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This paper is devoted to the study of hyperbolic systems of linear partial differential equations perturbed by a Brownian motion. The existence and uniqueness of solutions are proved by an energy method. The specific features of this class…
We develop a non-anticipating calculus of variations for functionals on a space of laws of continuous semi-martingales, which extends the classical one. We extend Hamilton's least action principle and Noether's theorem to this generalized…
The autor considers an initial-boundary value problem for the nonstationary Stokes system in an angle, where Dirichlet and Neumann conditions are prescribed on the diferent sides of the angle. The major part of the paper deals with the…
The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate…
In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and…
In this paper, we establish existence of solutions to an indefinite coupled non-linear system. We use partial coercivity to establish existence of a critical point to an indefinite functional and thus the existence of solutions on both…
Differential passivity is a property that allows to check with a pointwise criterion that a system is incrementally passive, a property that is relevant to study interconnected systems in the context of regulation, synchronization, and…
Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…
The reasons which restrict opportunities of classical mechanics at the description of nonequilibrium systems are discussed. The way of overcoming of the key restrictions is offered. This way is based on an opportunity of representation of…
In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting…
In this work, we study the existence and nonexistence of nonnegative solutions to a class of nonlocal elliptic systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of type $u_i\mapsto d_i(-\Delta)^{s_i}u_i$…
We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of field-theoretic models. Classical…
We present several recent results concerning the transition between quantum and classical mechanics, in the situation where the underlying dynamical system has an hyperbolic behaviour. The special role of invariant manifolds will be…
The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity…
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts…
A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order…
We study the existence of non-zero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with…
Small thermodynamic systems exhibit peculiar behavior different from that observed in long-scale systems. Non-equilibrium processes taking place in those systems are strongly influenced by the presence of fluctuations which can be large.…
Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and…
Bifurcations of dynamical systems, described by a second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to…