Nikolay Gusev

International School for Advanced Studies | SISSA · Functional Analysis Group

We consider the continuity equation $\partial_t \mu_t + \mathop{\mathrm{div}}(b \mu_t) = 0$, where $\{\mu_t\}_{t \in \mathbb R}$ is a measurable family of (possibily signed) Borel measures on $\mathbb R^d$ and $b \colon \mathbb R \times \mathbb R^d \to \mathbb R^d$ is a bounded Borel vector field (and the equation is understood in the sense of dist...

An elliptic-type equation with variable coefficients is considered. An overview is given of the definitions of boundary values of generalized solutions to this equation. Conditions for the existence of boundary values as well as conditions for the existence and uniqueness of solutions to the corresponding Dirichlet problem are analyzed.

We consider an initial–boundary value problem for the continuity equation in a class of non-negative measure-valued solutions. We prove that any solution in the considered class can be represented as a superposition of elementary solutions, associated with the solutions of the corresponding ordinary differential equation.

Let b: [0, T] × ℝd → ℝd be a bounded Borel vector field, T > 0 and let µ be a non-negative Radon measure on ℝd. We prove that a µ-measurable flow of b exists if and only if the corresponding continuity equation has a non-negative measure-valued solution with the initial condition µ.

For smooth vector fields the classical method of characteristics provides a link between the ordinary differential equation and the corresponding continuity equation (or transport equation). We study an analog of this connection for merely bounded Borel vector fields. In particular we show that, given a non-negative Borel measure $\bar \mu$ on $\ma...

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) \times \mathbb R^d \to \mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system). It is well known that in the generic multi-dimensional...

Given a bounded autonomous vector field $b \colon \mathbb R^d \to \mathbb R^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u= 0. \end{equation*} We are interested in the case where $b$ is of class BV and it is nearly incompressible. Ass...

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely cont...

In this note we provide new non-uniqueness examples for the continuity equation by constructing infinitely many weak solutions with prescribed energy.

Given bounded vector field $b : \mathbb R^d \to \mathbb R^d$, scalar field $u : \mathbb R^d \to \mathbb R$ and a smooth function $\beta : \mathbb R \to \mathbb R$ we study the characterization of the distribution $\mathrm{div}(\beta(u)b)$ in terms of $\mathrm{div}\, b$ and $\mathrm{div}(u b)$. In the case of $BV$ vector fields $b$ (and under some f...

Initial-boundary value problem for linearized equations of motion of viscous barotropic fluid in a bounded domain is considered. Existence, uniqueness and estimates of weak solutions to this problem are derived. Convergence of the solutions towards the incompressible limit when compressibility tends to zero is studied.

The nonequivalence of filtration equations and the Darcy set are proved with the help of theorems. First of all, the properties of the Steklov averaging are proposed for further consideration. It was proved when using the method of two scales that the Darcy law is the first term of the asymptotic expansion of steady-state Stokes and Navier-Stokes e...

We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of the special form (Minkowski plane with a handle) containing closed timelike curves (time machines). We prove that the classical solution of the Cauchy problem exists and is unique if and only if the initial data satisfy to some set of additional conditions...

Initial–boundary value problem for the linearized equations of viscous barotropic fluid motion in a bounded domain is considered. Existence, uniqueness and estimates of weak solutions to this problem are derived. Convergence of the solutions towards the incompressible limit when com-pressibility tends to zero is studied.