Dynamics of the closed hypersurfaces in central force fields (Q6540605)

The paper deals with a novel geometric flow which simulates the motion of closed surfaces with friction in central force fields and generalizes geometric flows considered earlier in [\textit{O.C. Schnürer} and \textit{K. Smoczyk}, J. Reine. Angew. Math. 550, 77--95 (2002; Zbl 1019.53033; \textit{P.G. Le Floch} and \textit{K. Smoczyk}, J. Math. Pures Appl. 90, 591--614 (2008; Zbl 1159.53025); \textit{T. Notz}, Comm. Pure. Appl.Math. 66, 790--819 (2013; Zbl 1263.53064); \textit{C.Y. Shao}, Arch. Ration. Mech. Anal. 243, 501--557 (2022; Zbl 1481.35068)].\N\NThe motion is described by a smooth family of immersions \(F: [0, T ) \times\mathbb S^2 \to \mathbb R^3\), whose evolution in time \(t\) is governed by the equation \N\[\N\partial_{tt}F +a F_t - \mathrm{div} \left( b(x)\nabla F_t\right) = \frac{d\mu_t}{d\mu} \left( - H(F) + \varphi\langle F, \nu(F)\rangle +\frac{\rho}{\mathrm{Vol}(F)}\right) \nu (F), \N\]\Nwhere \(a\) and \(\rho\) are non-negative constants, \(b:\mathbb S^2 \to \mathbb R_+\) is a non-negative smooth function, \(d\mu\) is a reference measure on \(\mathbb S^2\), \(d\mu_t\) stands for the induced area measure of the evolving surface, \(H\) is the mean curvature of the surface with respect to its outer unit normal \(\nu\), \(\mathrm{Vol}(F)\) is the volume enclosed by the surface, \(\langle,\rangle\) denotes the inner product in \(\mathbb R^3\), and \(\varphi\) is a specific function depending on \(\vert F\vert\) only. The equation is equipped with the initial data \(F(0,x)=F_0(x)\), \(F_t(0,x)=F_1(x)\).\N\NThe authors analyze the dynamical stability of solutions for the equation in question and demonstrate that if a special global smooth solution exists, then its behavior is stable in the sense that a sufficiently small (with respect to the Sobolev space \(H^s(\mathbb S^2)\)) perturbation of initial data results in a solution which converges smoothly to the given one.