On the complement of the unit disk $B$ we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field $u$ is equal to zero provided ${u|}_{\partial B}=0$ and ${lim}_{\left|x\right|\to \infty }{\left|x\right|}^{1/3}\left|u\left(x\right)\right|=0$ uniformly. For slow flows the latter condition can be replaced by ${lim}_{\left|x\right|\to \infty }\left|u\left(x\right)\right|=0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.

We prove existence and a representation formula for solutions to the equations describing steady flows of an isothermal, viscous, compressible gas having a positive infimum for the density $\varrho$, moving in an exterior domain, when the speed of the obstacle and the external forces are sufficiently small.

We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like ${\mu \left(D\right)\sim \left|D\right|}^{p-2}$ (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.

This paper deals with the global well-posedness of the $3$D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+\frac{3}{p}}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity .

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2\left(ℝ\right)$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ${∥f∥}_1\le 1/5$. Previous results of this...

We consider the motion of a viscous compressible heat conducting fluid in ℝ³ bounded by a free surface which is under constant exterior pressure. Assuming that the initial velocity is sufficiently small, the initial density and the initial temperature are close to constants, the external force, the heat sources and the heat flow vanish, we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

We study the $N$-dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.

We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $\gamma$-law pressure, $\gamma \ge 1$, admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2\times 2$$p$-system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann...

In this paper we prove the global well-posedness of the two-dimensional Boussinesq system with zero viscosity for rough initial data.

We analyse the effect of the mechanical response of the solid phase during liquid/solid phase change by numerical simulation of a benchmark test based on the well-known and debated experiment of melting of a pure gallium slab counducted by Gau & Viskanta in 1986. The adopted mathematical model includes the description of the melt flow and of the solid phase deformations. Surprisingly the conclusion reached is that, even in this case of pure material, the contribution of the solid phase to the...

Linear Force-free (or Beltrami) fields are three-components divergence-free fields solutions of the equation curlB = αB, where α is a real number. Such fields appear in many branches of physics like astrophysics, fluid mechanics, electromagnetics and plasma physics. In this paper, we deal with some related boundary value problems in multiply-connected bounded domains, in half-cylindrical domains and in exterior domains.

We study a linear system of equations arising from fluid motion around a moving rigid body, where rotation is included. Originally, the coordinate system is attached to the fluid, which means that the domain is changing with respect to time. To get a problem in the fixed domain, the problem is rewritten in the coordinate system attached to the body. The aim of the present paper is the proof of the existence of a strong solution in a weighted Lebesgue space. In particular, we prove the existence...

We consider a model for the viscoelastic fluid which has recently been studied in [4] and [1]. We show the local-in-time existence of a strong solution to the corresponding system of partial differential equations under less regularity assumptions on the initial data than in the above mentioned papers. The main difference in our approach is the use of the $L^p$ theory for the Stokes system.

Studiamo l'evoluzione temporale di un fluido bidimensionale incomprimibile non viscoso quando la vorticità iniziale è concentrata in $N$ regioni di diametro $ϵ$ e mostriamo che la vorticità evoluta temporalmente è anche lei concentrata in $N$ piccole regioni di diametro $d$, $d\le \text{const}{ϵ}^{\alpha }$ per qualunque $\alpha <1/3$. Noi chiamiamo questa proprietà "localizzazione". Come conseguenza abbiamo una connessione rigorosa tra il modello dei vortici puntiformi e l'Equazione di Eulero.