June  2016, 9(3): 847-868. doi: 10.3934/dcdss.2016032

Nonlocal elliptic problems in infinite cylinder and applications

1. 

Peoples' Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198, Russian Federation

Received  March 2015 Revised  August 2015 Published  April 2016

We consider a unique solvability of nonlocal elliptic problems in infinite cylinder in weighted spaces and in Hölder spaces. Using these results we prove the existence and uniqueness of classical solution for the Vlasov--Poisson equations with nonlocal conditions in infinite cylinder for sufficiently small initial data.
Citation: Alexander L. Skubachevskii. Nonlocal elliptic problems in infinite cylinder and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 847-868. doi: 10.3934/dcdss.2016032
References:
[1]

M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type,, Uspekhi Mat. Nauk, 19 (1964), 53.   Google Scholar

[2]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations,, Zhurnal Vychisl. Matem. i Mat. Phys., 15 (1975), 136.   Google Scholar

[3]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics,, J. Differential Equations, 25 (1977), 342.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

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O. V. Besov, Some properties of the space $H_p^{(r_1, \ldots, r_m)}$,, Izv. Vysch. Uchebn. Zaven. Mat., 1 (1960), 16.   Google Scholar

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A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems,, Dokl. Akad. Nauk SSSR, 185 (1969), 739.   Google Scholar

[6]

P. M. Blekher, Operators that depend meromorphically on a parameter,, Vestnik Moscov. Univ., 24 (1969), 30.   Google Scholar

[7]

R. J. Di Perna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Comm. Pure Appl. Math., 42 (1989), 729.  doi: 10.1002/cpa.3160420603.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224 Springer-Verlag, 224 (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

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Y. Guo, Regularity for the Vlasov equations in a half-space,, Indiana Univ. Math. J., 43 (1994), 255.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[10]

P. L. Gurevich, Elliptic problems with nonlocal boundary conditions and Feller semigroups,, Sovrem. Mat. Fundam. Napravl., 38 (2010), 3.  doi: 10.1007/s10958-012-0746-y.  Google Scholar

[11]

A. K. Gushchin and V. P. Mikhailov, On the solvability of nonlocal problems for a second-order elliptic equation,, Math. Sb., 185 (1994), 121.  doi: 10.1070/SM1995v081n01ABEH003617.  Google Scholar

[12]

H. J. Hwang and J. J. L. Velázquez, On global existence for the Vlasov-Poisson system in a half-space,, J. Differential equations, 247 (2009), 1915.  doi: 10.1016/j.jde.2009.06.004.  Google Scholar

[13]

K. Yu. Kishkis, The index of the Bitsadze-Samarskii problem for harmonic functions,, Differentsial'nye Uravneniya, 24 (1988), 105.   Google Scholar

[14]

V. A. Kondratiev, Boundary value problems for elliptic equations in domains with conical and angular points,, Trudy Moskov. Mat. Obshch., 16 (1967), 209.   Google Scholar

[15]

V. A. Kondratiev and O. A. Oleinik, On asymptotics of solutions of nonlinear second order elliptic equations in cylindrical domains,, in Partial differential equations and functional analysis, 22 (1996), 160.   Google Scholar

[16]

V. V. Kozlov, The generalized Vlasov kinetic equation,, Uspekhi Mat. Nauk 63 (2008), 63 (2008), 93.  doi: 10.1070/RM2008v063n04ABEH004549.  Google Scholar

[17]

A. M. Krall, The development of general differential and general differential-boundary systems,, Rocky Mountain J. Math., 5 (1975), 493.  doi: 10.1216/RMJ-1975-5-4-493.  Google Scholar

[18]

K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fusion,, Iwanami Book Service Centre, (1997).   Google Scholar

[19]

S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries,, de Guyter Exp. Math., 13 (1994).  doi: 10.1515/9783110848915.525.  Google Scholar

[20]

I. Pankratova and A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 935.  doi: 10.3934/dcdsb.2009.11.935.  Google Scholar

[21]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Differential Equations, 95 (1992), 281.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[22]

A. A. Samarskii, Some problems of the theory of differential equations,, Differentsial'nye Uravneniya, 16 (1980), 1925.   Google Scholar

[23]

A. A. Skovoroda, Magnetic Systems for Plasma Confinement,, Fizmatlit, (2009).   Google Scholar

[24]

A. L. Skubachevskii, Nonclassical boundary-value problems. I,, Sovrem. Mat. Fundam. Napravl., 26 (2007), 3.  doi: 10.1007/s10958-008-9218-9.  Google Scholar

[25]

A. L. Skubachevskii, Nonclassical boundary-value problems. II,, Sovrem. Mat. Fundam. Napravl., 33 (2009), 3.  doi: 10.1007/s10958-010-9873-5.  Google Scholar

[26]

A. L. Skubachevskii, Vlasov-Poisson equations for a two-component plasma in a homogeneous magnetic field,, Uspekhi Mat. Nauk, 69 (2014), 107.   Google Scholar

[27]

