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.

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...

This article surveys results on the global surjectivity of linear partial differential operators with constant coefficients on the space of real analytic functions. Some new results are also included.

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $${\widehat{H}}_s^r\left(ℝ\right)$$ defined by the norm $${∥v_0∥}_{{\widehat{H}}_s^r\left(ℝ\right)}:={∥{〈\xi 〉}^s\widehat{v_0}∥}_{L_{\xi }^{r^{\text{'}}}},〈\xi 〉={\left(1+{\xi }^2\right)}^{\frac{1}{2}},\frac{1}{r}+\frac{1}{r^{\text{'}}}=1$$ . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ $$\frac{2j-1}{2r^{\text{'}}}$$ . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < $$\frac{2j-1}{2r^{\text{'}}}$$ . The results for r = 2 - so far in...

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta –1}V\left(B^{\beta }\right)$. For the interaction parameter $\beta \in (0,2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N\to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0,3/5)$ in [28]. This allows, in the case $\beta \in (0,3/5)$, for the application of the Klainerman–Machedon uniqueness theorem...