An inverse problem for Kelvin–Voigt equations perturbed by isotropic diffusion and damping

Abstract

In this paper, we consider an inverse problem of finding a coefficient of right hand side of the following system of Kelvin–Voigt equations perturbed by an isotropic diffusion and damping terms vt+∇π−ϰΔvt−νdiv|D(v)|p−2D(v)=γ|v|m−2v+f(t)g(x,t), divv(x,t)=0. The damping term γ|v|m − 2v in the momentum equation realizes an absorbtion (sink) if γ ≤ 0, and a source if γ > 0. We show how the exponents p, m, the coefficients ν, ϰ, γ, the dimension of the space d, and data of the problem should interact each other for the existence of weak solutions to the problem. We also establish the conditions for uniqueness of the solutions to this problem.

An inverse problem for generalized Kelvin–Voigt equation with p-Laplacian and damping term

“…Regarding the convergence of the terms X 5 n and X 6 n , we first note that (see, for example [6,25]) Thus, we can use integration by parts to write X 6 n in the form…”

Inverse problems for a Boussinesq system for incompressible viscoelastic fluids

“…But, there are many works on inverse source problems of hydrodynamics, in particulary for Navier-Stokes system, we refer to the literature [28,7,9,11], and the references therein. To our best knowledge, an inverse source problem for heat convection for Kelvin-Voigt system has not been studied; however, there are several inverse problems for Kelvin-Voigt equations, which one can find in the literature [5,12,15,16,21].…”

Inverse Problems for Nonlinear Navier-Stokes-Voigt System With Memory

“…However, all above approaches are applicable only in the case when the corresponding direct problems are unique solvable and their solutions have additional extra smoothness. But, inverse problems for non-Newtonian hydrodynamics are not sufficiently studied from a mathematical point of view, see for instance [2,5,6,15,25,26], and in although they have important applications in physical point of view. Thus, in this paper, we study the unique solvability of some inverse source problems for a system of integro-differential Navier-Stokes-Voigt system (or so called Kelvin-Voigt system) describing the motion of incompressible non-Newtonian fluids, which taken into account viscoelastic properties.…”