On the solvability of a quasistationary problem for the Navier-Stokes equations (Q2747832)

The problem dealt with in this paper is to determine, for all \(t>0\), a bounded domain \(\Omega_t\subset \mathbb R^3\) rotationally symmetrical with respect to the \(x_3\)-axis, and of a rotationally symmetric vector field \(\vec{v}(x,t)\) and a rotationally symmetric function \(p(x,t)\) satisfying NEWLINE\[NEWLINE-\nabla^2\vec{v}+\nabla p=0,\quad \nabla \vec{v}=0,\quad x\in \Omega_t,\;t>0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE T(\vec{v},p)\vec{n}=\left(H+\frac{\gamma^2\left|\vec{h}(x)\right|^2}{2I_t^2}\right)\vec{n},\quad x\in \Gamma_t=\partial\Omega_t,\;\Omega_0=\Omega,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \int_{\Omega_t} \vec{v} dx=\int_{\Omega_t} \left[\vec{v}\times \vec{x}\right] dx=0,NEWLINE\]NEWLINE where \(\Omega\) is a given bounded rotationally symmetric domain, \(T(\vec{v},p)=-pI+S(\vec{v})\), \(S(\vec{v})=\nabla \vec{v}+\left(\nabla\vec{v}\right)^T\), \(H\) is the doubled mean curvature of \(\Gamma_t\), \(\gamma\) is a positive constant, \(\vec{h}(x)=(-x_2,x_1,0)\) and NEWLINE\[NEWLINE I_t=\int_{\Omega_t} \left|\vec{h}(x)\right|^2 dx. NEWLINE\]NEWLINE These equations describe the evolution of an isolated liquid mass in a quasistationary approximation and were stated by \textit{V. V. Pukhnachov} [Prikl. Math. Mech. 62, 1002-1013 (1998)]. NEWLINENEWLINENEWLINEThe main theorem of the present paper asserts that, if \(\Gamma=\partial\Omega\) is given by the equation NEWLINE\[NEWLINE \left|x\right|=R(\omega,0),\;\omega=x/\left|x\right|, NEWLINE\]NEWLINE with \(R(.,0)\) rotationally symmetric belonging to the Hölder space \(C^{l+3}(S_1)\) \((l\notin \mathbb N)\), and if \(\|R(.,0)-R_{\infty}\|_{C^{l+3}(S_1)}\) and \(\gamma\) are small enough, then the above problem has a unique rotationally symmetric solution \((R,\vec{v},p)\) defined for all \(t>0\), with \(R\in C^{l+3}(S_1)\), \(\vec{v}\in C^{l+2}(\Omega_t)\), \(p\in C^{l+1}(\Omega_t)\), with norms controlled by \(\left\|R(.,0)-R_{\infty}\right\|_{C^{l+3}(S_1)} e^{-bt}\) where \(b>0\).