Existence theorems for regular solutions to the Cauchy problem for the Navier-Stokes equations in R^3

Abstract: We consider the Cauchy problem for the Navier-Stokes equations over ${\mathbb R}^3 \times [0,T]$ with a positive time $T$ over a specially constructed scale of function spaces of Bochner-Sobolev type. We prove that the problem induces an open both injective and surjective mapping of each space of the scale. In particular, intersection of these classes gives a uniqueness and existence theorem for smooth solutions to the Navier-Stokes equations for smooth data with a prescribed asymptotic behaviour at the infinity with respect to the time and the space variables. Actually, we propose the following modified scheme of the proof of the existence theorem, based on apriori estimates and operator approach in Banach spaces:

1. We prove that the Navier-Stokes equations induce continuous injective OPEN mapping between the chosen Banach spaces.

2. Next, the standard topological arguments immediately imply that a nonempty open connected set in a topological vector space coincides with the space itself if and only if the set is closed. This reduces the proof of the existence theorem to an $L^\mathfrak{s} ([0,T], L^\mathfrak{r} ({\mathbb R^3}))$ a priori estimate for the INVERSE IMAGE OF PRECOMPACT SETS in the target Banach space where $\mathfrak{s}$, $\mathfrak{s}$ are Ladyzhenskaya-Prodi-Serrin numbers satisfying $2/\mathfrak{s} + 3/\mathfrak{r} = 1$ and $\mathfrak{r} > 3$. In this way we avoid proving a GLOBAL $L^\mathfrak{s} ([0,T], L^\mathfrak{r} ({\mathbb R^3}))$ a priori estimate.

3. To prove the weak $L^\mathfrak{s} ([0,T], L^\mathfrak{r} ({\mathbb R^3}))$ a priori estimate with $\mathfrak{r} > 3$ we calculate precisely the excess between the left hand side and the right hand side of the corresponding energy inequality, that equals to $2r$ when expressed in terms of the Lebesgue integrability index $r$. Then we operate with absolutely convergent series involving Lebesgue norms that gives the possibility to group together summands in a suitable way, using the energy type inequalities, interpolation inequalities and matching the asymptotic behaviour in order to exclude the unbounded sequences in the inverse image of a precompact set.

An early version of the paper is uploaded on arxiv.org: arxiv.org/abs/2009.10530 A similar approach can be used for investigation of the Navier-Stokes equations in the periodic setting: arxiv.org/abs/2007.14911

analysis of PDEs

Audience: researchers in the topic

Mathematical models and integration methods