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Kim Dang Phung

Publications and source records attributed to Kim Dang Phung.

13 recordsLinked to original sources
arXiv2024

On the equilibriation of chemical reaction-diffusion systems with degenerate reactions

The trend to equilibrium for reaction-diffusion systems modelling chemical reaction networks is investigated, in the case when reaction processes happen on subsets of the domain. We prove the convergence to equilibrium by directly showing functional inequalities in terms of entropy method. Our approach allows us to deal with nonlinearities of arbitrary orders, for which only global renormalised solutions are known to globally exist. For bounded solutions, we also prove the convergence to equilibrium when the diffusion as well as the reaction are degenerate, that is both diffusion and reaction processes only act on specific subsets of the domain.

math.AP↗
arXiv2023

An optimal spectral inequality for degenerate operators

In this paper we establish a Lebeau-Robbiano spectral inequality for a degenerate one dimensional elliptic operator. Carleman techniques and moment method are combined. Application to null controllability on a measurable set in time for the degenerated heat equation is described.

math.AP↗
arXiv2021

Exponential decay toward equilibrium via log convexity for a degenerate reaction-diffusion system

We consider a system of two reaction-diffusion equations coming out of reversible chemistry. When the reaction happens on the totality of the domain, it is known that exponential convergence to equilibrium holds. We show in this paper that this exponential convergence also holds when the reaction holds only on a given open set of a ball, thanks to an observation estimate deduced by logarithmic convexity.

math.AP↗
arXiv2021

Observation estimate for the heat equations with Neumann boundary condition via logarithmic convexity

We prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies with a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.

math.AP↗
arXiv2018

A spectral inequality for degenerated operators and applications

In this paper we establish a Lebeau-Robbiano spectral inequality for a degenerated one dimensional elliptic operator and show how it can be used to impulse control and finite time stabilization for a degenerated parabolic equation. R{é}sum{é} .-Dans cet article, on s'int{é}r{è}ss{è} a l'in{é}galit{é} spectrale de type Lebeau-Robbiano sur la somme de fonctions propres pour une famille d'op{é}rateurs d{é}g{é}n{é}r{é}s. Les applications sont donn{é}es en th{é}orie du contr{ô}le comme le contr{ô}le impulsionnel et la stabilisation en temps fini.

math.AP↗
arXiv2018

Carleman commutator approach in logarithmic convexity for parabolic equations

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.

math.AP↗
arXiv2016

Impulse output rapid stabilization for heat equations

The main aim of this paper is to provide a new feedback law for the heat equations in a bounded domain $Ω$ with Dirichlet boundary condition. Two constraints will be compulsory: First, The controls are active in a subdomain of $Ω$ and at discrete time points; Second, The observations are made in another subdomain and at different discrete time points. Our strategy consists in linking an observation estimate at one time, minimal norm impulse control, approximate inverse source problem and rapid output stabilization.

math.AP↗
arXiv2016

Observation estimate for kinetic transport equation by diffusion approximation

We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker-Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on the smallness of the mean free path and the frequency of the velocity average of the initial data. The proof relies on the well known diffusion approximation under convenience scaling and on basic properties of this diffusion. Eventually we propose a direct proof for the observation at one time of parabolic equations. It is based on the analysis of the heat kernel.

math.AP↗
arXiv2011

An observability for parabolic equations from a measurable set in time

This paper presents a new observability estimate for parabolic equations in $Ω\times(0,T)$, where $Ω$ is a convex domain. The observation region is restricted over a product set of an open nonempty subset of $Ω$ and a subset of positive measure in $(0,T)$. This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

math.AP↗
arXiv2009

Waves, damped wave and observation

We consider the wave equation in a bounded domain (eventually convex). Two kinds of inequality are described when occurs trapped ray. Applications to control theory are given. First, we link such kind of estimate with the damped wave equation and its decay rate. Next, we describe the design of an approximate control function by an iterative time reversal method.

math.AP↗
arXiv2005

Polynomial decay rate for the dissipative wave equation

We study the dissipative linear wave equation in a bounded domain. The exponential decay rate of the energy was established by Bardos, Lebeau and Rauch under a geometrical hypothesis linked with the geodesics. Furthermore such condition called geometric control condition is almost necessary to get a uniform exponential decay. In another hand, Lebeau proved a logarithmic decay rate for smooth solutions when no particular geometric condition is required. In this paper we give for some particular geometries a polynomial decay rate when the geometric control condition is not fulfilled.

math.AP↗