Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations

Alexander Shlapunov; Nikolai Tarkhanov

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arXiv · 2007.14911

Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations

Alexander Shlapunov·Nikolai Tarkhanov

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Abstract

We consider the initial value problem for the Navier-Stokes equations over $R^{3} \times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on $R^{3}$ with functions on the three-dimensional torus $T^{3}$, we prove that the problem induces an open both injective and surjective mapping of specially constructed function spaces of Bochner-Sobolev type. This gives a uniqueness and existence theorem for regular solutions to the Navier-Stokes equations. Our techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics.

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BibTeX RIS

Alexander Shlapunov, Nikolai Tarkhanov. 2020-07-29. Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations. https://arxiv.org/abs/2007.14911

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