Laurent Levi

• Senior Lecturer
• Professor (Assistant) at Université de Pau et des Pays de l'Adour

This paper is devoted to the existence and the uniqueness of the entropy solution for a general scalar conservation law associated with a forced bilateral obstacle condition in a bounded domain of ℝp, p≥1. The method of penalization is used with a view to obtaining an existence result. However, the former only gives uniform L∞-estimates and so lead...

We deal with the mathematical analysis of the coupling problem in a bounded domain of ℝ n , n≥1, between a purely quasilinear first-order hyperbolic equation set on a subdomain and a parabolic one, set on its complementary. We start by providing the definition of a weak solution through an entropy inequality on the whole domain. The uniqueness prop...

We deal with the mathematical analysis of the coupling problem in a bounded domain of $\R^n$, $n \geq 1$, between a purely quasilinear first-order hyperbolic equation set on a subdomain and a parabolic one, set on its complementary. We start by providing the definition of a weak solution through an entropy inequality on the whole domain. The unique...

This paper deals with the coupling of a quasilinear parabolic problem with a first order hyperbolic one in a multidimensional bounded domain $\Omega $. In a region $\Omega _{p}$ a diffusion-advection-reaction type equation is set while in the complementary $\Omega _{h}\equiv \Omega \backslash \Omega _{p}$, only advection-reaction terms are taken in...

In this paper, the mathematical analysis of a quasilinear parabolic–hyperbolic problem in a multidimensional bounded domain Ω is carried out. In a region Ω p a diffusion–advection–reaction-type equation is set, while in the complementary Ω h ≡ Ω\ Ω p , only advection–reaction terms are taken into account. First, the definition of a weak solution u...

Ce travail regroupe un ensemble de résultats concernant essentiellement l'analyse mathématique de problèmes d'obstacles intérieurs pour une classe d'opérateurs quasi linéaires hyperboliques du premier ordre ou paraboliques du second ordre. Dans un premier chapitre on s'intéresse aux problèmes d'obstacles pour des opérateurs du premier ordre. On do...

This paper deals with the coupling of a quasilinear parabolic problem with a first order hyperbolic one in a multidimensional bounded domain $Omega$. In a region $Omega_{p}$ a diffusion-advection-reaction type equation is set while in the complementary $Omega_hequiv Omega ackslash Omega_{p}$, only advection-reaction terms are taken into account. Su...

We establish the existence and uniqueness of the solution to some inner obstacle problems for a coupling of a multidimensional quasilinear first-order hyperbolic equation set in a region Ω h with a quasilinear parabolic one set in the complement Ω p =Ω∖Ω h . The mathematical problem is motivated by physical models for infiltration processes with sa...

We carry out the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain Ω=Ω h ∪Ω p , where Ω p =Ω∖Ω h . We start by providing the definition of a weak solution u through an entropy inequality on the whole Ω by using the classical Kuzhkov pairs. The uniqueness proof begins by focusing on the behavior...

We investigate some inner bilateral obstacle problems for a class of strongly degenerate parabolic-hyperbolic quasilinear operators associated with homogeneous Dirichlet data in a multidimensional bounded domain. We first introduce the concept of an entropy process solution, more convenient and generalizing the notion of an entropy solution. Moreov...

This paper deals with the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain Ω. In a region Ωp a diffusion-advection-reaction type equation is set, while in the complementary Ωh ≡ Ω\Ωp, only advection-reaction terms are taken into account. To begin we provide a definition of a weak solution thro...

This paper deals with the mathematical analysis of a quasilinear parabolichyperbolic problem in a multidimensional bounded domain W. In a region Wp a diffusionadvection- reaction type equation is set while in the complementary Wh W\Wp, only advection-reaction terms are taken into account. To begin we provide the definition of a weak solution throug...

We establish the existence and uniqueness of the solution to some inner obstacle problems for a coupling of a multidimensional quasilinear first-order hyperbolic equation set in a region ­$\Omega_h$ with a quasilinear parabolic one set in the complementary ­$\Omega_p =\Omega \backslash \Omega _h$. We start by providing the definition of a weak solu...

We investigate some inner bilateral obstacle problems for a class of strongly degenerate parabolic-hyperbolic quasilinear operators associated with homogeneous Dirichlet data in a multidimensional bounded domain. We first introduce the concept of an entropy process solution, more convenient and generalizing the notion of an entropy solution. Moreov...

