Continuous dependence of the entropy solution of general parabolic equation

Mohamed Maliki[1]

  • [1] Université Hassan II, F.S.T. Mohammedia, B.P. 146 Mohammedia, Maroc.

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 3, page 589-598
  • ISSN: 0240-2963

Abstract

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We consider the general parabolic equation : u t - Δ b ( u ) + d i v F ( u ) = f in Q = ] 0 , T [ × ℝ N , T > 0 with u 0 ∈ L ∞ ( ℝ N ) , for a . e t ∈ ] 0 , T [ , f ( t ) ∈ L ∞ ( ℝ N ) and ∫ 0 T f ( t ) L ∞ ( ℝ N ) d t < ∞ . 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.

How to cite

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Maliki, Mohamed. "Continuous dependence of the entropy solution of general parabolic equation." Annales de la faculté des sciences de Toulouse Mathématiques 15.3 (2006): 589-598. <http://eudml.org/doc/10012>.

@article{Maliki2006,
abstract = {We consider the general parabolic equation :$ u_t - \Delta b(u) + div \ F(u)= f$ in $Q=]0,T[\times \mathbb\{R\}^N , \ T&gt;0 $ with $\ \ \displaystyle u_0 \in L^\infty (\mathbb\{R\}^N),$$\ \ \hbox\{for\}\ \ a.e\ \ t\in ]0,T[, \ \ f(t) \in L^\infty (\mathbb\{R\}^N) $ and $ \displaystyle \int _0^T \left\Vert f(t)\right\Vert _\{\displaystyle L^\infty ( \mathbb\{R\}^N)\}dt&lt; \infty .$We prove the continuous dependence of the entropy solution with respect to $F,$$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 $\mathbb\{R\}^n$ instead of a bounded domain and considering a bounded data instead of integrable data.},
affiliation = {Université Hassan II, F.S.T. Mohammedia, B.P. 146 Mohammedia, Maroc.},
author = {Maliki, Mohamed},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {bounded data},
language = {eng},
number = {3},
pages = {589-598},
publisher = {Université Paul Sabatier, Toulouse},
title = {Continuous dependence of the entropy solution of general parabolic equation},
url = {http://eudml.org/doc/10012},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Maliki, Mohamed
TI - Continuous dependence of the entropy solution of general parabolic equation
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 3
SP - 589
EP - 598
AB - We consider the general parabolic equation :$ u_t - \Delta b(u) + div \ F(u)= f$ in $Q=]0,T[\times \mathbb{R}^N , \ T&gt;0 $ with $\ \ \displaystyle u_0 \in L^\infty (\mathbb{R}^N),$$\ \ \hbox{for}\ \ a.e\ \ t\in ]0,T[, \ \ f(t) \in L^\infty (\mathbb{R}^N) $ and $ \displaystyle \int _0^T \left\Vert f(t)\right\Vert _{\displaystyle L^\infty ( \mathbb{R}^N)}dt&lt; \infty .$We prove the continuous dependence of the entropy solution with respect to $F,$$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 $\mathbb{R}^n$ instead of a bounded domain and considering a bounded data instead of integrable data.
LA - eng
KW - bounded data
UR - http://eudml.org/doc/10012
ER -

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