scholarly journals Boundary Estimates for a Degenerate Parabolic Equation with Partial Dirichlet Boundary Conditions

2017 ◽  
Vol 30 (3) ◽  
pp. 2377-2421
Author(s):  
Charles L. Epstein ◽  
Camelia A. Pop
Keyword(s):  
Parabolic Equation ◽  
Boundary Conditions ◽  
Degenerate Parabolic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Degenerate Parabolic ◽  
Boundary Estimates

scholarly journals Global in Time Results for a Parabolic Equation Solution in Non-rectangular Domains

2020 ◽  
pp. 257-274 ◽  
Author(s):  
Louanas Bouzidi ◽  
Arezki Kheloufi
Keyword(s):  
Parabolic Equation ◽  
Sobolev Space ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Dirichlet Boundary ◽  
Global Regularity ◽  
Interpolation Inequality ◽  
Dirichlet Boundary Conditions ◽  
Energy Estimates ◽  
Equation Solution

This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f ∈ L2(D) there exists a unique solution w such that w, ∂tw, ∂jw ∈ L2(D), j = 1, 2. Notice that the case of bounded non-rectangular domains is studied in [9]. The proof is based on energy estimates after transforming the problem in a strip region combined with some interpolation inequality. This work complements the results obtained in [19] in the case of Cauchy-Dirichlet boundary conditions

Complete blow-up for quasilinear degenerate parabolic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 877-908 ◽  
Author(s):  
R. Suzuki
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Image Position

Non-negative post-blow-up solutions of the quasilinear degenerate parabolic equation in RN (or a bounded domain with Dirichlet boundary condition) are studied. Various sufficient conditions for complete blow-up of solutions are given.

scholarly journals A MATHEMATICAL MODEL FOR THE USE OF ENERGY RESOURCES: A SINGULAR PARABOLIC EQUATION

2020 ◽  
Vol 25 (1) ◽  
pp. 88-109
Author(s):  
Daniel López-García ◽  
Rosa Pardo
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Initial Data ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Fixed Time ◽  
Dirichlet Boundary Conditions ◽  
Boundary Layer Flows ◽  
Singular Parabolic Equation

We consider a singular parabolic equation tβut − ∆u = f, for (x,t)∈ Ω × (0,T), arising in symmetric boundary layer flows. Here Ω ⊂ RN is a bounded domain with C2 boundary ∂Ω,β ≤ 1,f = f(t,x) is bounded, and T > 0 is some fixed time. We establish sufficient conditions for the existence and uniqueness of a weak solution of this singular parabolic equation with Dirichlet boundary conditions, and we investigate its regularity. There are two different cases depending on β. If β < 1, for any initial data u0 ϵ L2(Ω), there exists a unique weak solution, which in fact is a strong solution. The singularity is removable when β < 1. While if β = 1, there exists a unique solution of the singular parabolic problem tut − ∆u = f. The initial data cannot be arbitrarily chosen. In fact, if f is continuous and f(t) → f0, as t → 0, then, this solution converges, as t → 0, to the solution of the elliptic problem −∆u = f0, for x ∈ Ω, with Dirichlet boundary conditions. Hence, no initial data can be prescribed when β = 1, and the singularity in that case is strong.

Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝN+1

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Arezki Kheloufi
Keyword(s):  
Parabolic Equation ◽  
Boundary Conditions ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Maximal Regularity ◽  
Dirichlet Boundary Conditions ◽  
Regular Domain

Abstract.New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

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