The incompressible limit of the weak solutions to a fluid-particle interaction model is studied in this paper. By using the relative entropy method and refined energy analysis, we show that, for well-prepared initial data, the weak solutions of the compressible fluid-particle interaction model converge to the strong solution of the incompressible Navier-Stokes equations as long as the Mach number goes to zero. Furthermore, the desired convergence rates are also obtained.

We consider the existence of infinitely many solutions to the boundary value problem $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left({\frac{1}{2}}_0{D}_t^{-\beta }\left(u^{\text{'}}\left(t\right)\right)+{\frac{1}{2}}_t{D}_T^{-\beta }\left(u^{\text{'}}\left(t\right)\right)\right)+\nabla F(t,u\left(t\right))=0\text{a.e.}t\in [0,T],\\ u\left(0\right)=u\left(T\right)=0.\end{array}$$ Under more general assumptions on the nonlinearity, we obtain new criteria to guarantee that this boundary value problem has infinitely many solutions in the superquadratic, subquadratic and asymptotically quadratic cases by using the critical point theory.

We mainly study initial boundary value problems for the Degasperis-Procesi equation on the half line and on a compact interval. By the symmetry of the equation, we can convert these boundary value problems into Cauchy problems on the line and on the circle, respectively. Applying thus known results for the equation on the line and on the circle, we first obtain the local well-posedness of the initial boundary value problems. Then we present some blow-up and global existence results for strong solutions....

The present paper is devoted to study the behaviour at infinity of some solutions of non autonomous and non homogeneous second order linear systems in semi-infinite strips.

This paper develops various estimates for solutions of a nonlinear, fouth order PDE which corresponds to prescribing the scalar curvature of a toric Kähler metric. The results combine techniques from Riemannian geometry and from the theory of Monge-Ampère equations.

On définit des invariants entiers mesurant l’irrégularité d’un point singulier d’un système différentiel. Les propriétés de ces invariants, l’étude de la variation de l’ordre de la singularité par perturbation linéaire ainsi qu’une généralisation d’un théorème de W. Jurkat et D.A. Lutz permettent de donner une méthode de calcul de cet ordre.