Concerning the Navier-Stokes problem

Alexander G. Ramm Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA.; ramm@math.ksu.edu Received: 16 June 2020; Accepted: 27 August 2020; Published: 31 August 2020. Abstract: The problem discussed is the Navier-Stokes problem (NSP) in R3. Uniqueness of its solution is proved in a suitable space X. No smallness assumptions are used in the proof. Existence of the solution in X is proved for t ∈ [0, T], where T > 0 is sufficiently small. Existence of the solution in X is proved for t ∈ [0, ∞) if some a priori estimate of the solution holds.


Introduction
T here is a large literature on the Navier-Stokes problem (NSP) in R 3 ( see [1], Chapter 5) and references therein). The global existence and uniqueness of a solution in R 3 was not proved. The goal of this paper is to prove uniqueness of the solution to NSP in a suitable functional space. No smallness assumptions are used in our proof.
The NS problem in R 3 consists of solving the equations Vector-functions v = v(x, t), f = f (x, t) and the scalar function p = p(x, t) decay as |x| → ∞ uniformly with respect to t ∈ R + := [0, ∞), v := v t , ν = const > 0 is the viscosity coefficient, the velocity v and the pressure p are unknown, v 0 and f are known, ∇ · v 0 = 0. Equations (1) describe viscous incompressible fluid with density ρ = 1.
We use the integral equation Equation (2) is equivalent to (1), see [2]. Formula for the tensor G is derived in [2], see also [1], p.41. The term F = F(x, t) depends only on the data f and v 0 (see equation (18) in [2] or formula (5.42) in [1]): ( We assume throughout that f and v 0 are such that F is bounded in all the norms we use. Let X be the Banach space of continuous functions with respect to t with the norm where t > 0, andṽ := (2π) −3 R 3 v(x, t)e −iξ·x dx. Taking the Fourier transform of (2) yields where denotes the convolution in R 3 and for brevity we omitted the tensorial indices: instead ofG mpṽj (iξ j )ṽ p , where one sums up over the repeated indices, we wroteG(ξ, t − s)ṽ (iξṽ). From formula (5.9) in [1] it follows that |G| ≤ ce −νξ 2 (t−s) .
By c > 0 we denote various constants independent of t and ξ. Let S(R 3 × R + ) and S(R 3 ) be the L.Schwartz spaces. Our results are: Theorem 1. Assume that f and v 0 are in S(R 3 × R + ) and S(R 3 ) respectively. Then there is at most one solution to NSP in X.
where c a > 0 is a constant depending only on the data.

Lemma 1. Inequality (13) has only the trivial non-negative solution u(t) = 0.
Proof of Lemma 1. Denote Then The kernel K(t, s) > 0 is weakly singular. Any solution q ≥ 0 to (14) satisfies the estimate 0 ≤ q ≤ Q, where Q ≥ 0 solves the Volterra equation This equation has only the trivial solution Q = 0. Lemma 1 is proved.
Proof of Theorem 2. From (5) after multiplying by 1 + |ξ|, integrating over R 3 and using calculations similar to the ones in equation (12), one gets The corresponding equation U = A 1 U is a linear Volterra integral equation. It has a unique solution defined for all t ≥ 0, and 0 ≤ u(t) ≤ U(t). Theorem 3 is proved.