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Master's Dissertation
DOI
10.11606/D.55.2017.tde-16112017-160410
Document
Author
Full name
Alexandre do Nascimento Oliveira Sousa
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Carvalho, Alexandre Nolasco de (President)
Pereira, Antonio Luiz
Planas, Gabriela Del Valle
Soares, Sérgio Henrique Monari
Title in Portuguese
Equações de Navier-Stokes: o problema de um milhão de dólares sob o ponto de vista da continuação de soluções
Keywords in Portuguese
Boa colocação local e global
Continuação
Crescimento crítico
Navier-Stokes
Abstract in Portuguese

Neste trabalho consideramos o problema de Navier-Stokes em RN

ut = Δu — ∇π + f (t) — (u .∇)u,   x∈ Ω
div(u) = 0,    x ∈ Ω
u = 0,    x ∈ ∂ Ω
u(0, x) = u0 (x),

onde u0 ∈ LN (Ω)N e Ω é um subconjunto aberto, limitado e suave de RN. Provamos que o problema acima é localmente bem colocado e fornecemos condições para obter que estas soluções existem para todo t ≥ 0. Utilizamos técnicas de equações parabólicas semilineares considerando não linearidades com crescimento crítico desenvolvidas em (ARRIETA; CARVALHO, 1999).

Title in English
Navier Stokes equations: The one million dollar problem from the point of view of continuation of solutions
Keywords in English
Continuation
Critical growth
Local and global well posedness
Navier-Stokes
Abstract in English
In this work we we consider the Navier-Stokes problem on RN
ut = Δu — ∇π + f (t) — (u .∇)u,   x∈ Ω
div(u) = 0,    x ∈ Ω
u = 0,    x ∈ ∂ Ω
u(0, x) = u0 (x),
where u0 ∈ LN (Ω)N and Ω is an open, bounded and smooth subset of RN. We prove that the above problem is locally well posed and give conditions to obtain that these solutions exist for all t ≥ 0. We used techniques of semilinear parabolic equations considering nonlinearities with critical grouth developed in (ARRIETA; CARVALHO, 1999).
 
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Publishing Date
2017-11-16
 
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