The Navier-Stokes equations in the weak- $L^n$ space with time-dependent external force

Abstract

We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension n ≥ 3. We give conditions on the external force sufficient for the unique existence of small solutions in the weak-L n space bounded for all time.In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak-L n space with time-independent external force.

On the uniqueness for sub-critical quasi-geostrophic equations

“…The estimates below can be found in [3] (see also [15] for (3.3) in the case γ = 1 and n ≥ 3), but we include their proofs for the convenience of the reader. Lemma 3.2.…”

Time-Periodic Solutions to the Navier-Stokes Equations

“…Galdi and Sohr showed existence of strong solutions for sufficiently "small" data. These solutions are more regular than the mild solutions established by Yamazaki in [61], whence, since both results require a restriction on the "size" of the data, their result constituted an important improvement. Similar to [61], Galdi and Sohr introduced a function space that contains…”

“…Observe that for a three-dimensional exterior domain Ω the space X (Ω) is slightly larger than L 3 (Ω) 3 , but smaller than the space L 3,∞ (Ω) 3 used by Yamazaki in [61]. Galdi and Sohr established existence of a time-periodic solution in L ∞ 0, T ; X (Ω) , provided, as already mentioned, the data is sufficiently "small".…”

“…A few years later, Yamazaki was able to employ in [61] the Kozono-Nakao method to a three-dimensional exterior domain with u ∞ = 0. The new idea of Yamazaki was to use the space L ∞ 0, T ; L σ (Ω).…”

On Stability of Steady Circular Flows in a Two-Dimensional Exterior Disk

“…When Ω ⊂ R n is an n-dimensional exterior domain we also recall that the unique existence of small global solutions in L ∞ (0, ∞; L n,∞ σ (Ω)) for given small initial data in L n,∞ σ (Ω) is proved by Kozono and Yamzaki [31] for n ≥ 2, while its local stability in the framework of the Lorentz space is achieved for the case n ≥ 3 by Yamazaki [45]. Contrary to the higher dimensional cases, less is known so far for the case of two-dimensional exterior domains.…”