On the existence and uniqueness of the stationary solution to equations of natural convection with boundary data in L ²

Abstract

By M. S a n t o s d a R o c h a 1 , M a r k o A. R o j a s-M e d a r 1 a n d M. D r i n a R o j a s-M e d a r 2

An alternative theoretical approach for the derivation of analytic and numerical solutions to thermal Marangoni flows

“…Most of these theoretical studies have been directed to the case in which the driving force is a volume force (as implicit in the problem statement per se). For the case of buoyancy flow, as an example, the interested reader may consider [61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79]. Surprisingly, however, no results specific for the case of Marangoni flow have been obtained or presented so far.…”

A note on regularity and uniqueness of natural convection with effects of viscous dissipation in 3D open channels

“…has the Carathéodory property (see [ By [8,Theorem 12.10] we deduce that the Němytskiȋ operator N is a continuous operator from L r (Ω) into L r (Ω) (recall (8)). Since we assume θ n → θ in W 2−ε,r (Ω) and W 2−ε,r (Ω) ֒→ L r (Ω), we get (25).…”

“…Coefficient α 1 reflects the dissipation effect, which is omitted frequently in many mathematical models [19,24,25,26]. Otherwise, taking into account the dissipative term α 1 νe(u) : e(u) with quadratic growth of the gradient, the equations ( 1)-( 7) represent the elliptic system with strong nonlinearities without appropriate general existence and regularity theory.…”

“…This makes the present problem quite different than in the case of Dirichlet-type boundary conditions for u on the whole boundary frequently studied in the literature (cf. [6,19,22,23,25,26]).…”

Soluções ultra fracas, fracas, brandas e fortes para equações do tipo Navier-Stokes

“…No sistema linearizado, ele aplicou os resultados obtidos por Conca [21], [22] e definiu uma contração que permite aplicar o Teorema do ponto fixo de Banach. A extensão para as equações de Boussinesq foram dadas por Santos et al [114] os quais analisaram o sistema (2.1), mas com condições de fronteira do tipo Dirichlet não somente para a velocidade u mas também para a temperatura θ. Em outros modelos da Hydrodinâmica, Guillén-González et al [ 5 1 ]e s t udaram o conceito de solução ultra fraca para o sistema de Stokes hidrostático com condições de Neumann não suaves sobre a superfície rígida e condições do tipo Dirichlet sobre o resto da fronteira. Em [115], são consideradas as equações estacionárias de Navier-Stokes com dados de fronteira em algum espaço W 1− 1 q ,q (Γ).…”