On variants of HH-measures and compensated compactness

We introduce new variant of $H$-measures defined on spectra of general algebra of test symbols and derive the localization properties of such $H$-measures. Applications for the compensated compactness theory are given. In particular, we present new compensated compactness results for quadratic functionals in the case of general pseudo-differential constraints. The case of inhomogeneous second order differential constraints is also studied

On the Cauchy problem for scalar conservation laws on the Bohr compactification of Rn\R^n

We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We propose also the necessary and sufficient condition for the decay of entropy solutions as time $t\to+\infty$

On decay of entropy solutions to multidimensional conservation laws

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux

On one criterion of the uniqueness of generalized solutions for linear transport equations with discontinuous coefficients

We study generalized solutions of multidimensional transport equation with bounded measurable solenoidal field of coefficients $a(x)$. It is shown that any generalized solution satisfies the renormalization property if and only if the operator $a\cdot\nabla u$, $u\in C_0^1(\mathbb{R}^n)$ in the Hilbert space $L^2(\mathbb{R}^n)$ is an essentially skew-adjoint operator, and this is equivalent to the uniqueness of generalized solutions. We also establish existence of a contractive semigroup, which provides generalized solutions, and give a criterion of its uniqueness

Decay of periodic entropy solutions to degenerate nonlinear parabolic equations

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation

On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws

We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws

On long time behavior of periodic entropy solutions of a degenerate non-linear parabolic equation

We prove the asymptotic convergence of a space-periodic entropy solution of a one-dimensional degenerate parabolic equation to a traveling wave. It is also shown that on a segment containing the essential range of the limit profile the flux function is linear (with the slope equaled to the speed of the traveling wave) and the diffusion function is constant

On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: global well-posedness and decay property

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We propose also the necessary and sufficient condition for the decay of almost periodic entropy solutions as time $t\to+\infty$

On decay of almost periodic viscosity solutions to Hamilton-Jacobi equations

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.Comment: arXiv admin note: text overlap with arXiv:1707.0014

Vibronic Mechanism of the Isotope and Pressure Effects in Cuprates

The doped cuprates are considered to be a system of local spinless bosons moving in a lattice of the hole pseudo-Jahn-Teller centers CuO_{4}^{5-}. A detailed qualitative and quantitative description of the vibronic mechanism determining both the isotope substitution and the external pressure effect on T_c is given. This mechanism can properly interpret the principal peculiarities of isotopic and baric effects in cuprates except the region of the well developed percolation phenomenaComment: 6 pages; to appear in Proceedings of XIV Int. Symposium on Electron-Phonon Dynamics and Jahn-Teller Effec