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

A. Tsionskiy, Mikhail Tsionskiy|arXiv (Cornell University)|Jan 8, 2012

Navier-Stokes equation solutions17 references3 citations

TL;DR

This paper proves the existence, uniqueness, and smoothness of solutions to the 3D Navier-Stokes equations for any smooth, compactly supported initial velocity field using Fourier and Laplace transforms, followed by a fixed-point argument in a function space of rapidly decreasing functions. The solution is shown to be globally defined in time and infinitely differentiable, with finite energy for all time, resolving a key aspect of the Navier-Stokes existence problem under smooth initial conditions.

ABSTRACT

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.

Motivation & Objective

• To establish the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations for any smooth, divergence-free initial velocity field.
• To demonstrate that the solution remains smooth and globally defined for all time t ∈ [0, ∞).
• To show that the solution satisfies the energy bound and decays rapidly at infinity, consistent with physical expectations.
• To extend the result to the Euler equations in the limit ν → 0.
• To provide a rigorous analytical framework using integral equations and function space theory to address the Navier-Stokes problem.

Proposed method

• Transform the Navier-Stokes equations into a system of linear PDEs with constant coefficients by moving nonlinear terms to the right-hand side.
• Apply Fourier transforms in space and Laplace transforms in time to derive an integral equation representation of the velocity field.
• Work within the Schwartz space S of rapidly decreasing functions to ensure smoothness and decay properties of the solution.
• Introduce a normalized equivalent integral equation through variable substitution to facilitate fixed-point analysis.
• Apply the Caccioppoli-Banach fixed-point theorem in a complete metric space to prove existence and uniqueness of the solution.
• Use a priori energy estimates to confirm that the total energy remains finite for all time t ≥ 0.

Experimental results

Research questions

• RQ1 Does a smooth, globally defined solution exist for the 3D Navier-Stokes equations with any smooth initial velocity field?
• RQ2 Can the solution be proven unique and infinitely differentiable for all time t ∈ [0, ∞)?
• RQ3 Is the solution stable under small perturbations and does it satisfy the required decay and regularity conditions at infinity?
• RQ4 Can the same method be extended to the Euler equations in the inviscid limit ν → 0?
• RQ5 Does the solution maintain finite energy for all time, consistent with physical constraints?

Key findings

• The solution to the 3D Navier-Stokes Cauchy problem with smooth, compactly supported initial velocity is shown to exist and be unique for all time t ∈ [0, ∞).
• The solution is infinitely differentiable and decays faster than any inverse power of |x| as |x| → ∞.
• The energy of the fluid system remains bounded for all time, with a finite value uniformly over [0, ∞).
• The fixed-point argument based on the Caccioppoli-Banach principle confirms continuous dependence on time and uniqueness in the function space of rapidly decreasing functions.
• The method applies to the Euler equations in the limit ν → 0, preserving the existence and smoothness of the solution.
• The inverse Fourier transform of the constructed solution belongs to the Schwartz space S, confirming its smoothness and rapid decay.