Existence of a weak solution to the Navier–Stokes equation in a general time‐varying domain by the Rothe method

Abstract

SUMMARYWe assume that t is a domain in R 3 , arbitrarily (but continuously) varying for 0 t T . We impose no conditions on smoothness or shape of t . We prove the global in time existence of a weak solution of the Navier-Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition inThe solution satisfies the energy-type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall.

The Navier–Stokes equations with Navier's boundary condition around moving bodies in presence of collisions

“…The same result in a time variable domain Ω t with a prescribed form at each time t was proved by H. Fujita and N. Sauer [1] and it was recently generalized by J. Neustupa [5]. In [1], the boundary of Ω t consisted of a finite number of moving simple closed surfaces of the class C 3 so that the distance of any two of these surfaces was never less than d > 0.…”

“…It is important to mention that the validity of (a4), namely inequality (7), leads to the restriction, that |δ t | is "sufficiently small" in comparison with coefficients ν and γ for t in a certain neighbourhood of the instant of collision t c . Then we can apply Theorem 3.1 and obtain the statement on the global in time existence of a weak solution to the problem (1)- (5).…”

“…(a1) a and ∂ t a are continuous in Q [0,T ] T c , (a2) θ 1 (t) := a(., t) 1,2;Ω t ∈ L 2 (0, T ), (a3) θ 2 (t) := a(., t) − V(., t) 2;Γ t ∈ L 2 (0, T ), (a4) there exist functions θ 3 ∈ L 1 (0, T ), θ 4 ∈ L 2 (0, T ) and θ 5 …”

“…In [1], the boundary of Ω t consisted of a finite number of moving simple closed surfaces of the class C 3 so that the distance of any two of these surfaces was never less than d > 0. In [5], Ω t has an arbitrary shape and smoothness, the assumptions on Ω t involve simulation of collisions of bodies moving in a fluid. The existence and uniqueness of a strong solution in domain Ω t with given smooth moving boundaries was proved by O.A.…”

Boundary characteristic point regularity for Navier–Stokes equations: Blow-up scaling and Petrovskii-type criterion (a formal approach)

“…In 1968, Ladyzhenskaya [39] proved local existence (global for N = 2 and for small initial data if N = 3) and uniqueness of strong solutions for timedependent domains using a different method. See Neustupa [59] for more recent results, references, and other related problems.…”