This paper deals with the evolution Fokker-Planck-Smoluchowski configurational probability diffusion equation for the FENE dumbbell model in dilute polymer solutions. We prove the exponential convergence in time of the solution of this equation to a corresponding steady-state solution, for arbitrary velocity gradients.

We study the one-dimensional motion of the viscous gas represented by the system $v_t-u_x=0$, $u_t+p{\left(v\right)}_x=\mu {(u_x/v)}_x+f\left({\int }_0^{x}v\ddot{x},t\right)$, with the initial and the boundary conditions $(v(x,0),u(x,0))=(v_0\left(x\right),u_0\left(x\right))$, $u(0,t)=u(X,t)=0$. We are concerned with the external forces, namely the function $f$, which do not become small for large time $t$. The main purpose is to show how the solution to this problem behaves around the stationary one, and the proof is based on an elementary $L^2$-energy method.

The asymptotic behaviour of solutions of a class of free-boundary problems arising in vortex theory is discussed.

This article studies the asymptotic behavior of the number $N\left(\lambda \right)$ of the negative eigenvalues $\<-\lambda$ as $\lambda \to +0$ of the two dimensional Pauli operators with electric potential $V\left(x\right)$ decaying at $\infty$ and with nonconstant magnetic field $b\left(x\right)$, which is assumed to be bounded or to decay at $\infty$. In particular, it is shown that $N\left(\lambda \right)=(1/2\pi ){\int }_{V\left(x\right)\>\lambda }b\left(x\right)dx(1+o\left(1\right))$, when $V\left(x\right)$ decays faster than $b\left(x\right)$ under some additional conditions.

We consider the double-diffusive convection phenomenon and analyze the governing equations. A system of partial differential equations describing the convective flow arising when a layer of fluid with a dissolved solute is heated from below is considered. The problem is placed in a functional analytic setting in order to prove a theorem on existence, uniqueness and continuous dependence on initial data of weak solutions in the class $([0,\infty );H)\cap L{²}_{loc}({ℝ}^{+};V)$. This theorem enables us to show that the infinite-dimensional...

We prove a priori estimates for a linear system of partial differential equations originating from the equations for the flow of a barotropic compressible viscous fluid under the influence of the gravity it generates. These estimates will be used in a forthcoming paper to prove the nonlinear stability of the motionless, spherically symmetric equilibrium states of barotropic, self-gravitating viscous fluids with respect to perturbations of zero total angular momentum. These equilibrium states as...