New regularity results for the heat equation in triangular domains

International audience We study the free boundary problem in a nonstandard setting of infinitesimal discretisations of the heat equation. In particular we derive regularity results of solutions and the free boundary, in terms of S-continuity and S-differentiability Nous étudions le problème de la frontière libre dans le contexte non-standard de discrétisations infinitésimales de l'équation de la chaleur. En particulier nous prouvons des résultats de régularité des solutions et de la frontière libre, en termes de S-continuité et de S-dérivabilité.

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Some regularity results on the ‘relativistic’ heat equation

Journal of Differential Equations ◽

10.1016/j.jde.2008.06.024 ◽

2008 ◽

Vol 245 (12) ◽

pp. 3639-3663 ◽

Cited By ~ 18

Author(s):

F. Andreu ◽

V. Caselles ◽

J.M. Mazón

Keyword(s):

Heat Equation ◽

Regularity Results

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Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

Journal of Siberian Federal University Mathematics & Physics ◽

10.17516/1997-1397-2019-12-3-355-370 ◽

2019 ◽

pp. 355-370

Author(s):

Tahir Boudjeriou ◽

Arezki Kheloufi

Keyword(s):

Sobolev Space ◽

Boundary Conditions ◽

Heat Equation ◽

Decomposition Method ◽

Global Regularity ◽

Domain Decomposition Method ◽

Time Varying ◽

Neumann Type ◽

Type Boundary ◽

Regularity Results

This article deals with the heat equation @tu @2x u = f in D; D = {(t; x) 2 R2 : a < t < b; (t) < x < +1} with the function satisfying some conditions and the problem is supplemented with boundary conditions of Robin-Neumann type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f 2 L2(D) there exists a unique solution u such that u; @tu; @jx u 2 L2 (D) ; j = 1; 2: The proof is based on the domain decomposition method. This work complements the results obtained in [10].

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Sample paths of white noise in spaces with dominating mixed smoothness

Banach Journal of Mathematical Analysis ◽

10.1007/s43037-021-00136-8 ◽

2021 ◽

Vol 15 (3) ◽

Author(s):

Felix Hummel

Keyword(s):

Heat Equation ◽

White Noise ◽

Space Dimension ◽

Regularity Of Solutions ◽

Time Direction ◽

Sample Paths ◽

Regularity Results ◽

Boundary Noise ◽

Dominating Mixed Smoothness ◽

Mixed Smoothness

AbstractThe sample paths of white noise are proved to be elements of certain Besov spaces with dominating mixed smoothness. Unlike in isotropic spaces, here the regularity does not get worse with increasing space dimension. Consequently, white noise is actually much smoother than the known sharp regularity results in isotropic spaces suggest. An application of our techniques yields new results for the regularity of solutions of Poisson and heat equation on the half space with boundary noise. The main novelty is the flexible treatment of the interplay between the singularity at the boundary and the smoothness in tangential, normal and time direction.

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