Continuous dependence of the entropy solution of general parabolic equation

Maliki, Mohamed

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 589-598.

On considère l’équation parabolique générale :

$u_t-\Delta b\left(u\right)+divF\left(u\right)=f\text{dans}Q=]0,T[\times {ℝ}^N,T>0$ avec ${\displaystyle u_0\in L^{\infty }\left({ℝ}^N\right),f\in L_{Loc}^1\left(Q\right)}$ pour $p.pt\in ]0,T[f\left(t\right)\in L^{\infty }\left({ℝ}^N\right)$,

et ${\displaystyle {\int }_0^T{∥f\left(t\right)∥}_{{\displaystyle L^{\infty }\left({ℝ}^N\right)}}dt<\infty .}$

On montre la dépendance continue de la solution entropique du problème de Cauchy associé, par rapport aux données $F$, $b$ $f$ et la donnée initiale $u_0$. Ce type de solution a été introduit et étudié dans [MT3].

On commence le travail par un rappel de la définition de la solution faible et entropique ainsi que les résultats importants obtenus dans [MT3]. Ensuite on montre le résultat principal du travail en utilisant le lemme abstrait (Théorème 2.3). La contribution du travail est de traiter le problème dans ${ℝ}^N$ ainsi que de considérer des données bornées au lieu des données intégrables utilisées dans la littérature.

We consider the general parabolic equation :

$u_t-\Delta b\left(u\right)+divF\left(u\right)=f$ in $Q=]0,T[\times {ℝ}^N,T>0$ with ${\displaystyle u_0\in L^{\infty }\left({ℝ}^N\right),}$ $\text{for}a.et\in ]0,T[,f\left(t\right)\in L^{\infty }\left({ℝ}^N\right)$ and ${\displaystyle {\int }_0^T{∥f\left(t\right)∥}_{{\displaystyle L^{\infty }\left({ℝ}^N\right)}}dt<\infty .}$

We prove the continuous dependence of the entropy solution with respect to $F,$ $b,$ $f$ and the initial data $u_0$ of the associated Cauchy problem.

This type of solution was introduced and studied in [MT3]. We start by recalling the definition of weak solution and entropy solution. By applying an abstract result (Theorem 2.3), we get the continuous dependance of the entropy solution. The contribution of the present work consists of considering the equation in the whole space ${ℝ}^n$ instead of a bounded domain and considering a bounded data instead of integrable data.

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