A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

Fast Reaction Limit with Nonmonotone Reaction Function

We analyse the fast reaction limit in the reaction-diffusion system with nonmonotone reaction function and one nondiffusing component. As speed of reaction tends to infinity, the concentration of the nondiffusing component exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component, a result that is not obvious from a priori estimates. Our work is based on an analysis of regularization for forward-backward parabolic equations by Plotnikov. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions, and provides a weak formulation for the evolution of the Young measure. © 2022 Wiley Periodicals LLC.     

Applications of the Kinetic Formulation for Scalar Conservation Laws with a Zero-Flux Type Boundary Condition

The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws.Firstly,a kinetic formulation of entropy solutions is established.Secondly,by using the kinet...     

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and then we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame and Simeoni (2001) [1] and Botchorishvili et al. (2003) [2] using an energetic balance at microscopic level. The validation is lastly performed in the case of a water hammer in an uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.     

Kinetic formulation for the Graetz-Nusselt ultra-parabolic equation

The well-posedness of the Cauchy problems for a quasilinear ultra-parabolic equation with partial diffusion and discontinuous convection coefficients is established for both entropy and kinetic formulations. The kinetic formulation is set up and solved by means of studying of the Young measures, associated with sequences of solutions of parabolic approximations. The kinetic equation appears as the linear scalar equation, which describes the evolution of the distribution functions of the Young measures in time and space, and which involves an additional ‘kinetic’ variable. The proofs of the principal results of the paper are based on the originally constructed renormalization procedure for the kinetic equation.     

非自治退化抛物-双曲方程的柯西问题

非自治退化抛物-双曲型方程可以描述许多自然现象.主要研究这类方程的柯西问题,建立了动力学公式,在对流函数、扩散函数适当光滑性的基础上,证明了该问题动力学解的存在唯一性.     

Well-posedness for entropy solutions to multidimensional scalar conservation laws with a strong boundary condition

In this paper we give a simple proof of well-posedness of multidimensional scalar conservations laws with a strong boundary condition. The proof is based on a result of strong trace for solutions of scalar conservation laws and kinetic formulation.     

A priori estimates for rough PDEs with application to rough conservation laws

We introduce a general weak formulation for PDEs driven by rough paths, as well as a new strategy to prove well-posedness. Our procedure is based on a combination of fundamental a priori estimates with (rough) Gronwall-type arguments. In particular this approach does not rely on any sort of transformation formula (flow transformation, Feynman–Kac representation formula etc.) and is therefore rather flexible. As an application, we study conservation laws driven by rough paths establishing well–posedness for the corresponding kinetic formulation.     

The tail process revisited

The tail measure of a regularly varying stationary time series has been recently introduced. It is used in this contribution to reconsider certain properties of the tail process and establish new ones. A new formulation of the time change formula is used to establish identities, some of which were indirectly known and some of which are new.     

退化抛物-双曲方程动力学解的唯一性

主要研究系数显含有时间和空间变量的退化抛物-双曲型方程柯西问题动力学解的唯一性.首先推广了这种类型方程的动力学公式,在给定系数适当的光滑性条件下,得到了动力学解的唯一性.     

Diffusion-dispersion limits for multidimensional scalar conservation laws with source terms

In this paper we consider conservation laws with diffusion and dispersion terms. We study the convergence for approximation applied to conservation laws with source terms. The proof is based on the Hwang and Tzavaras's new approach [Seok Hwang, Athanasios E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (5-6) (2002) 1229-1254] and the kinetic formulation developed by Lions, Perthame, and Tadmor [P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1) (1994) 169-191].     

Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial–boundary value problem. The proof utilizes the kinetic formulation and the averaging lemma. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial–boundary value problem for the Degasperis–Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.     

Scalar conservation laws with stochastic forcing

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.     

On the construction and investigation of solutions of boundary value problems of thermoviscoelasticity

The uncoupled mixed boundary value problem of thermoviscoelasticity is considered in a quasistatic formulation. The temperature distribution is assumed nonstationary and inhomogeneous. The influence of the temperature on the viscoelastic properties of the material is taken into account by the introduction of a reduced time. The equations of state of the material are written in differential form as a system of kinetic equations in some tensor-type strain parameters. The system mentioned is equivalent to a Volterra integral equation with kernel in the form of a sum of exponents. The differential approach used is apparently more convenient for numerical realization /1/ (especially in nonuniform problems) and results in a substantially different mathematical formulation as compared with that based on the integral form of writing the equations of state investigated in /2,3/. Precisely for going over to the boundary value problem are the kinetic differential equations converted into an operator differential equation in Hubert space. The existence, uniqueness, and stability of the solution of the problem formulated are established, and conditions for the convergence of the Galerkin approximations and the stability of the difference approximations in time are formulated.     

Optimal Besov Differentiability for Entropy Solutions of the Eikonal Equation

In this paper we study the eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production, and the validity of a kinetic formulation. © 2019 Wiley Periodicals, Inc.     

On the kinetic equations of a lithium-ion battery model

In this Note we provide a Hamilton–Poisson realization of the system obtained from the kinetic equations of a well known lithium-ion battery model, explicitly integrate the Poincaré compactification of this system and give a Lax formulation.     

THE APPLICATION OF THE KINETIC FORMULATION TO A SYSTEM OF QUADRATIC FLUX

In this article, the author uses the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global entropy solutions to a system of quadratic flux. The fire-new method of reduction of Young measures is a pith of this work.     

A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions

We study the comparison principle for degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L¹ contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kru?kov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic-hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L¹ contractive property.     

A kinetic scheme for the Savage–Hutter equations

The Savage–Hutter (SH) equations describe the motion of granular material under the influence of friction. Based on the kinetic formulation of the SH equations, we present a kinetic scheme in one dimension, which describes the deformation of the mass profile and allows it to start and to stop. Moreover the method is able to preserve the steady states of granular masses at rest. The method is tested on several numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.     

非完整动力学逆问题的一种提法和解法*

本文给出非完整动力学逆问题的一种提法和解法:已知某些积分,来求施加在系统上的非完整约束的形式;进而在已知系统动能表达式的情况下,来求加在系统上的广义约束反力;最后,给出例子说明解法的应用.