Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted Hölder spaces of functions

Bizhanova, G. I.

doi:10.1007/bf02399935

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…These inequalities are well known for the spaces C l 1 ,l 2 0 (see [20], [24]), and for the spaces C 5) where x N = max{x N , x N }, and here and throughout we, without loss of generality, assume that…”

Section: Auxiliary Results On the Spacesmentioning

confidence: 99%

Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

Degtyarev

2014

Nonlinear Differ. Equ. Appl.

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We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.Key words: free boundary, Stefan problem, classical solvability, porous medium equation, degenerate parabolic equations. To the memory of Professor B.V.BazaliyThe final publication is available at Springer via http://dx.doi.org/10.1007/s00030-014-0280-3 1 Statement of the problem and the main result Classical solvability of the Stefan problem for uniformly parabolic equations has been well studied -see for example papers [1] -[6] and the references therein. At the same time, as has long been known, the heat transfer model based on uniformly parabolic equations, has some properties which can not be observed in the reality, in particular, the infinite speed of propagation of disturbances. We also know that more accurate model of the heat transfer is the model which is based on degenerate parabolic equations, such as equations of the form where β(u) is a discontinuous function of the form 1

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Section: Auxiliary Results On the Spacesmentioning

confidence: 99%

Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

Degtyarev

2014

Nonlinear Differ. Equ. Appl.

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“…(b) We refer to [6,7,8,9,10,11,12,13,33,45,49,52,53] for different results and approaches for the linearization of the classical two-phase Stefan problem.…”

Section: The Transformed Problemmentioning

confidence: 99%

“…Classical solutions for the two-phase Stefan problem in Hölder spaces and weighted Hölder spaces (without loss of regularity) were obtained by Bazaliȋ, Bizhanova, Degtyarev, Solonnikov, and Radkevich, see [6,7,8,9,10,11,12,13,49] for more information.…”

Section: Introductionmentioning

confidence: 99%

Existence of analytic solutions for the classical Stefan problem

2007

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“…The problems in weighted Hölder spaces were studied by Bizhanova and Solonnikov (2000) and Bizhanova (1994) and others. Bizhanova and Solonnikov (2000) have investigated in this space, the problem with a time derivative in the boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Bizhanova and Solonnikov (2000) have investigated in this space, the problem with a time derivative in the boundary conditions. In Bizhanova (1994) was studied in the weighted Hölder space the solvability Stefan and Florin problems with the condition u 1 = u 2 = g(x, t) on the free boundary. In the present study the linear problem for the second order parabolic equations in the bounded domain is considered.…”

Section: Introductionmentioning

confidence: 99%

Investigation of the Solvability in a Weighted Hölder Spaces of the Linearized Problem

Sarsekeyev¹

2015

Asian J. of Scientific Research

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In the study, the linear problem for the second order general parabolic equations in the bounded domain is considered. This problem arises in the linearization of multidimensional two-phase problem with the condition on the free boundary in which the unknown temperature at the free surface depending on the curvature and the velocity of the free surface. The proof of the existence and uniqueness of solutions of the linear problem and derivation of estimate of its solution is realized with Schauder method and construction of regularizer. The unique solvability of the linear problem in weighted Hölder spaces is proved, the coercive estimates of the solutions is established.Key words: Second order parabolic equation, weighted hölder space, existence, uniqueness of the solutions, coercive estimate INTRODUCTIONBazaliy and Degtyarev (1992) investigated the three-dimensional Stefan problem for the heat equation with the condition u 1 = u 2 = αk-βV v on the free boundary, where k-the curvature of free boundary and V v -velocity of free boundary on the direction of the normal ν. Radkevich (1992) studied such a problem for a divergent parabolic equation with conormal derivative in the boundary condition for the space of dimensions n = 2, 3. In these works in the classical Hölder space were considered the problems in domains of dimension 2 and 3 at overstated smoothness of the given surface and the initial data.The weighted Hölder spaces , introduced by Belonosov and Zelenyak (1975), permits

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Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted Hölder spaces of functions

Abstract: Let index k = 1 correspond to the Stefan problem, and index k = 0 to the Florin problem.

Cited by 8 publications

References 11 publications

Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

Existence of analytic solutions for the classical Stefan problem

Investigation of the Solvability in a Weighted Hölder Spaces of the Linearized Problem

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