We consider a coupled system of PDEs for the scalar functions u and k in a cylinder ( bounded domain, ). This system represents a simplified version of Prandtlʼs (1945) model of turbulence in the case of an unsteady motion of a fluid through a pipe with cross-section Ω (u = one-dimensional velocity, k = turbulent kinetic energy). We prove the existence of weak solutions to the problem under consideration with homogeneous Dirichlet conditions on u and homogeneous Neumann conditions on k along , and initial conditions on u and k in .
On considère un système couplé dʼéquations aux dérivées partielles pour des fonctions scalaires u et k dans un cylindre ( domaine borné, ). Ce système représente une version simplifiée du modèle de turbulence de Prandtl (1945) dans le cas de lʼécoulement non stationnaire dʼun liquide dans une conduite de section Ω ( à une dimension, k = énergie cinétique de la turbulence). Nous démontrons lʼexistence de solutions faibles pour le système envisagé avec des conditions homogènes de Dirichlet pour u et des conditions de Neumann pour k sur , et des conditions initiales pour des fonctions u et k dans .
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@article{CRMATH_2013__351_11-12_451_0, author = {Naumann, Joachim}, title = {On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow}, journal = {Comptes Rendus. Math\'ematique}, pages = {451--456}, publisher = {Elsevier}, volume = {351}, number = {11-12}, year = {2013}, doi = {10.1016/j.crma.2013.06.011}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2013.06.011/} }
TY - JOUR AU - Naumann, Joachim TI - On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow JO - Comptes Rendus. Mathématique PY - 2013 SP - 451 EP - 456 VL - 351 IS - 11-12 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2013.06.011/ DO - 10.1016/j.crma.2013.06.011 LA - en ID - CRMATH_2013__351_11-12_451_0 ER -
%0 Journal Article %A Naumann, Joachim %T On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow %J Comptes Rendus. Mathématique %D 2013 %P 451-456 %V 351 %N 11-12 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2013.06.011/ %R 10.1016/j.crma.2013.06.011 %G en %F CRMATH_2013__351_11-12_451_0
Naumann, Joachim. On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 451-456. doi : 10.1016/j.crma.2013.06.011. http://archive.numdam.org/articles/10.1016/j.crma.2013.06.011/
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