Applied Mathematics Research eXpress - recent issues

Complete Solution of a Differential Game with Linear Dynamics and Bounded Controls

Glizer, V. Y., Turetsky, V. — 2008-01-30

A zero-sum finite-horizon differential game with linear dynamics and bounded controls is considered. The target set is a given hyperplane in the state space. The cost function is the distance between the terminal state and this hyperplane. The complete game solution is obtained in two classes of controls: open-loop and feedback-based controls.

On a New Family of Degenerate Parabolic Equations

Ulusoy, S. — 2008-04-10

In this paper we consider the equation\[ {h}_{t}+(p-1)[{h}^{n}[({h}_{x}^{2}{)}^{\frac{p}{2}-1}{h}_{\mathrm{xx}}{]}_{x}{]}_{x}=0,\]which was first derived in (Ulusoy, Nonlinearity 20 (2007): 685–712). We prove results on the regularity of non-negative solutions. In Ulusoy, an entropy dissipation–entropy estimate was provided for the p = 3 and n = 2 case using the energy functional $${K}_{q}:=\int \hspace{1em}\frac{{h}_{x}^{2}}{{h}^{q}}\hspace{1em}dx$$. Here, we extend our calculations to include various other p and n values. After establishing some results on the support properties of solutions, we finally complete the analysis of the long-time behavior of non-negative weak solutions.

Convergence of Leray-type Solutions of Inhomogeneous MHD Systems

Ghazel, M., Benameur, J. — 2008-04-08

In this paper, we study the convergence of "Leray-type" solutions of a magneto-hydro-dynamic system as the Rossby number approaches zero. The proofs are based on weak compactness arguments, interpolation estimates, and linear algebra methods.