\(L^q\)-solvability for an equation of viscoelasticity in power type material
type: Item
instance of: scholarly article
title: \(L^q\)-solvability for an equation of viscoelasticity in power type material
zbMATH Open document ID: 1472.45002
author: Bruno de Andrade
author: Clessius Silva
author: Arlúcio Viana
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2021-05-19T00:00:00Z
review text: The authors study the existence, uniqueness, regularity, continuous dependence, unique continuation, a blow-up alternative for mild solutions, and global well-posedness of the nonlinear Volterra equation \[ u_t = \int_0^t dg_\alpha(s) \Delta u(t-s,x)- \nabla p +h - (u\cdot \nabla u),\quad \textrm{div}(u)=0, \] in \((0,\infty)\times \Omega\), where \(u=0\) on \((0,\infty)\times \partial \Omega\) and \(u(0,x)=u_0(x)\) in \(\Omega\). Here the kernel is taken to be \(g_\alpha (t)= t^{\alpha}/\Gamma(\alpha+1)\) with \(0\leq \alpha <1\) and a mild solution is a solution to the equation \[ u(t)= S_\alpha (tA)u_0 + \int_0^t S_\alpha((t-s)A)(F(u)(s)+Ph(s))\, ds, \] where \(P\) is the Leray projection on divergence free functions, \(F(u)= P(u\cdot \nabla)u\), \(A=P\Delta\) and \[S_\alpha(tA)= \frac 1{2\pi i} \int_{Ha}e^{\lambda t}\lambda^\alpha(\lambda^{\alpha+1}I+A)^{-1}\, d\lambda,\] where \(t>0\) and \(Ha\) is a suitable path. The existence results show that the mild solutions have more spatial regularity in terms of estimates on norms in fractional power spaces when \(\alpha\) is closer to \(0\), the case of the Navier-Stokes equations. The linear estimates needed are stated in an abstract setting for sectorial operators which makes it possible to restate the results for some other equations as well.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76A10
Mathematics Subject Classification ID: 76A05
Mathematics Subject Classification ID: 76D03
zbMATH DE Number: 7349777
zbMATH Keywords: PDE
zbMATH Keywords: viscoelasticity
zbMATH Keywords: existence of solutions
zbMATH Keywords: regularity of solutions
zbMATH Keywords: power type materials
zbMATH Keywords: nonlinear Volterra equations
MaRDI profile type: Publication
full work available at URL: s00033-020-01443-0
OpenAlex ID: W3118763399
cites work: Vector-Valued Tauberian Theorems and Asymptotic Behavior of Linear Volterra Equations
cites work: Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations
cites work: Navier-Stokes equation with hereditary viscosity
cites work: Global smooth solutions for a quasilinear fractional evolution equation
cites work: On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations
cites work: Fractional Navier-Stokes equations
cites work: Self-similarity, symmetries and asymptotic behavior in Morrey spaces for a fractional wave equation.
cites work: Global existence in critical spaces for compressible Navier-Stokes equations
cites work: Navier-Stokes equation and its fractional approximations
cites work: On the well‐posedness of a Volterra equation with applications in the Navier‐Stokes problem
cites work: Abstract Volterra integrodifferential equations with applications to parabolic models with memory
cites work: Global smooth solutions for a class of parabolic integrodifferential equations
cites work: Q3998295
cites work: Global Solutions to the Isentropic Compressible Navier--Stokes Equations with a Class of Large Initial Data
cites work: On the Navier-Stokes initial value problem. I
cites work: Analyticity of the semigroup generated by the Stokes operator in \(L_r\) spaces
cites work: Solutions in \(L_ r\) of the Navier-Stokes initial value problem
cites work: An introduction to continuum mechanics
cites work: Geometric theory of semilinear parabolic equations
cites work: Pullback asymptotic behavior of solutions for a 2D non-autonomous non-Newtonian fluid
cites work: Lectures on viscoelasticity theory
cites work: THE VISCOELASTIC WAVE PRODUCED BY RETRACTION OF A STRETCHED REGION
cites work: SHOCK FORMATION IN NEARLY ELASTIC MATERIALS
cites work: Q4276447
cites work: A hereditary partial differential equation with applications in the theory of simple fluids
cites work: Resolvent estimates in \(L ^{p }\) for the Stokes operator in Lipschitz domains
cites work: Existence of global strong solution to the micropolar fluid system in a bounded domain
cites work: Q4629102
DOI: 10.1007/S00033-020-01443-0
description: scientific article; zbMATH DE number 7349777