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M. Tsionskiy

Publications and source records attributed to M. Tsionskiy.

7 recordsLinked to original sources
arXiv2022

Existence, uniqueness and smoothness of a solution for 3D Navier-Stokes equations with any smooth initial velocity. A priori estimate of this solution

Solutions of the Navier-Stokes and Euler equations with initial conditions for 2D and 3D cases were obtained in the form of converging series, by an analytical iterative method using Fourier and Laplace transforms \cite{TT10,TT11}. There the solutions are infinitely differentiable functions, and for several combinations of parameters numerical results are presented. This article provides a detailed proof of the existence, uniqueness and smoothness of the solution of the Cauchy problem for the 3D Navier-Stokes equations with any smooth initial velocity. When the viscosity tends to zero, this proof applies also to the Euler equations. A priori estimate of this solution is presented.

math.AP↗
arXiv2013

Existence, uniqueness and smoothness of solution for 3D Navier-Stokes equations with any smooth initial velocity

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.

math.AP↗
arXiv2012

An effective approach to the solution of a system of nonlinear differential equations in partial derivatives

There are few approaches to the solution of a system of nonlinear differential equations in partial derivatives, for example $\cite{NK87} - \cite{EK98}$. In our paper we propose an approach that was used to solve the Navier-Stokes equations in three dimensional space. This solution is described in details in article "Existence, uniqueness and smoothness of solution for 3D Navier-Stokes equations with any smooth initial velocity" $\cite{TT12}$. The authors expect that it can be successfully applied to other systems of nonlinear differential equations in partial derivatives.

math.AP↗
arXiv2011

The Cauchy problem for the 3D Navier - Stokes equations. New approach to the solution and its justification

Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of solutions for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed that the Navier-Stokes equations in three space dimensions 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 Bertozzi and Majda, 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 convergence series form by analytical iterative method using Fourier and Laplace transforms in paper $\cite{TT10}$. These solutions were received in a form of infinitely differentiable functions, and that allows us to analyze all aspects of the problem on a much deeper level and with more details. Also such smooth solutions satisfy the conditions required in $\cite{CF06}$ for the problem of Navier-Stokes equations. For several combinations of problem parameters numerical results were obtained and presented as graphs $\cite{TT10}$,$\;\cite{TT11}$. This paper describes detailed proof of convergence of the analitical iterative method for solution of the Cauchy problem for the 3D Navier - Stokes equations. The convergence is shown for wide ranges of the problem's parameters. Estimated formula for the border of convergence area of the iterative process in the space of system parameters is obtained. Also we have provided justification of the analytical iterative method solution for Cauchy problem for the 3D Navier-Stokes equations.

math.AP↗
arXiv2011

Research of convergence of the iterative method for solution of the Cauchy problem for the Navier - Stokes equations based on estimated formula

Solution of the Navier-Stokes equations with initial conditions (Cauchy problem) for 2D and 3D cases was obtained in the convergence series form by iterative method using Fourier and Laplace transforms in paper $\cite{TT02}$. For several combinations of problem parameters numerical results were obtained and presented as graphs. Estimated formula for the border of the parameter area of convergence of the iterative method was obtained in paper $\cite{TT03}$. This paper describes numerical proof of convergence of the iterative method for solution of the Cauchy problem for the Navier - Stokes equations. Usage of estimated formula for the border of the parameter area of convergence of the iterative method is shown for wide ranges of the problem's parameters.

math.AP↗
arXiv2010

Solution of the Cauchy problem for the Navier - Stokes and Euler equations

Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed that the Navier-Stokes equations in three space dimensions have a weak solution. Scheffer and Shnirelman obtained weak solution of the Euler equations with compact support in spacetime. Caffarelli-Kohn-Nirenberg improved Scheffer's results, and F.-H. Lin simplified the proof of the results of J. Leray. Many problems and conjectures about the behavior of solutions of the Euler and Navier-Stokes equations are described in the book of Bertozzi and Majda or Constantin. Solutions of the Navier-Stokes and Euler equations with initial conditions (Cauchy problem) for two and three dimensions are obtained in the convergence series form by the iterative method using the Fourier and Laplace transforms in this paper. For several combinations of problem parameters numerical results were obtained and presented as graphs.

math.AP↗