Time-dependent incompressible viscous flows around a rigid body: Estimates of spatial decay independent of boundary conditions

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

“…In the next theorem, we introduce a pressure Π associated with the velocity part U of a weak solution to the Oseen system ($$ \lambda =0$$) or the Oseen resolvent system ($$\lambda \ne 0$$) in $$\mathbb {R}^3$$. The case $$\lambda \ne 0$$ is included in view of an application in [8]. Theorem Let $$A \subset \mathbb {R}^3$$ be open, $$q \in (1, \infty ),\; \lambda \in \mathbb {C} ,\; U \in W^{1,q}_{loc}(A)^3$$ with $$$$\begin{eqnarray} \int _{ A}\bigl (\, \nabla U \cdot \nabla \vartheta + (\tau \, \partial _1U +\lambda \,U -F )\cdot \vartheta \,\bigr ) \, dx =0 \;\; \mbox{for}\;\; \vartheta \in C ^{ \infty } _{0, \sigma }(A),\;\; \mbox{div}\, U=0.$$…”

“…Again due to the equation 𝑊 = ∑ 𝑘 0 𝑗=1 (1 − 𝜑) 𝑉 (𝑗) and the integrability properties of (1 − 𝜑) 𝑉 (𝑗) In the next theorem, we introduce a pressure Π associated with the velocity part 𝑈 of a weak solution to the Oseen system (𝜆 = 0) or the Oseen resolvent system (𝜆 ≠ 0) in ℝ 3 . The case 𝜆 ≠ 0 is included in view of an application in [8].…”

“…The case $$m_0>1$$ or $$ n_0>1$$ and the distinction between the functions $$G ^{(l)}$$ on $$ \overline{ B_{S_0}}^c \times (0, \infty )$$ with $$u(t)| \overline{ B_{S_0}}^c= \sum _{l = 1}^{m_0}G ^{(l)} (t)$$ on the one hand and $$u| \Omega _{S_0}\times (0,T_0)$$ on the other one are introduced in order to take account of some technical difficulties arising when the theory in this work is applied to a nonlinear problem in [8] and when it is used in the proof of Theorem 6.1, and in order to avoid any assumptions on $$\partial \Omega$$ stronger than Lipschitz regularity. Any reader who wants to avoid these technicalities may consider the case that $$\partial \Omega$$ is smooth, $$u \in C^0 \bigl (\, [0, T_0),\, L^{q_0}(\overline{ ...$…”

“…This work is the basis of two further articles. In [8], we start from the representation formula (5.21), obtaining convergence estimates for $$|x|\rightarrow \infty$$ of L 2 ‐strong solutions to a generalization of the nonlinear system (1.2). The focus in this latter work is on improving the decay results in [32], [36] and [16] for nonlinear problems without requiring specific boundary conditions.…”

“…In fact, we fix numbers 𝑆 0 , 𝑅 0 ∈ (0, ∞) with Ω ⊂ 𝐵 𝑆 0 and 𝑆 0 < 𝑅 0 , abbreviate 𝐴 𝑅 0 ,𝑆 0 ∶= 𝐵 𝑅 0 ∖𝐵 𝑆 0 , and require there are parameters 𝛾 1 , 𝛾 2 , 𝛾 3 ∈ [0, ∞], 𝑞 ∈ (1, ∞) such that the restriction 𝑢|𝐴 𝑅 0 ,𝑆 0 × (0, 𝑇 0 ) belongs to 𝐿 𝛾 1 ( 0, 𝑇 0 , 𝐿 𝑞 (𝐴 𝑅 0 ,𝑆 0 ) 3 ) , ∇ 𝑥 𝑢|𝐴 𝑅 0 ,𝑆 0 × (0, 𝑇 0 ) to 𝐿 𝛾 2 ( 0, 𝑇 0 , 𝐿 𝑞 (𝐴 𝑅 0 ,𝑆 0 ) 3 ) , and 𝑓|𝐴 𝑅 0 ,𝑆 0 × (0, 𝑇 0 ) to 𝐿 𝛾 3 ( 0, 𝑇 0 , 𝐿 𝑞 (𝐴 𝑅 0 ,𝑆 0 ) 3 ) . Actually our assumptions are somewhat more general (see at the beginning of Section 5 and Theorem 5.11), in view of applications in [8] to a nonlinear problem with (1.2) as special case. But with the preceding conditions, all the main difficulties of our proofs would already be present.…”