A. P. Soldatov, The The Bitsadze-Samarskii problem for functions analytic in the sense of Douglis,, Differ. Uravn., 41 (2005), 396.  doi: 10.1007/s10625-005-0173-7.  Google Scholar

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Math. Library, 18 (1978).   Google Scholar

show all references

References:
[1]

M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type,, Uspekhi Mat. Nauk, 19 (1964), 53.   Google Scholar

[2]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations,, Zhurnal Vychisl. Matem. i Mat. Phys., 15 (1975), 136.   Google Scholar

[3]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics,, J. Differential Equations, 25 (1977), 342.  doi: 10.1016/0022-0396(77)90049-3.  Google Scholar

[4]

O. V. Besov, Some properties of the space $H_p^{(r_1, \ldots, r_m)}$,, Izv. Vysch. Uchebn. Zaven. Mat., 1 (1960), 16.   Google Scholar

[5]

A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems,, Dokl. Akad. Nauk SSSR, 185 (1969), 739.   Google Scholar

[6]

P. M. Blekher, Operators that depend meromorphically on a parameter,, Vestnik Moscov. Univ., 24 (1969), 30.   Google Scholar

[7]

R. J. Di Perna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems,, Comm. Pure Appl. Math., 42 (1989), 729.  doi: 10.1002/cpa.3160420603.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224 Springer-Verlag, 224 (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[9]

Y. Guo, Regularity for the Vlasov equations in a half-space,, Indiana Univ. Math. J., 43 (1994), 255.  doi: 10.1512/iumj.1994.43.43013.  Google Scholar

[10]

P. L. Gurevich, Elliptic problems with nonlocal boundary conditions and Feller semigroups,, Sovrem. Mat. Fundam. Napravl., 38 (2010), 3.  doi: 10.1007/s10958-012-0746-y.  Google Scholar

[11]

A. K. Gushchin and V. P. Mikhailov, On the solvability of nonlocal problems for a second-order elliptic equation,, Math. Sb., 185 (1994), 121.  doi: 10.1070/SM1995v081n01ABEH003617.  Google Scholar

[12]

H. J. Hwang and J. J. L. Velázquez, On global existence for the Vlasov-Poisson system in a half-space,, J. Differential equations, 247 (2009), 1915.  doi: 10.1016/j.jde.2009.06.004.  Google Scholar

[13]

K. Yu. Kishkis, The index of the Bitsadze-Samarskii problem for harmonic functions,, Differentsial'nye Uravneniya, 24 (1988), 105.   Google Scholar

[14]

V. A. Kondratiev, Boundary value problems for elliptic equations in domains with conical and angular points,, Trudy Moskov. Mat. Obshch., 16 (1967), 209.   Google Scholar

[15]

V. A. Kondratiev and O. A. Oleinik, On asymptotics of solutions of nonlinear second order elliptic equations in cylindrical domains,, in Partial differential equations and functional analysis, 22 (1996), 160.   Google Scholar

[16]

V. V. Kozlov, The generalized Vlasov kinetic equation,, Uspekhi Mat. Nauk 63 (2008), 63 (2008), 93.  doi: 10.1070/RM2008v063n04ABEH004549.  Google Scholar

[17]

A. M. Krall, The development of general differential and general differential-boundary systems,, Rocky Mountain J. Math., 5 (1975), 493.  doi: 10.1216/RMJ-1975-5-4-493.  Google Scholar

[18]

K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fusion,, Iwanami Book Service Centre, (1997).   Google Scholar

[19]

S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries,, de Guyter Exp. Math., 13 (1994).  doi: 10.1515/9783110848915.525.  Google Scholar

[20]

I. Pankratova and A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 935.  doi: 10.3934/dcdsb.2009.11.935.  Google Scholar

[21]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data,, J. Differential Equations, 95 (1992), 281.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[22]

A. A. Samarskii, Some problems of the theory of differential equations,, Differentsial'nye Uravneniya, 16 (1980), 1925.   Google Scholar

[23]

A. A. Skovoroda, Magnetic Systems for Plasma Confinement,, Fizmatlit, (2009).   Google Scholar

[24]

A. L. Skubachevskii, Nonclassical boundary-value problems. I,, Sovrem. Mat. Fundam. Napravl., 26 (2007), 3.  doi: 10.1007/s10958-008-9218-9.  Google Scholar

[25]

A. L. Skubachevskii, Nonclassical boundary-value problems. II,, Sovrem. Mat. Fundam. Napravl., 33 (2009), 3.  doi: 10.1007/s10958-010-9873-5.  Google Scholar

[26]

A. L. Skubachevskii, Vlasov-Poisson equations for a two-component plasma in a homogeneous magnetic field,, Uspekhi Mat. Nauk, 69 (2014), 107.   Google Scholar

[27]

A. P. Soldatov, The The Bitsadze-Samarskii problem for functions analytic in the sense of Douglis,, Differ. Uravn., 41 (2005), 396.  doi: 10.1007/s10625-005-0173-7.  Google Scholar

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Math. Library, 18 (1978).   Google Scholar

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