Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains

2012 ◽  
Vol 75 (14) ◽  
pp. 5561-5588 ◽  
Author(s):  
Mahamadi Warma
Keyword(s):  
Boundary Conditions ◽  
Parabolic Problems ◽  
Well Posedness ◽  
Wentzell Boundary Conditions ◽  
Nonsmooth Domains

The non-autonomous wave equation with general Wentzell boundary conditions

2005 ◽  
Vol 135 (2) ◽  
pp. 317-329 ◽  
Author(s):  
Angelo Favini ◽  
Ciprian G. Gal ◽  
Gisèle Ruiz Goldstein ◽  
Jerome A. Goldstein ◽  
Silvia Romanelli
Keyword(s):  
Hilbert Space ◽  
Wave Equation ◽  
Boundary Conditions ◽  
Abstract Cauchy Problem ◽  
Regularity Conditions ◽  
Well Posedness ◽  
One Dimensional ◽  
Wentzell Boundary Conditions ◽  
Image Position ◽  
Dimensional Wave

We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε > 0 in [0, 1] × [0, + ∞) and βj(t) > 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

scholarly journals Stability of parabolic problems with nonlinear Wentzell boundary conditions

2009 ◽  
Vol 246 (6) ◽  
pp. 2434-2447 ◽  
Author(s):  
Giuseppe M. Coclite ◽  
Gisèle R. Goldstein ◽  
Jerome A. Goldstein
Keyword(s):  
Boundary Conditions ◽  
Parabolic Problems ◽  
Wentzell Boundary Conditions

scholarly journals Stability estimates for parabolic problems with Wentzell boundary conditions

2008 ◽  
Vol 245 (9) ◽  
pp. 2595-2626 ◽  
Author(s):  
Giuseppe M. Coclite ◽  
Gisèle R. Goldstein ◽  
Jerome A. Goldstein
Keyword(s):  
Boundary Conditions ◽  
Parabolic Problems ◽  
Stability Estimates ◽  
Wentzell Boundary Conditions

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

scholarly journals Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

2021 ◽  
Author(s):  
Tim Binz
Keyword(s):  
Boundary Conditions ◽  
Elliptic Operator ◽  
Compact Manifold ◽  
Analytic Semigroup ◽  
Continuous Functions ◽  
Analytic Semigroups ◽  
Abstract Theory ◽  
Neumann Operator ◽  
Wentzell Boundary Conditions ◽  
Dirichlet To Neumann Operator

AbstractWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

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