We deal with the scalar conservation law in a one dimensional bounded domain : $\Omega: \partial_t u + \partial_x(k(x)g(u)) = 0$, associated with a bounded initial value $u_0$. The function $k$ is supposed to be bounded, discontinuous at ${x_0 = 0}$, and with bounded variation. A weak entropy formulation for the Cauchy problem has been introduced b...

We study the limit as ɛ goes to 0+ of the sequence (uɛ)ɛ>0 of solutions to the Dirichlet problem for the weakly degenerate quasilinear parabolic operators Hɛ(t,x,.):u→∂tu+∑i=1p∂xifi(t,x,u)+g(t,x,u)−ɛΔϕ(u), subject to an inner bilateral constraint in an open bounded domain of Rp, 1⩽p+∞. We first establish the existence of uɛ by coupling the method o...

We study inner obstacle problems for a class of strongly degenerate parabolic–hyperbolic quasilinear operators associated with Dirichlet data in an open bounded subset of Rp, p≥1. We first give the definition of a weak entropy solution that warrants uniqueness; the boundary conditions are expressed by using the framework of divergence measure field...

We establish a singular perturbation property for a class of quasi-linear parabolic degenerate equations associated with a mixed Dirichlet-Neumann boundary condition in a bounded domain of R p , 1 < p < +∞. In order to prove the L 1 -convergence of viscous solutions toward the entropy solution of the corresponding first-order hyperbolic problem, we...

We study the limit as goes to 0 + for the sequence (u) >0 of solutions to the Dirichlet problem for the quasilinear parabolic operators H (t, x, .) : u → ∂ t u + p i=1 ∂ x i ϕ i (t, x, u) + ψ(t, x, u) − ∆φ(u), where φ is a nondecreasing function, associated with a positiveness condition in an open bounded domain of R p , 1 ≤ p < +∞. The positive pa...

We are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method...

In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy...

This paper is devoted to the existence and the uniqueness of the entropy solution for a general scalar conservation law associated with a forced bilateral obstacle condition in a bounded domain of ℝp, p ≥ 1. The method of penalization is used with a view to obtaining an existence result. However, the former only gives uniform L∞-estimates and so le...

On etablit un resultat d'existence et d'unicite pour une loi de conservation scalaire associee a des conditions de bord de Dirichlet homogenes et a une condition d'obstacle unilateral simple. Considerant le probleme: ...Formula math.... nous avons developpe deux methodes, soit par penalisation de l'equation (E) soit en approchant U par une suite de...

We study the limit asgoes to 0+ for the sequence (u� )�> 0 of solutions to the Dirichlet problem for the quasilinear parabolic operators where φ is a nondecreasing function, associated with a positiveness condition in an open bounded domain of Rp, 1 ≤ p< +∞. The positive parameterbeing fixed, we first pro- pose the definition of a weak entropy solu...

This paper deals with the mathematical analysis of a quasilinear parabolic- hyperbolic problem in a multidimensional bounded domain ›. In a region ›p a difiusion- advection-reaction type equation is set while in the complementary ›h · ›n›p, only advection-reaction terms are taken into account. To begin we provide the deflnition of a weak solution t...

We study the limit as † goes to 0+ of the sequence (u†)†>0 of solutions to the Dirichlet problem for the weakly degenerate quasilinear parabolic operators p X i=1 @xifi(t;x;u) + g(t;x;u) ¡ †¢`(u); associated with an inner bilateral constraint in an open bounded domain of Rp;1 • p < +1. After setting the positive parameter †, we flrst establish the...

This deals with the Cauchy problem for a general first-order quasilinear operator related to a forced bilateral obstacle condition, namely a bounded control function. Firstly, a weak entropy formulation is given that guarantees the uniqueness of a solution. The latter is obtained through the penalization method, by using some specific properties of...

Ce travail regroupe un ensemble de résultats concernant essentiellement l’analyse mathématique de problèmes d’obstacles intérieurs pour une classe d’opérateurs quasi linéaires hyperboliques du premier ordre ou paraboliques du second ordre. Dans un premier chapitre on s’intéresse aux problèmes d’obstacles pour des opérateurs du premier ordre. On do...