The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations with damping is proved for large data in three dimensional space. The model consists of the compressible Navier-Stokes equations with degenerate viscosity, and a nonlinear third-order differential operator, with the quantum Bohm potential, and the damping terms. The global weak solution...

The fractional reaction diffusion equation ∂tu+Au = g(u) is discussed, where A is a fractional differential operator on R of order α ∈ (0, 2), the C function g vanishes at ζ = 0 and ζ = 1 and either g ≥ 0 on (0, 1) or g < 0 near ζ = 0. In the case of non-negative g, it is shown that solutions with initial support on the positive half axis spread into the left half axis with unbounded speed if g...

In this paper we study the full system of incompressible liquid crystals, as modelled in the Qtensor framework. Under certain conditions we prove the global existence of weak solutions in dimension two or three and the existence of global regular solutions in dimension two. We also prove the weak-strong uniqueness of the solutions, for suficiently regular initial data.

In this article we obtain a Pohozaev-type inequality for Sobolev spaces with variable exponents. This inequality is used for proving the nonexistence of nontrivial weak solutions for the Dirichlet problem −∆p(x)u = |u|q(x)−2u, x ∈ Ω u(x) = 0, x ∈ ∂Ω, with non-standard growth. Our results extend those obtained by Ôtani [16].

We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we are able to prove the existence for all...

In this paper we prove that an operator which projects weak solutions of the two-or three-dimensional Navier-Stokes equations onto a nite-dimensional space is determining if it annihilates the diierence of two \nearby" weak solutions asymptotically, and if it satisses a single appoximation inequality. We then apply this result to show that the long-time behavior of weak solutions to the Navier-...

In this paper we prove that an operator which projects weak solutions of the twoor three-dimensional Navier-Stokes equations onto a finitedimensional space is determining if it annihilates the difference of two “nearby” weak solutions asymptotically, and if it satisfies a single appoximation inequality. We then apply this result to show that the long-time behavior of weak solutions to the Navie...