Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...

Nous démontrons l’unicité des solutions faibles pour une classe d’équations de transport dont les vitesses sont partiellement à variations bornées. Nous nous intéressons à des champs de vecteurs du type$$a_1\left(x_1\right)·{\partial }_{x_1}+a_2(x_1,x_2)·{\partial }_{x_2},a_1\in BV\left({ℝ}_{x_1}^{N_1}\right),a_2\in L_{x_1}^1\left(BV\left({ℝ}_{x_2}^{N_2}\right)\right),$$avec une borne sur la divergence de chacun des champs $a_1,a_2$. Ce modèle a été étudié récemment dans [LL] par C. Le Bris et P.-L. Lions avec une régularité $W^{1,1}$ ; nous montrons ici également que, dans le cas $W^{1,1}$, le contrôle $L^{\infty }$ de la divergence totale du champ est suffisant. Notre méthode consiste à démontrer...

Whenever nonlinear problems have to be solved through approximation methods by solving related linear problems a priori estimates are very useful. In the following this kind of estimates are presented for a variety of equations related to generalized first order Beltrami systems in the plane and for second order elliptic equations in ${ℝ}^m$. Different types of boundary value problems are considered. For Beltrami systems these are the Riemann-Hilbert, the Riemann and the Poincaré problem, while for elliptic...

This paper is devoted to the existence and uniqueness of solutions for gradient systems of evolution which involve gradients taken with respect to time-variable inner products. The Gelfand triple $(V,H,V^{\text{'}})$ considered in the setting of this paper is such that the embedding $V\hookrightarrow H$ is only continuous.

We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand side depends on the derivative of unknown function with deviating argument.

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers' equation. The approximate solution...

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation $$\dot{x}\left(t\right)=ℒ\left(t\right)x\left(t\right)+f(t,x\left(t\right)),t\in ℝ\left(\mathrm{P}\right)$$ where $\left\{ℒ\right(t):t\in ℝ\}$ is a family of linear operators from a Banach space $E$ into itself and $f:ℝ\times E\to E$. By $L\left(E\right)$ we denote the space of linear operators from $E$ into itself. Furthermore, for $a<b$ and $d>0$, we let $C\left(\right[-d,0],E)$ be the Banach space of continuous functions from $[-d,0]$ into $E$ and $f^d:[a,b]\times C([-d,0],E)\to E$. Let $\widehat{ℒ}:[a,b]\to L\left(E\right)$ be a strongly measurable and Bochner integrable operator on $[a,b]$ and for $t\in [a,b]$ define ${\tau }_{t}x\left(s\right)=x(t+s)$ for each $s\in [-d,0]$. We prove that, under certain conditions,...

We consider the spatially inhomogeneous Bekker-Döring infinite-dimensional kinetic system describing the evolution of coagulating and fragmenting particles under the influence of convection and diffusion. The simultaneous consideration of opposite coagulating and fragmenting processes causes many additional difficulties in the investigation of spatially inhomogeneous problems, where the space variable changes differently for distinct particle sizes. To overcome these difficulties, we use a modified...

Despite their deficiencies, continuous second-order traffic flow models are still commonly used to derive discrete-time models that help traffic engineers to model and predict traffic oflow behaviour on highways. We brie fly overview the development of traffic flow theory based on continuous flow-density models of Lighthill-Whitham-Richards (LWR) type, that lead to the second-order model of Aw-Rascle. We will then concentrate on widely-adopted discrete approximation to the LWR model by Daganzo's...