One-phase parabolic free boundary problem in a convex ring

Poghosyan, Michael; Teymurazyan, Rafayel

doi:10.3103/s1068362309030054

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“…Problem 3 was studied by Petrosyan in [19] and by Pogosyan and Teymurazyan in [20]. They established the existence of a classical solution of Problem 3 in small times.…”

Section: Problemmentioning

confidence: 99%

Hölder space solutions of free boundary problems that arise in combustion theory

Bizhanova

2016

St. Petersburg Math. J.

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Multidimensional single-phase problems with free boundary are studies for the heat equation with derivative in the direction of the gradient of the unknown function in differential equations on the free boundary. The unique solvability of such problems is established in Hölder spaces for small times, and coercive estimates are obtained for the solutions. §1. Introduction. Setting of the problem. Main resultsThis paper is devoted to solution of multidimensional one-phase free boundary problems. In such problems, one needs to find the boundary of a domain and the solution of a differential equation that satisfies certain initial and boundary conditions and is defined in this, unknown, domain.In his investigations devoted to the process of crystallization of a liquid (see [1]), J. Stefan stated a free boundary problem with with unknown temperature of the liquid and the solid matter, and also with unknown boundary separating them. In that problem, the free boundary γ(t) is determined by an additional differential equation involving the time-derivative of the function that describes γ(t), i.e., the free boundary is given in an explicit form.In Florin's problem (a "degenerate" Stefan-type problem), which models filtration processes for liquid or gas in a porous medium under unknown pressure [2], the free boundary is given implicitly in the form of the composition of γ(t) and an unknown function that solves a parabolic equation.Problems with free boundary are mathematical models for many physical processes, such as, e.g., melting or crystallization [3], melting with "supercooling", when the temperature of liquid near the free boundary becomes less than that of crystallization [4,5], changing the concentration of admixture in a substance in the process of its melting, which affects substantially the melting process [6], combustion [7], filtration of liquids and gazes in porous media [8,2,9]. Therefore, these mathematical models have important applications in mechanics, metallurgy [10], geology, geothermy [11, 12, 13], perfrostology, melioration, in the study of processes of extraction and transformation of petrol [14], in biology [15], in medicine [16].It should also be noted that the problems with free boundary are nonlinear, which can easily be traced after transforming the unknown domains into given ones. Moreover,

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“…Problem 3 was studied by Petrosyan in [19] and by Pogosyan and Teymurazyan in [20]. They established the existence of a classical solution of Problem 3 in small times.…”

Section: Problemmentioning

confidence: 99%

Hölder space solutions of free boundary problems that arise in combustion theory

Bizhanova

2016

St. Petersburg Math. J.

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show abstract

One-phase parabolic free boundary problem in a convex ring

Abstract: In this paper we study a free boundary problem for the heat equation in a convex ring. Here we prove that under some conditions on initial data, the considered problem has unique solution.

Cited by 1 publication

References 5 publications

Hölder space solutions of free boundary problems that arise in combustion theory

Hölder space solutions of free boundary problems that arise in combustion theory

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