Existence theorems for regular solutions to the Cauchy problem for the Navier-Stokes equations in ${\mathbb R}^3$

Alexander Shlapunov; Nikolai Tarkhanov

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arXiv · 2009.10530

Existence theorems for regular solutions to the Cauchy problem for the Navier-Stokes equations in ${\mathbb R}^3$

Alexander Shlapunov·Nikolai Tarkhanov

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Abstract

We consider the initial problem for the Navier-Stokes equations over ${\mathbb R}^3 \times [0,T]$ with a positive time $T$ over 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.

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Alexander Shlapunov, Nikolai Tarkhanov. 2020-09-21. Existence theorems for regular solutions to the Cauchy problem for the Navier-Stokes equations in ${\mathbb R}^3$. https://arxiv.org/abs/2009.10530

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