The backward Euler algorithm for the multidimensional nonhomogeneous heat equation is analyzed, based on the finite element method. The existence and uniqueness of the numerical solution is investigated. Also, the convergence of the numerical solutions is studied.

We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.

We investigate the existence of solutions for the Dirichlet problem including the generalized balance of a membrane equation. We present a duality theory and variational principle for this problem. As one of the consequences of the duality we obtain some numerical results which give a measure of a duality gap between the primal and dual functional for approximate solutions.

The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms: $$\left\{\begin{array}{cc}{(-\Delta )}^{s}u+V\left(x\right)u+\phi u-\frac{1}{2}u{(-\Delta )}^s{u}^2=f(x,u),\hfill & x\in {ℝ}^3,\hfill \\ {(-\Delta )}^t\phi =u^2,\hfill & x\in {ℝ}^3,\hfill \end{array}\right.$$ where ${(-\Delta )}^{\alpha }$ is the fractional Laplacian for $\alpha =s$, $t\in (0,1]$ with $s<t$ and $2t+4s>3$. Under assumptions on $V$ and $f$, we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem.

We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by $\frac{n}{2}+4+ϵ$ improving thus the bound $2n+4+ϵ$ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type $S_{0,0}^0$, we show that this number is bounded by $n+4+ϵ$; more precisely, for a non negative symbol $a$, the Fefferman-Phong inequality holds if ${\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }a(x,\xi )$ are bounded for, roughly, $4\le \left|\alpha \right|+\left|\beta \right|\le n+4+ϵ$. To obtain such results and others, we first prove an abstract result which says that...

The article studies a second-order linear differential operator of the type $-L=$$X_1^2+...$

We describe qualitative behaviour of solutions of the Gross-Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain rather detailed picture of the vortex...

We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like ${\mu \left(D\right)\sim \left|D\right|}^{p-2}$ (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2\left(ℝ\right)$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ${∥f∥}_1\le 1/5$. Previous results of this...