ABSTRACT

Some new estimations of scalar products of vector fields in unbounded domains are investigated. L_{p}-estimations for the vector fields were proved in special weighted functional spaces. The paper generalizes our earlier results for bounded domains. Estimations for scalar products make it possible to investigate wide classes of mathematical physics problems in physically inhomogeneous domains. Such estimations allow studying issues of correctness for problems with non-smooth coefficients. The paper analyses solvability of stationary set of Maxwell equations in inhomogeneous unbounded domains based on the proved L_{p}-estimations.

Some new estimations of scalar products of vector fields in unbounded domains are investigated. L

Cite this paper

A. Kalinin, A. Tyukhtina and A. Zhidkov, "*L*_{p}-Estimations of Vector Fields in Unbounded Domains," *Applied Mathematics*, Vol. 3 No. 1, 2012, pp. 45-51. doi: 10.4236/am.2012.31008.

A. Kalinin, A. Tyukhtina and A. Zhidkov, "

References

[1] E. Byhovskii and N. Smirnov, “Orthogonal Decomposition of the Space of Vector Functions square-Summable on a Given Domain, and the Operators of Vector Analysis,” (Russian) Trudy Matematicheskogo Instituta Steklov, Vol. 59, 1960, pp. 5-36.

[2] G. Duvaut and J. Lions, “Inequalities on Mechanics and Physics,” Springer-Verlag, Berlin, 1976. doi:10.1007/978-3-642-66165-5

[3] R. Temam, “Navier-Stokes Equations: Theory and Numerical Analysis,” North-Holland Publishing Company, Amsterdam, 1977.

[4] H. Weil, “The Method of Orthogonal Projection in Potential Theory,” Duke Mathematics Journal, Vol. 7, No. 1, 1940, pp. 411-444. doi:10.1215/S0012-7094-40-00725-6

[5] V. Girault and P.-A. Raviart, “Finite Element Approximation of the Navier-Stokes Equations,” Springer-Verlag, New-York, 1979.

[6] V. Maslennikova, “Lp Estimates, and the Asymptotics at t→∞ of the Solution of the Cauchy Problem for a Sobolev System,” (Russian) Trudy Matematicheskogo Instituta Steklov, Vol. 103, 1968, pp. 117-141.

[7] A. Kalinin, “Some Estimations in the Theory of Vector Fields,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, Nizhny Novgorod, No. 20, 1997, pp. 32-38.

[8] A. Kalinin and A. Kalinkina, “Estimates of Vector Fields and Stationary Set of Maxwell Equations,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, No. 1, 2002, pp. 95-107.

[9] A. Kalinin and A. Kalinkina, “Lp-Estimates for Vector Fields,” Russian Mathematics (Izvestiya Uchebnykh Zavedenii Matematika), Vol. 48, No. 3, 2004, pp. 23-31.

[10] A. Kalinin, S. Morozov, “Stationary Problems for the Set of Maxwell Equations in Heterogeneous Areas,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, No. 20, 1997, pp. 24-31.

[11] A. Kalinin, “Estimations of Scalar Products for Vector Fields and Their Application in Some Problems of Mathematical Physics,” (Russian) Izvestiya of Institution of Mathematics and Infomatics UdSU, Vol. 3, No. 37, 2006, pp. 55-56.

[12] A. Zhidkov, “Estimates of the Scalar Products of Vector Fields in Unbounded Regions,” (Russian) Vestnik UNN, Nizhny Novgorod, No. 1, 2007, pp. 162-166.

[13] P. Lax and A. Milgram, “Parabolic Equations,” Annals of Mathematics Studies, Vol. 33, 1954, pp. 167-190.

[1] E. Byhovskii and N. Smirnov, “Orthogonal Decomposition of the Space of Vector Functions square-Summable on a Given Domain, and the Operators of Vector Analysis,” (Russian) Trudy Matematicheskogo Instituta Steklov, Vol. 59, 1960, pp. 5-36.

[2] G. Duvaut and J. Lions, “Inequalities on Mechanics and Physics,” Springer-Verlag, Berlin, 1976. doi:10.1007/978-3-642-66165-5

[3] R. Temam, “Navier-Stokes Equations: Theory and Numerical Analysis,” North-Holland Publishing Company, Amsterdam, 1977.

[4] H. Weil, “The Method of Orthogonal Projection in Potential Theory,” Duke Mathematics Journal, Vol. 7, No. 1, 1940, pp. 411-444. doi:10.1215/S0012-7094-40-00725-6

[5] V. Girault and P.-A. Raviart, “Finite Element Approximation of the Navier-Stokes Equations,” Springer-Verlag, New-York, 1979.

[6] V. Maslennikova, “Lp Estimates, and the Asymptotics at t→∞ of the Solution of the Cauchy Problem for a Sobolev System,” (Russian) Trudy Matematicheskogo Instituta Steklov, Vol. 103, 1968, pp. 117-141.

[7] A. Kalinin, “Some Estimations in the Theory of Vector Fields,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, Nizhny Novgorod, No. 20, 1997, pp. 32-38.

[8] A. Kalinin and A. Kalinkina, “Estimates of Vector Fields and Stationary Set of Maxwell Equations,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, No. 1, 2002, pp. 95-107.

[9] A. Kalinin and A. Kalinkina, “Lp-Estimates for Vector Fields,” Russian Mathematics (Izvestiya Uchebnykh Zavedenii Matematika), Vol. 48, No. 3, 2004, pp. 23-31.

[10] A. Kalinin, S. Morozov, “Stationary Problems for the Set of Maxwell Equations in Heterogeneous Areas,” (Russian) Vestnik UNN Series Mathematical Modeling and Optimal Control, No. 20, 1997, pp. 24-31.

[11] A. Kalinin, “Estimations of Scalar Products for Vector Fields and Their Application in Some Problems of Mathematical Physics,” (Russian) Izvestiya of Institution of Mathematics and Infomatics UdSU, Vol. 3, No. 37, 2006, pp. 55-56.

[12] A. Zhidkov, “Estimates of the Scalar Products of Vector Fields in Unbounded Regions,” (Russian) Vestnik UNN, Nizhny Novgorod, No. 1, 2007, pp. 162-166.

[13] P. Lax and A. Milgram, “Parabolic Equations,” Annals of Mathematics Studies, Vol. 33, 1954, pp. 167-190.