2012/01/08 by A. Tsionskiy, Tsionskiy, A., Mikhail Tsionskiy +2 · 1 citation
Engineering · Mathematics · #35Q30 #76D05 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #math.AP #msc:35Q30 #msc:76D05
paper · pdf · doi:10.48550/arxiv.1201.1609
18 pages. arXiv admin note: text overlap with arXiv:1101.0801
openalex publication_date 2012/01/08 · arxiv created 2013/09/01 · arxiv updated 2013/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Different authors had received a lot of results regarding the Euler and Navier-Stokes equations. Existence and smoothness of solution for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed that the Navier-Stokes equations in three dimensional space have a weak solution. Scheffer, and Shnirelman, obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli, Kohn and Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about behavior of weak solutions of the Euler and Navier-Stokes equations are described in the books of Ladyzhenskaya, Bertozzi and Majda, Temam, Constantin or Lemarié-Rieusset. Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for 2D and 3D cases were obtained in the converging series form by analytical iterative method using Fourier and Laplace transforms in a paper by Tsionskiy. These solutions were received as infinitely differentiable functions. That allowed us to analyze essential aspects of the problem on a much deeper level and with more details. For several combinations of problem parameters numerical results were obtained and presented as graphs by Tsionskiy. This paper describes detailed proof of existence and uniqueness of the solution of the Cauchy problem for the 3D Navier-Stokes equations with any smooth initial velocity. This solution satisfies the conditions required in Fefferman for the problem of Navier-Stokes equations. When viscosity tends to zero this proof is correct for the Euler equations also.