Related papers: Conservation Laws With Random and Deterministic Da…
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…
In this paper, we consider a general form of nonlinear Schr\"{o}dinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schr\"{o}dinger equation is identified by…
For systems of partial differential equations in three spatial dimensions, dynamical conservation laws holding on volumes, surfaces, and curves, as well as topological conservation laws holding on surfaces and curves, are studied in a…
We analyze the stochastic scaling laws arising in the invicid limit of the decaying solutions of the Burgers equation. The linear scaling of the velocity structure functions is shown to reflect the domination by shocks of the long-time…
In this paper we analyze the long-time behavior of solutions to conservation laws with nonlinear diffusion terms of different types: saturating dissipation (monotone and non monotone) and singular nonlinear diffusions are considered. In…
The probability density functions (PDFs) for the solution of the incompressible Navier-Stokes equation can be represented by a hierarchy of linear equations. This article develops new hierarchical evolution equations for PDFs of a scalar…
Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an $L^{1}\cap L^{2}$ setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove…
The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more…
All the Doubly Special Relativity (DSR) models studied in literature so far involve a deformation of the energy conservation rule that forces us to release the hypothesis of the additivity of the energy for composite systems. In view of the…
Parameterization (closure) schemes in numerical weather and climate prediction models account for the effects of physical processes that cannot be resolved explicitly by these models. Methods for finding physical parameterization schemes…
A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal…
In quantum experiments the acquisition and representation of basic experimental information is governed by the multinomial probability distribution. There exist unique random variables, whose standard deviation becomes asymptotically…
We establish local-in-time existence and uniqueness results for nonlocal conservation laws with a nonlinear mobility, in several space dimensions, under weak assumptions on the kernel, which is assumed to be bounded and of finite total…
We investigate the question of how the knowledge of sufficiently many local conservation laws for a model can be utilized to solve the model. We show that for models where the conservation laws can be written in one-sided forms, like…
This note is concerned with the study of the initial boundary value problem for systems of conservation laws from the point of view of control theory, where the initial data is fixed and the boundary data are regarded as control functions.…
Linearisation is often used as a first step in the analysis of nonlinear initial boundary value problems. The linearisation procedure frequently results in a confusing contradiction where the nonlinear problem conserves energy and has an…
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such…
I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their…
We consider complex Mandelbrot multiplicative cascades on a random weigh\-ted tree. Under suitable assumptions, this yields a dynamics $\T$ on laws invariant by random weighted means (the so called fixed points of smoothing transformations)…
Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about…