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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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Volume 23, Issue 6

Issues

On a mixed problem for the parabolic Lamé type operator

Roman Puzyrev
  • Institute of Mathematics and Computer Science, Siberian Federal University, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia
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/ Alexander Shlapunov
  • Institute of Mathematics and Computer Science, Siberian Federal University, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia
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Published Online: 2014-12-04 | DOI: https://doi.org/10.1515/jiip-2014-0043

Abstract

We consider a boundary value problem for a Lamé type operator, which corresponds to a linearisation of the Navier–Stokes' equations for compressible flow of Newtonian fluids in the case where pressure is known. It consists of recovering a vector function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the integral representation's method we obtain a uniqueness theorem and solvability conditions for the problem. We also describe conditions, providing dense solvability of the problem.

Keywords: Boundary value problems for parabolic equations; ill-posed problems; integral representation's method

MSC: 35K40

About the article

Received: 2014-06-05

Accepted: 2014-07-24

Published Online: 2014-12-04

Published in Print: 2015-12-01


Funding Source: Russian Foundation for Basic Research

Award identifier / Grant number: 14-01-00544

Funding Source: Russian Federation Government

Award identifier / Grant number: 14.Y26.31.0006


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 23, Issue 6, Pages 555–570, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2014-0043.

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