A Time-Periodic Bifurcation Theorem and its Application to Navier-Stokes Flow Past an Obstacle (Mathematical Analysis of Viscous Incompressible Fluid)
Giovanni Paolo Galdi · Institutional Repositories DataBase (IRDB) · 2015
We show an abstract time-periodic bifurcation theorem in Banach spaces.The key point as well as the novelty of the method is to split the original evolution equation into two different coupled equations, one for the time-average of the sought solution and the other for the "purely periodic"' component.This approach may be particularly useful in studying physical phenomena occurring in unbounded spatial regions.Actually, we furnish a significant application of the theorem, by providing suffcient conditions for time-periodic bifurcation from a steady-state flow of a Navier-Stokes liquid past a three-dimensional obstacle.Also, we denote by $R_{*}>0$ a number such that the closure of $\Omega_{0}$ is strictly contained in $\{x\in \mathbb{R}^{3}:(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{\frac{1}{2}}<R_{*}\}.$For $R\geq R_{*}$ , we let $\Omega_{R}=\Omega\cap\{x\in \mathbb{R}^{2}:(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{\frac{1}{2}}<R\}, \Omega^{R}=\Omega-\overline{\Omega_{R}},$ where the bar denotes closure.We set $u_{t}:=\partial u/\partial t,$ $\partial_{1}u:=\partial u/\partial x_{1}$ , and indicate by $D^{2}u$ the matrix of the second derivatives of $u.$For an open and connected, stand for the usual Lebesgue and Sobolev classes, respectively, of real or complex functions.(2) Norms in $L^{q}(A)$ and $W^{m,q}(A)$ are indicated by $\Vert.\Vert_{q,A}$and $\Vert.\Vert_{m,q,A}$ .The scalar product of functions $u,$ $v\in L^{2}(A)$ will be denoted by $\langle u,$ $v\rangle_{A}$ .In the above notation, the symbol $A$ will be omitted, unless confusion arises.As customary, for $q\in[1, \infty]$ we let $q'=q/(q-1)$ be its H\"older conjugate.By $D^{1,q}(\Omega)$ , $1<q<\infty$ , we denote the space of (equivalence classes of) functions $u$ such that $\Vert abla u\Vert_{q}<\infty$ .Moreover, setting, $\mathcal{D}(\Omega) :=\{u\in C_{0}^{\infty}(\Omega) : divu=0\}$ we let $\mathcal{D}_{0}^{1,2}(\Omega)$ be the completion of $\mathcal{D}(\Omega)$ in the norm $\Vert abla(\cdot)\Vert_{2}$ , and set $Z^{2,2}(\Omega):=W^{2,2}(\Omega)\cap \mathcal{D}_{0}^{1,2}(\Omega)$ .Furthermore, we denote by $H_{q}(\Omega)$ , $1<q<\infty,$ $(H_{2}(\Omega)\equiv H(\Omega))$ the completion of $\mathcal{D}(\Omega)$ in the norm $L^{q}(\Omega)$ and let $P_{q}$ be the (Helmholtz) pro- jection from $L^{q}(\Omega)$ onto $H_{q}(\Omega)$ .$P_{q}$ is independent of $q$ [ $6$ , \S III.I], so that we shall simply denote it by P. We define $X^{2,\frac{4}{3}}(\Omega):=\{u:u\in L^{4}(\Omega)\cap D^{1,2}(\Omega)\cap D^{1,\frac{12}{5}}(\Omega), \partial_{1}u, D^{2}u\in L^{\frac{4}{3}}(\Omega)\}$ and $X_{0}^{2,\frac{4}{3}}(\Omega)$ $:=\{u\in X^{2,\frac{4}{3}}(\Omega)$ : $divu=0,$ $u|_{\partial\Omega}=0\}.$As is known, $X^{2,q}(\Omega)$ and $X_{0}^{2,q}(\Omega)$ become Banach spaces when endowed with the "natural norm $\Vert u\Vert_{x^{2},\#}:=\Vert u\Vert_{4}+\Vert abla u\Vert_{2}+\Vert abla u\Vert_{\frac{12}{5}}+\Vert\partial_{1}u\Vert_{\frac{4}{3}}+\Vert D^{2}u\Vert_{\frac{4}{3}}$ ; see [9].(2) $We$ shall use the same font style to denote scalar, vector and tensor function spaces.