Blow-up of the hyperbolic Burgers equation

The memory effects on microscopic kinetic systems have been sometimes modelled by means of the introduction of second order time derivatives in the macroscopic hydrodynamic equations. One prototypical example is the hyperbolic modification of the Burgers equation, that has been introduced to clarify the interplay of hyperbolicity and nonlinear hydrodynamic evolution. Previous studies suggested the finite time blow-up of this equation, and here we present a rigorous proof of this fact

On effective compactness and sigma-compactness

Using the Gandy -- Harrington topology and other methods of effective descriptive set theory, we prove several theorems on compact and sigma-compact pointsets. In particular we show that any $\Sigma^1_1$ set $A$ of the Baire space $N^N$ either is covered by a countable union of compact $\Delta^1_1$ sets, or $A$ contains a subset closed in $N^N$ and homeomorphic to $N^N$ (and then $A$ is not covered by a sigma-compact set, of course)

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page

Аліментні обов'язки інших членів сім'ї та родичів

Виявлено проблеми у врегулюванні аліментних обов’язків інших членів сім’ї та родичів, вироблено рекомендації щодо їх вирішення. Проаналізовано специфіку правового регулювання аліментних зобов’язань зазначених суб’єктів, сімейне законодавство та міжнародний досвід. Ключові слова: аліментні обов’язки, правове регулювання, сімейне законодавство.Выявлены проблемы в урегулировании алиментных обязанностей других членов семьи и родственников, выработаны рекомендации по их решению. Проанализирована специфика правового регулирования алиментных обязательств указанных субъектов, семейное законодательство и международный опыт. Ключевые слова: алиментные обязанности, правовое регулирование, семейное законодавствоThis article is dedicated to identifying problems in the regulation of the alimentary obligations of other family members and relatives, and to making recommendations and proposing solutions. Studing the specificity of the legal regulation of alimentary obligations of these entities, analysing the current family law and international experience are very important. Key words: alimentary obligations, legal regulation, family law

Nonexistence of self-similar singularities for the 3D incompressible Euler equations

We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.Comment: This version refines the previous one by relaxing the condition of compact support for the vorticit

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Four conjectures in Nonlinear Analysis

In this chapter, I formulate four challenging conjectures in Nonlinear Analysis. More precisely: a conjecture on the Monge-Amp\`ere equation; a conjecture on an eigenvalue problem; a conjecture on a non-local problem; a conjecture on disconnectedness versus infinitely many solutions.Comment: arXiv admin note: text overlap with arXiv:1504.01010, arXiv:1409.5919, arXiv:1612.0819

Uniform regularity for the Navier-Stokes equation with Navier boundary condition

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in $L^\infty$. This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument

Relative entropies, suitable weak solutions, and weak strong uniqueness for the compressible Navier-Stokes system

We introduce the notion of relative entropy for the weak solutions of the compressible Navier-Stokes system. We show that any finite energy weak solution satisfies a relative entropy inequality for any pair of sufficiently smooth test functions. As a corollary we establish weak-strong uniqueness principle for the compressible Navier-Stokes system

On Ultrabarrelled Spaces, their Group Analogs and Baire Spaces

Let E and F be topological vector spaces and let G and Y be topological abelian groups. We say that E is sequentially barrelled with respect to F if every sequence (un)n∈N of continuous linear maps from E to F which converges pointwise to zero is equicontinuous. We say that G is barrelled with respect to F if every set H of continuous homomorphisms from G to F, for which the set H(x) is bounded in F for every x∈E, is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every H⊆CHom(G,Y) which is compact in the product topology of YG is equicontinuous. We prove that - a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space, - a topological group which is a Baire space is barrelled with respect to any topological vector space, - a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group, - a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to R/Z)