In this paper we study the acoustic properties of a microstructured material such as glass, wool, or foam. In our model, the solid matrix is governed by linear elasticity and the surrounding fluid obeys the Stokes equations. The microstructure is assumed to be periodic at some small scale ${\varepsilon}$ and the viscosity coefficient of the fluid is assumed to be of order ${\varepsilon}^2$. We consider the time-harmonic regime forced by vibrations applied on a part of the boundary. We use the two-scale convergence theory to prove the convergence of the displacements to the solution of a homogeneous problem as the size of the microstructure shrinks to 0.

We consider an area-minimizing scheme for anisotropic mean-curvature flow originally due to Chambolle (2004). We show the convergence of the scheme to anisotropic mean-curvature flow in the sense of Hausdorff distance by the level-set method provided that no fattening occurs.

This paper is dedicated to the study of both viscous compressible barotropic fluids and Navier-Stokes equations with dependent density, when the viscosity coefficients are variable, in dimension $d\geq2$. We aim at proving the local and global well-posedness for respectively large and {small initial data having critical Besov regularity, and more precisely we are interested in extending the class of initial-data velocity when we consider the shallow-water system, improving the results in [17] and [15]. Our result relies on the fact that the velocity $u$ can be written as the sum of the solution $u_{L}$ of the associated linear system and a remainder velocity term $\bar{u}$; then in the specific case of the shallow-water system the remainder term $\bar{u}$ is more regular than $u_{L}$ by taking into account the regularizing effects induced on the bilinear convection term. In particular we are able to deal with initial velocity in $\dot{H}^{\N-1}$ as Fujita and Kato for the incompressible Navier-Stokes equations (see [12]) with an additional condition of type $u_{0}\in B^{-1}_{\infty,1}$. We would like to point out that this type of result is of particular interest when we want to deal with the problem of the convergence of the solution of compressible system to the incompressible system when the Mach number goes to $0$.

We study $\Gamma$-convergence of graph-based Ginzburg--Landau functionals, both the limit for zero diffusive interface parameter $\varepsilon \to 0$ and the limit for infinite nodes in the graph $m \to \infty$. For general graphs we prove that in the limit $\varepsilon \to 0$ the graph cut objective function is recovered. We show that the continuum limit of this objective function on 4-regular graphs is related to the total variation seminorm and compare it with the limit of the discretized Ginzburg--Landau functional. For both functionals we also study the simultaneous limit $\varepsilon \to 0$ and $m \to \infty$, by expressing $\varepsilon$ as a power of $m$ and taking $m \to \infty$. Finally we investigate the continuum limit for a nonlocal means-type functional on a completely connected graph.

We study the homogeneous Dirichlet problem for the anisotropic parabolic equation with double degeneracy \begin{align*} \frac{d}{dt} (|v|^{m(x,t)} \operatorname{sign}\,v ) = \sum_{i=1}^{n}D_{i} ( a_{i}(x,t)|D_{i} v|^{p_{i}(x,t)-2}D_{i} v ) +b(x,t)|v|^{\sigma(x,t)-2}v+g(x,t). \end{align*} The exponents of nonlinearity $m(x,t)>0$, $p_{i}(x,t)>1$, and $\sigma(x,t)>1$ are given bounded continuous functions. It is proved that the problem has a bounded solution in a variable-exponent Sobolev space. The main existence result is local in time and is established under minimal restrictions on the low-order terms. It is shown that under further restrictions on $b$ and $\sigma(x,t)$ the constructed solution can be continued to the arbitrary time interval. The energy estimates are derived.