Related papers: Non linear diffusion and wave damped propagation :…
We analyze mechanisms and regimes of wave packet spreading in nonlinear disordered media. We predict that wave packets can spread in two regimes of strong and weak chaos. We discuss resonance probabilities, nonlinear diffusion equations,…
By performing two parallel numerical experiments -- solving the dynamical Hamiltonian equations and solving the Hasselmann kinetic equation -- we examined the applicability of the theory of weak turbulence to the description of the time…
We consider a tsunami wave equation with singular coefficients and prove that it has a very weak solution. Moreover, we show the uniqueness results and consistency theorem of the very weak solution with the classical one in some appropriate…
The problem of characterizing weak limits of sequences of solutions for a non-linear diffusion equation of $p$-laplacian type is addressed. It is formulated in terms of certain moments of underlying Young measures associated with main…
This article is devoted to the proof of the well-posedness of a model describing waves propagating in shallow water in horizontal dimension $d=2$ and in the presence of a fixed partially immersed object. We first show that this…
We prove existence of weak solutions (in the probabilistic sense) for a general class of stochastic semilinear wave equations on bounded domains of $R^d$ driven by a possibly discontinuous square integrable martingale.
We obtain a general solution for the probability density function of wave intensities in non-stationary Wave Turbulence. The solution is expressed in terms of the wave action spectrum evolving according the the wave-kinetic equation. We…
We study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1-D Euclidean space. The diffusion coefficient depends on the gradient in analogy with the p-Laplacian and may be degenerate.…
A system of two-dimensional nonlinear equations of hydrodynamics is considered. It is shown that for the this system in the general case a solution with weak discontinuity-type singularity behaves as a square root of S(x,y,t), where…
We develop a theory of turbulence of weak random gravity waves on surface of deep water in which the main nonlinear process at high-frequency part of the spectrum is a nonlocal interaction with a strong low-frequency component. The latter…
This paper is devoted to study a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under suitable conditions, we prove that any weak solutions with negative initial energy…
The weak turbulence model, also known as the quasilinear theory in plasma physics, has been a cornerstone in modeling resonant particle-wave interactions in plasmas. This reduced model stems from the Vlasov-Poisson/Maxwell system under the…
The purpose of this article is numerical verification of the theory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial…
In this paper we consider a singular wave equation with distributional and more singular non-distributional coefficients and develop tools and techniques for the phase-space analysis of such problems. In particular we provide a detailed…
We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular…
The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a…
This article addresses linear hyperbolic partial differential equations with non-smooth coefficients and distributional data. Solutions are studied in the framework of Colombeau algebras of generalized functions. Its aim is to prove upper…
This paper provides a theoretical foundation for some common formulations of inverse problems in wave propagation, based on hyperbolic systems of linear integro-differential equations with bounded and measurable coefficients. The…
We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary…
We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of $p$-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others,…