Null Controllability of Three-dimensional Heat Equation in Singular Domains

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

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Null controllability of a nonlinear heat equation

Abstract and Applied Analysis ◽

10.1155/s1085337502002865 ◽

2002 ◽

Vol 7 (7) ◽

pp. 375-383 ◽

Cited By ~ 2

Author(s):

G. Aniculăesei ◽

S. Aniţa

Keyword(s):

Heat Equation ◽

Dirichlet Boundary ◽

Null Controllability ◽

Carleman Estimates ◽

Nonlinear Heat Equation ◽

Homogeneous Dirichlet Boundary Condition ◽

Linearized System ◽

Kakutani Fixed Point ◽

Exact Null Controllability ◽

Kakutani Fixed Point Theorem

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

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The Yang-Mills heat equation with finite action in three dimensions

Memoirs of the American Mathematical Society ◽

10.1090/memo/1349 ◽

2022 ◽

Vol 275 (1349) ◽

Author(s):

Leonard Gross

Keyword(s):

Heat Equation ◽

Three Dimensional ◽

Strong Solutions ◽

Three Dimensions ◽

Dimensional Manifold ◽

Uniqueness Of Solutions ◽

Reconstruction Procedure ◽

Content Type ◽

Yang Mills ◽

Gauge Functions

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 \mathbb {R}^3 and over a bounded open convex set in R 3 \mathbb {R}^3 . The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

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Null controllability of the heat equation in pseudo-cylinders by an internal control

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020048 ◽

2020 ◽

Vol 26 ◽

pp. 122

Author(s):

Jon Asier Bárcena-Petisco

Keyword(s):

Heat Equation ◽

Internal Control ◽

Cylindrical Part ◽

Null Controllability ◽

Carleman Estimate ◽

Main Difficulty ◽

Absorption Properties ◽

Weighted Function ◽

Cylindrical Structure ◽

Linear Heat

In this paper we prove the null controllability of the heat equation in domains with a cylindrical part and limited by a Lipschitz graph. The proof consists mainly on getting a Carleman estimate which presents the usual absorption properties. The main difficulty we face is the loss of existence of the usual weighted function in C2 smooth domains. In order to deal with this, we use its cylindrical structure and approximate the system by the same system stated in regular domains. Finally, we show some applications like the controllability of the semi-linear heat equation in those domains.

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