The Oseen and Navier-Stokes equations in a non-solenoidal framework
Rodríguez Bellido, María Ángeles
|Department||Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico|
|Abstract||The very weak solution for the Stokes, Oseen and Navier-Stokes equations has been studied by several authors in the last decades in domains of Rn, n ≥ 2. The authors studied the Oseen and Navier-Stokes problems assuming a ...
The very weak solution for the Stokes, Oseen and Navier-Stokes equations has been studied by several authors in the last decades in domains of Rn, n ≥ 2. The authors studied the Oseen and Navier-Stokes problems assuming a solenoidal convective velocity in a bounded domain Ω ⊂ R3 of class C1,1 for v ∈ Ls (Ω) for s ≥ 3 in some previous papers. The results for the Navier-Stokes equations were obtained by using a fixed-point argument over the Oseen problem. These results improve those of Galdi et al. , Farwig et al. and Kim for the Navier-Stokes equations, because a less regular domain Ω ⊂ R3 and more general hypothesis on the data are considered. In particular, the external forces must not be small. In this work, existence of weak, strong, regularised and very weak solution for the Oseen problem are proved, mainly assuming that v ∈ L3(Ω) and its divergence ∇ · v is sufficiently small in the W−1,3(Ω)-norm. In this sense, one extends the analysis made by the authors for a given solenoidal v in some previous papers. As a consequence, the existence of very weak solution for the Navier-Stokes problem (u, π) ∈ L3(Ω) × W−1,3(Ω)/R for a non-zero divergence condition is obtained in the 3D case.
|Funding agencies||Ministerio de Ciencia e Innovación (MICIN). España
Ministerio de Economía y Competitividad (MINECO). España
|Citation||Amrouche, C. y Rodríguez Bellido, M.Á. (2014). The Oseen and Navier-Stokes equations in a non-solenoidal framework. Mathematical Methods in the Applied Sciences, 39 (17), 5066-5090.|