The goal of this paper is to work out a thermodynamical setting for nonisothermal phase-field models with conserved and nonconserved order parameters in thermoelastic materials. Our approach consists in exploiting the second law of thermodynamics in the form of the entropy principle according to I. Müller and I. S. Liu, which leads to the evaluation of the entropy inequality with multipliers. As the main result we obtain a general scheme of phase-field models which involves an...

We study a two-particle quantum system given by a test particle interacting in three dimensions with a harmonic oscillator through a zero-range potential. We give a rigorous meaning to the Schrödinger operator associated with the system by applying the theory of quadratic forms and defining suitable families of self-adjoint operators. Finally we fully characterize the spectral properties of such operators.

A uniqueness criterion is given for the weak solution of the Navier-Stokes equations in the stationary case. Precisely, it is proved that, for a generic known term, there exists one and only one solution such that the mechanical power of the corresponding flow is maximum and that this maximum is "stable" in an appropriate sense.

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho }_i$ of the fluids and their velocity fields $u^{\left(i\right)}$ are prescribed at infinity: ${\rho }_i{|}_{\infty }={\rho }_{i\infty }>0$, $u^{\left(i\right)}{|}_{\infty }=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho }_i\equiv {\rho }_{i\infty }$, $u^{\left(i\right)}\equiv 0$, $i=1,2$.

It is proved that there can exist at most one solution of the homogeneous Dirichlet problem for the stationary Navier-Stokes equations in 3-dimensional space which is approximable by a given consistent and regular approximation scheme.

In questa Nota si fornisce un teorema di unicità per soluzioni regolari delle equazioni di Navier-Stokes in domini esterni. Tale teorema non richiede che le velocità tendano ad un prefissato limite all'infinito, mentre il gradiente di pressione è supposto essere di $q$-ma potenza sommabile nel cilindro spazio-temporale $(q\in (1,\mathrm{\infty }))$. Questo risultato non può essere ulteriormente generalizzato al caso $q=\mathrm{\infty }$, a causa di noti controesempi.

This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $-\Delta u\left(x\right)+\left(\omega +\frac{h^2\left(\right|x\left|\right)}{{\left|x\right|}^2}+{\int }_{\left|x\right|}^{+\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u\left(x\right)={\left|u\left(x\right)\right|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_0^{r}s{u}^2\left(s\right)ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit. From...

Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation div$\left(A_1{\Delta }_u\right)+A_{2}u=0$, where $A_1$ may have singularities in the domaind $U$ of definition. We study the case when $U$ is a half-plane and $u$ possesses high Fourier components, analyzing the changes brought about by the singularity $A_1=\infty$. We show that absorptions of energy takes...

This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.