Existence and uniqueness of very weak solution of the MHD type system

Abstract In this paper, we investigate the problem for optimal control of a viscous generalized \theta -type dispersive equation (VG \theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG \theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

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A piezoelectric contact problem with slip dependent friction and damage

Journal of Applied Analysis ◽

10.1515/jaa-2020-2034 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Abderrezak Kasri

Keyword(s):

Weak Solution ◽

Contact Problem ◽

Dry Friction ◽

Existence And Uniqueness ◽

Variational Formulation ◽

Normal Compliance ◽

Electrically Conductive ◽

Coulomb’S Law ◽

Electrical Condition ◽

Coulomb's Law

Abstract The aim of this paper is to study a quasistatic contact problem between an electro-elastic viscoplastic body with damage and an electrically conductive foundation. The contact is modelled with an electrical condition, normal compliance and the associated version of Coulomb’s law of dry friction in which slip dependent friction is included. We derive a variational formulation for the model and, under a smallness assumption, we prove the existence and uniqueness of a weak solution.

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EXISTENCE AND UNIQUENESS OF WEAK SOLUTION FOR A FLEXIBLE EULER-BERNOULLI BEAM WITH VARIABLE COEFFICIENTS

Far East Journal of Applied Mathematics ◽

10.17654/am110010065 ◽

2021 ◽

Vol 110 (1) ◽

pp. 65-80

Author(s):

Bomisso Gossrin Jean-Marc ◽

Touré Kidjégbo Augustin

Keyword(s):

Weak Solution ◽

Existence And Uniqueness ◽

Variable Coefficients ◽

Bernoulli Beam ◽

Euler Bernoulli Beam

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Existence and regularity of time-dependent pullback attractors for the non-autonomous nonclassical diffusion equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.65 ◽

2021 ◽

pp. 1-18

Author(s):

Yuming Qin ◽

Bin Yang

Keyword(s):

Weak Solution ◽

Existence And Uniqueness ◽

Nonlinear Term ◽

Time Dependent ◽

Diffusion Equations ◽

Nonlocal Diffusion ◽

Force Term ◽

Pullback Attractors ◽

Critical Exponential Growth ◽

Nonclassical Diffusion Equations

In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term $h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space $\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.

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On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

Mathematical Problems in Engineering ◽

10.1155/2018/3626543 ◽

2018 ◽

Vol 2018 ◽

pp. 1-18

Author(s):

Nguyen Huu Nhan ◽

Le Thi Phuong Ngoc ◽

Nguyen Thanh Long

Keyword(s):

Asymptotic Expansion ◽

Wave Equation ◽

Weak Solution ◽

Dirichlet Problem ◽

Small Parameter ◽

Nonlinear Wave ◽

Nonlinear Wave Equation ◽

Existence And Uniqueness ◽

Linearization Method ◽

Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

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ON THE ROBIN PROBLEM FOR STOKES AND NAVIER–STOKES SYSTEMS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202506001327 ◽

2006 ◽

Vol 16 (05) ◽

pp. 701-716 ◽

Cited By ~ 15

Author(s):

REMIGIO RUSSO ◽

ALFONSINA TARTAGLIONE

Keyword(s):

Boundary Layer ◽

Weak Solution ◽

Lipschitz Domain ◽

Stokes System ◽

Navier Stokes ◽

Robin Problem ◽

Layer Potentials ◽

Very Weak Solution ◽

Stokes Systems ◽

Compact Boundary

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

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