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Local Existence (local + existence)

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Mathematics and Statistics100%

Selected Abstracts

Local existence of solutions of a three phase-field model for solidification

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009
Bianca Morelli Calsavara Caretta
Abstract In this article we discuss the local existence and uniqueness of solutions of a system of parabolic differential partial equations modeling the process of solidification/melting of a certain kind of alloy. This model governs the evolution of the temperature field, as well as the evolution of three phase-field functions; the first two describe two different possible solid crystallization states and the last one describes the liquid state. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Local existence for the one-dimensional Vlasov,Poisson system with infinite mass

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2007
Stephen Pankavich
Abstract A collisionless plasma is modelled by the Vlasov,Poisson system in one dimension. We consider the situation in which mobile negative ions balance a fixed background of positive charge, which is independent of space and time, as ,x, , ,. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behaviour are shown to exist locally in time, and criteria for the continuation of these solutions are established. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Local existence for the free boundary problem for nonrelativistic and Relativistic compressible Euler equations with a vacuum boundary condition

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2009
Yuri Trakhinin
We study the free boundary problem for the equations of compressible Euler equations with a vacuum boundary condition. Our main goal is to recover in Eulerian coordinates the earlier well-posedness result obtained by Lindblad [11] for the isentropic Euler equations and extend it to the case of full gas dynamics. For technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain. We prove the local-in-time existence in Sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities [5, 12]. It contains, in particular, the reduction to a fixed domain, using the "good unknown" of Alinhac [1], and a suitable Nash-Moser-type iteration scheme. A certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic. This approach is still directly applicable to the relativistic version of our problem in the setting of special relativity, and we briefly discuss its extension to general relativity. © 2009 Wiley Periodicals, Inc. [source]

Local existence of solutions of a three phase-field model for solidification

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2009
Bianca Morelli Calsavara Caretta
Abstract In this article we discuss the local existence and uniqueness of solutions of a system of parabolic differential partial equations modeling the process of solidification/melting of a certain kind of alloy. This model governs the evolution of the temperature field, as well as the evolution of three phase-field functions; the first two describe two different possible solid crystallization states and the last one describes the liquid state. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains

MATHEMATISCHE NACHRICHTEN, Issue 12 2008
Annegret Glitzky
Abstract We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain ,0 of the domain of definition , of the energy balance equation and of the Poisson equation. Here ,0 corresponds to the region of semiconducting material, , \ ,0 represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a W1,p -regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

On the evolution of sharp fronts for the quasi-geostrophic equation

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2005
José Luis Rodrigo
We consider the problem of the evolution of sharp fronts for the surface quasi-geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two-dimensional Euler equation. The special interest of the quasi-geostrophic equation lies in its strong similarities with the three-dimensional Euler equation, while being a two-dimen-sional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the three-dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot-Savart law still needs to be understood. We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve. Finally, using a Nash-Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C, case. © 2004 Wiley Periodicals, Inc. [source]