On boundary-driven time-dependent Oseen flows

Abstract

Abstract. We consider the single layer potential associated to the fundamental solution of the time-dependent Oseen system. It is shown this potential belongs to L 2 (0, ∞, H 1 (Ω) 3 ) and to(Ω denotes the complement of a bounded Lipschitz set; V denotes the set of smooth solenoidal functions in H 1 0 (Ω) 3 .) This result means that the usual weak solution of the time-dependent Oseen function with zero initial data and zero body force may be represented by a single layer potential, provided a certain integral equation involving the boundary data may be solved.

Lq$L^q$‐weak solutions to the time‐dependent 3D Oseen system: Decay estimates

“…For the existence result, we refer to [15, Theorem 2.26], which yields a function $$u \in L ^{ 2}_{loc} \bigl (\, [0,T_0),\;H^1(\overline{ \Omega }^c)^3 \,\bigr )$$ such that $$\mbox{div}\, u(t)=0,\;u(t)| \partial \Omega = b(t)$$ for $$t \in (0,T_0)$$ and Equation (1.3) is valid with $$f=0,\; U_0=0$$. According to [15, (2.13)] and [11, Theorem 2.3], this function u additionally belongs to $$L ^{ \infty } \bigl (\, 0,T_0,\;L^2(\overline{ \Omega }^c)^3 \,\bigr )$$, and $$\nabla _xu \in L ^{2 } \bigl (\, 0,T_0,\, L^2(\overline{ \Omega }^c)^9 \,\bigr )$$. Reference [15] reduces its existence result to a solution theory for a boundary integral equation related to the time‐dependent Stokes system.…”

“…In addition ) 3 ) such that div 𝑢(𝑡) = 0, 𝑢(𝑡)|𝜕Ω = 𝑏(𝑡) for 𝑡 ∈ (0, 𝑇 0 ) and Equation (1.3) is valid with 𝑓 = 0, 𝑈 0 = 0. According to [15, (2.13)] and [11,Theorem 2.3], this function 𝑢 additionally belongs to 𝐿 ∞ ( 0, 𝑇 0 , 𝐿 2 (Ω 𝑐 ) 3 ) , and ∇ 𝑥 𝑢 ∈ 𝐿 2 ( 0, 𝑇 0 , 𝐿 2 (Ω 𝑐 ) 9 )…”

Pointwise Decay in Space and in Time for Incompressible Viscous Flow Around a Rigid Body Moving with Constant Velocity

“…Mizumachi [28] studied the spatial asymptotics of strong solutions of (1.4), (1.10), (1.6), but still under rather restrictive assumptions. The results of these two authors were improved in the articles [10], [13] (linear case) and [14] (nonlinear problem (1.9), (1.10), (1.6)), with predecessor papers [6], [7], [8], [9], [11], [12]. As concerns temporal decay of spatial L p -norms of solutions to the Oseen system (1.12) under side conditions (1.10), (1.6) and with f = 0 and b = 0, a basic study is due to Kobayashi, Shibata [25].…”