Abstract
Existence and uniqueness of solutions to the Navier-Stokes equations in dimension two with forces in the space Lq((0, T); W−1,p(Ω)) for p and q in appropriate parameter ranges are proven. The case of spatially measured-valued forces is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions for any 1 < p, q < ∞.
Highlights
In this paper we investigate the following Navier-Stokes system
We introduce the following spaces: Y = [L2(I; V) ∩ L∞(I; H)] + Lq(I; Wp(Ω)), Y = W(0, T ) + Wq,p(0, T )
We prove that yN ∈ C([0, T ]; L4(Ω)) if q = 8
Summary
With focus on low regularity assumptions on the inhomogeneity f. Keywords and phrases: Evolution Navier-Stokes equations, weak solutions, uniqueness clasess, sensitivity analysis, asymptotic stability. For the two-dimensional case the result in [24] guarantees the existence of a solution to (1.1) for f ∈ W 1,∞(I; M(Ω)) This regularity requirement with respect to time is not practical for control theory purposes. The focus of our work is the investigation of (1.1) for f ∈ Lq(I; W−1,p(Ω)) in the case Ω ⊂ R2, and p < 2 For this purpose we require results on the Stokes equation associated to (1.1). The analysis of the Stokes problem associated to (1.1) with forcing functions in the Bochner spaces Lq(I; Lp(Ω)), with 1 < p, q < ∞, has attracted much attention. C([0, T ]; H) holds; see Page 22, Proposition I-2.1 of [22] and Page 143, Remark 3 of [31]
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