Abstract
AbstractWe prove that a strong solutionuto the Navier-Stokes equations on (0,T) can be extended if eitheru∈Lθ(0,T;U˙∞,1/θ,∞−α$\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$) for 2/θ+α= 1, 0 <α< 1 oru∈L2(0,T;V˙∞,∞,20$\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$), whereU˙p,β,σs$\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$andV˙p,q,θs$\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$are Banach spaces that may be larger than the homogeneous Besov spaceB˙p,qs$\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$. Our method is based on a bilinear estimate and a logarithmic interpolation inequality.
Highlights
We prove that a strong solution u to the Navier-Stokes equations on (, T) can be extended if either u ∈ Lθ(, T; U ∞−α, /θ,∞) for /θ + α =, < α < or u ∈ L (, T; V ∞,∞, ), where Ups,β,σ and Vps,q,θ are Banach spaces that may be larger than the homogeneous
The aim of this paper is to improve the extension criterion (1.2) to the Navier-Stokes equations by means of Banach spaces which are larger than B −∞α,∞ in the same way that the condition (1.5) was relaxed to (1.6)
We introduce Banach spaces Vps,q,θ and Ups,β,σ which are larger than the homogeneous Besov spaces
Summary
The aim of this paper is to improve the extension criterion (1.2) to the Navier-Stokes equations by means of Banach spaces which are larger than B −∞α,∞ in the same way that the condition (1.5) was relaxed to (1.6). Ups ,β,σ is a Banach space introduced by De nition 2.2 and has the following continuous embeddings: B −∞α,∞ ⊂ V ∞−α,∞,θ ⊂ U ∞−α, /θ,∞ We prove that Ups,β,σ is the weakest normed space that satis es such a logarithmic interpolation inequality.
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