Narrow Results

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in $H^k(k\ge 2)$ in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established.

We study the plasma-vacuum interface problem in relativistic magnetohydrodynamics for the case when the plasma density does not go to zero continuously, but jumps. In the vacuum region, we consider the Maxwell equations for electric and magnetic fields. We show that a sufficiently large vacuum electric field can make the planar interface violently unstable. By using a suitable secondary symmetrization of the vacuum Maxwell equations, we find a sufficient condition that precludes violent instabilities. Under this condition, we derive an energy a priori estimate in the anisotropic weighted Sobolev space for the variable coefficients linearized problem for nonplanar plasma-vacuum interfaces.

We establish several a priori estimates of local or global nature in Sobolev spaces with general exponent $s\le 0$ for a class of second-order hyperbolic operators with double characteristics in presence of a transition in a domain of the Euclidian space ${ℝ}^3$.

A no-flux initial-boundary value problem for the cross-diffusion system

To appropriately cope with the considerably stronger cross-degeneracies thus allowed through $(\star )$ when $\alpha$ is large, in its core part the analysis relies on the use of the Moser–Trudinger inequality in controlling the respective diffusion rates $\varphi (v)$ from below.

In this note we present the Adomian decomposition method for solving a simple model for the diffusion and absorption of oxygen in tissue. The method is examined for computational efficiency and accuracy.

Nonlinear differential equations with moving singular points require emergence and development of new approximate methods of solution. In this paper, we give a solution to one of the problems of the analytical approximate method for solving nonlinear differential equations with moving singular points, and study the influence of the perturbation of the initial conditions on the analytical approximate solution in the analytic domain. Theoretical material was tested using a numerical experiment confirming its reliability. The theoretical material presented in this paper allows researchers to use nonlinear differential equations with moving singular points when designing mathematical models of building structures.

In a real Hilbert space H we study the behavior of solutions u_εδ to the following Cauchy problem