Z.-Y. Chen, J.-L. Chern, J. Shi and Y.-L. Tang, On the uniqueness and structure of solutions to a coupled elliptic system,, J. Differential Equations, 249 (2010), 3419.  doi: 10.1016/j.jde.2010.09.001.  Google Scholar

[5]

J.-L. Chern, Z.-Y. Chen, Y.-L. Tang and C.-S. Lin, Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system,, J. Differential Equations, 246 (2009), 3704.  doi: 10.1016/j.jde.2009.01.005.  Google Scholar

[6]

S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type,, Geom. Funct. Anal., 5 (1995), 924.  doi: 10.1007/BF01902215.  Google Scholar

[7]

S. Childress and J. K. Percus, Nonlinear aspects of Chemotaxis,, Math. Biosci., 56 (1981), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[8]

M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems,, J. Differential Equations, 140 (1997), 59.  doi: 10.1006/jdeq.1997.3316.  Google Scholar

[9]

G. Dunne, "Self-dual Chern-Simons Theories,", Lecture Notes in Physics, m36 (1995).   Google Scholar

[10]

P. Debye and E. Huckel, Zur theorie der electrolyte,, Phys. Zft, 24 (1923), 305.   Google Scholar

[11]

J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions,, Comm. Pure Appl. Math., 59 (2006), 526.  doi: 10.1002/cpa.20099.  Google Scholar

[12]

J. Jost and G. Wang, Classification of solutions of a Toda system in $\R^2$,, Int. Math. Res. Not., (2002), 277.  doi: 10.1155/S1073792802105022.  Google Scholar

[13]

J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality,, Comm. Pure Appl. Math., 54 (2001), 1289.  doi: 10.1002/cpa.10004.  Google Scholar

[14]

M. K.-H. Kiessling, Symmetry results for finite-temperature, relativistic Thomas-Fermi equations,, Comm. Math. Phys., 226 (2002), 607.  doi: 10.1007/s002200200625.  Google Scholar

[15]

M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary Plasmas: Kinetic Modeling Bennet Pinch, and generalizations,, Phys. Plasmas, 1 (1994), 1841.  doi: 10.1063/1.870639.  Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235.   Google Scholar

[17]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar

[18]

C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system,, Ann. I. H. Poincaré-AN, 27 (2010), 117.  doi: 10.1016/j.anihpc.2009.09.001.  Google Scholar

[19]

M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices,, J. Math. Anal. Appl., 49 (1975), 215.   Google Scholar

[20]

Y. Yang, "Solitons in Field Theory and Nonlinear Analysis,", Springer-Verlag, (2001).   Google Scholar

show all references

References:
[1]

W. H. Bennet, Magnetically self-focusing streams,, Phys. Rev., 45 (1934), 890.   Google Scholar

[2]

J.-L. Chern, Z.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles,, Comm. Math. Phys., 296 (2010), 323.  doi: 10.1007/s00220-010-1021-z.  Google Scholar

[3]

Z.-Y. Chen, J.-L. Chern and Y.-L. Tang, On the solutions to a Liouville-type system involving singularity,, Calc. Var. Partial Differential Equations, 43 (2012), 57.  doi: 10.1007/s00526-011-0403-1.  Google Scholar

[4]

Z.-Y. Chen, J.-L. Chern, J. Shi and Y.-L. Tang, On the uniqueness and structure of solutions to a coupled elliptic system,, J. Differential Equations, 249 (2010), 3419.  doi: 10.1016/j.jde.2010.09.001.  Google Scholar

[5]

J.-L. Chern, Z.-Y. Chen, Y.-L. Tang and C.-S. Lin, Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system,, J. Differential Equations, 246 (2009), 3704.  doi: 10.1016/j.jde.2009.01.005.  Google Scholar

[6]

S. Chanillo and M. K.-H. Kiessling, Conformally invariant systems of nonlinear PDE of Liouville type,, Geom. Funct. Anal., 5 (1995), 924.  doi: 10.1007/BF01902215.  Google Scholar

[7]

S. Childress and J. K. Percus, Nonlinear aspects of Chemotaxis,, Math. Biosci., 56 (1981), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[8]

M. Chipot, I. Shafrir and G. Wolansky, On the solutions of Liouville systems,, J. Differential Equations, 140 (1997), 59.  doi: 10.1006/jdeq.1997.3316.  Google Scholar

[9]

G. Dunne, "Self-dual Chern-Simons Theories,", Lecture Notes in Physics, m36 (1995).   Google Scholar

[10]

P. Debye and E. Huckel, Zur theorie der electrolyte,, Phys. Zft, 24 (1923), 305.   Google Scholar

[11]

J. Jost, C. S. Lin and G. Wang, Analytic aspects of the Toda system. II. Bubbling behavior and existence of solutions,, Comm. Pure Appl. Math., 59 (2006), 526.  doi: 10.1002/cpa.20099.  Google Scholar

[12]

J. Jost and G. Wang, Classification of solutions of a Toda system in $\R^2$,, Int. Math. Res. Not., (2002), 277.  doi: 10.1155/S1073792802105022.  Google Scholar

[13]

J. Jost and G. Wang, Analytic aspects of the Toda system. I. A Moser-Trudinger inequality,, Comm. Pure Appl. Math., 54 (2001), 1289.  doi: 10.1002/cpa.10004.  Google Scholar

[14]

M. K.-H. Kiessling, Symmetry results for finite-temperature, relativistic Thomas-Fermi equations,, Comm. Math. Phys., 226 (2002), 607.  doi: 10.1007/s002200200625.  Google Scholar

[15]

M. K.-H. Kiessling and J. L. Lebowitz, Dissipative stationary Plasmas: Kinetic Modeling Bennet Pinch, and generalizations,, Phys. Plasmas, 1 (1994), 1841.  doi: 10.1063/1.870639.  Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bands of Chemotactic Bacteria: A theoretical analysis,, J. Theor. Biol., 30 (1971), 235.   Google Scholar

[17]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.  doi: 10.1007/s002220050023.  Google Scholar

[18]

C.-S. Lin and L. Zhang, Profile of bubbling solutions to a Liouville system,, Ann. I. H. Poincaré-AN, 27 (2010), 117.  doi: 10.1016/j.anihpc.2009.09.001.  Google Scholar

[19]

M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices,, J. Math. Anal. Appl., 49 (1975), 215.   Google Scholar

[20]

Y. Yang, "Solitons in Field Theory and Nonlinear Analysis,", Springer-Verlag, (2001).   Google Scholar

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