Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.1088

G-convergence and homogenization of viscoelastic flows

Alexander Panchenko
View original source ↗Read source document ↗Export metadata ↓

Abstract

The paper is devoted to homogenization of two-phase incompressible viscoelastic flows with disordered microstructure. We study two cases. In the first case, both phases are modeled as Kelvin-Voight viscoelastic materials. In the second case, one phase is a Kelvin-Voight material, and the other is a viscous Newtonian fluid. The microscale system contains the conservation of mass and balance of momentum equations. The inertial terms in the momentum equation incorporate the actual interface advected by the flow. In the constitutive equations, a frozen interface is employed. The interface geometry is arbitrary: we do not assume periodicity, statistical homogeneity or scale separation. The problem is homogenized using G-convergence and oscillating test functions. Since the microscale system is not parabolic, previously known constructions of the test functions do not work here. The test functions developed in the paper are non-local in time and satisfy divergence-free constraint exactly. The latter feature enables us to avoid working with pressure directly. We show that the effective medium is a single phase viscoelastic material that is not necessarily of Kelvin-Voight type. The effective constitutive equation contains a long memory viscoelastic term, as well as instantaneous elastic and viscous terms.

Explore related subjects

math.APmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alexander Panchenko. 2007-06-22. G-convergence and homogenization of viscoelastic flows. https://arxiv.org/abs/0706.1088

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimizers in Sobolev-curl inequalities

We study a Sobolev-type inequality involving the $p$-curl operator in $\mathbb{R}^3$. We prove the existence of a minimizer $u:\mathbb{R}^3\to \mathbb{R}^3$ which yields a solution to the $p$-curl-curl equation in the critical case $$ \nabla\times (|\nabla\times u|^{p-2}\nabla\times u)= |u|^{p^*-2}u\quad \hbox{in }\mathbb{R}^3.$$ The cases $p=2$ and $p=3/2$ are motivated, respectively, by nonlinear Maxwell equations and zero modes of three-dimensional Dirac operators. The infinite-dimensional kernel of curl and the critical exponent prevent a direct application of standard compactness arguments. Our proof combines local compactness under the nonlinear constraint $\operatorname{div}(|u|^{p^*-2}u)=0$ and a new variational approach that allows to treat quasilinear strongly indefinite problems by direct minimization on a Nehari-type constraint. We also establish existence and compactness results in axially symmetric classes, including two types of solutions for $p=3/2$ distinct from the explicit Loss--Yau fields. Finally, the same minimization strategy gives a new proof of compactness modulo translations and dilations for minimizing sequences in the classical critical Sobolev inequality.

math.AP↗
arXiv2026

A remark on the log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation

We consider the question of showing a log-Sobolev inequality for the Gibbs measure of the focusing Schrödinger equation built by Lebowitz-Rose-Speer (1988), formally given by $$ dρ\propto \exp\big(\frac 1 p\int_{\mathbb T} |u|^p d x - \frac 12\int_{\mathbb T} |\nabla u|^2 d x - \frac 12\int_{\mathbb T} |u|^2 d x\big) \mathbf 1_{\| u \|_{L^2(\mathbb T)}^2 \le K}dud\overline{u}. $$ When $2 \le p \le 4$, we show that these measures indeed satisfy a log-Sobolev inequality. When $p> 4$, we show a lower bound for the Hessian of the potential, which implies that the known techniques to show these inequalities cannot apply to the measure $ρ$.

math.AP↗
arXiv2026

On the inhomogeneous discounted Hamilton-Jacobi equations

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} λ(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $λ(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c c_0$. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of $λ$ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as $c$ tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for $c=c_0$.

math.AP↗