Singularities of certain finite energy solutions to the Navier-Stokes system

We continue and supplement studies from [G. Karch and X. Zheng, Discrete Contin. Dyn. Syst. 35 (2015), 3039-3057] on solutions to the three dimensional incompressible Navier-Stokes system which are regular outside a curve in \begin{document}$ \big(\gamma(t), t\big)\in \mathbb{R}^3\times [0, \infty) $\end{document} and singular on it. We revisit some of the existence results as well as some of the asymptotic estimates obtained in that work in order prove that those solutions belongs to the space \begin{document}$ C\big([0, \infty), L^2( \mathbb{R}^3)^3\big) $\end{document} .


Introduction.
Statement of the main result. In this paper, we extend results from the work [9] on singular solutions of the Navier-Stokes system ∂ t u − ∆u + (u · ∇)u + ∇p = 0, div u = 0 (1) in the three dimensional space. Here, the vector u = u 1 (x, t), u 2 (x, t), u 3 (x, t) denotes the unknown velocity field and the scalar function p = p(x, t) stands for the unknown pressure. The main result of this paper is contained in the following theorem on the existence of singular solutions in the energy space L 2 (R 3 ) 3 to the Navier-Stokes system. Theorem 1.1. Consider a curve Γ = (γ(t), t) ∈ R 3 × R + : t ≥ 0 , where γ : R + → R 3 is an arbitrary Hölder continuous function with a Hölder exponent α ∈ ( 1 2 , 1]. There exists a divergence-free vector field u ∈ C [0, ∞), L 2 (R 3 ) 3 and the corresponding pressure p = p(x, t) which are regular and solve system (1) pointwise for all (x, t) ∈ (R 3 × R + )\Γ and which are singular on the curve Γ.
Solutions to system (1) with singularities on the curve Γ were constructed in paper [9] and the main contribution of this work is to show that those solutions have a finite energy and belong to the space C [0, ∞), L 2 (R 3 ) 3 .
The problem considered in [9] is a time dependent generalized version of the Landau stationary problem [11] which models a fluid into which a thin pipe discharges a jet oriented along the positive part of the x-axis. The solution studied by Landau seems to have been found originally by Slëzkin [19] and a translation from the Russian of Slëzkin's original notes on this solution can be found as appendices in [5]. The stationary solution in question, to be called the Slëzkin-Landau solution , Q c is defined for each real constant |c| > 1 by the following formulas where |x| = x 2 1 + x 2 2 + x 2 3 . It is immediate to check that these locally integrable functions are in C ∞ R 3 \{0} and V c 1 , V c 2 , V c 3 are homogeneous of degree −1. One can prove (see e.g. [1, p. 209] and [2,Prop. 2.1]) that V c , Q c satisfies in the sense of distributions the following stationary Navier-Stokes system with a singular force where e 1 = (1, 0, 0), the symbol δ 0 denotes as usual the Dirac measure at the origin, and We will be using the following main property of this relation: lim |c|→∞ κ(c) = 0. The authors of the paper [9] developed tools to study analogous time dependent solutions of the following singular Navier-Stokes equations with a Hölder continuous function γ : [0, ∞) → R 3 , the shifted Dirac delta function δ γ(t) = δ(· − γ(t)), and a sufficiently small constant κ. The paper [9] recalls the result from [2] on the existence of a solution (u, p) to problem (4), however, the main goal in [9] is to show that u = u(x, t) and p = p(x, t) are locally bounded away from the graph of the curve Γ = {(γ(t), t) : t ≥ 0} and are singular along this curve. Moreover, the couple (u, p) satisfies problem (4) in the sense of distributions. In ad- ) with the Riesz transforms R 1 , R 2 , R 3 . These results only require γ to be continuous, but the authors of [9] do assume κ = κ(c) to be sufficiently small. Then, assuming γ to be Hölder continuous with exponent α ∈ ( 1 2 , 1], the paper [9] compares the solution (u, p) of (4) to the Slëzkin-Landau solution made time dependent by translating the origin to γ(t) proving that for all t > 0, and In this work, we revisit some of the existence results as well as some of the asymptotic estimates obtained in paper [9]. We repeat some of the proofs, with slight variations, because we need to have solutions to problem (4) and their estimates expressed in a slightly different form from how it appeared in [9]. Here, estimates of the L 2 (R 3 )-norms are essentially new and this is done in Sections 2 and 3. Section 4 contains our main result: we prove that the solution of problem (4) satisfies Related results. The Slëzkin-Landau solution has appeared in several recent works. It is proved in [2] that they are asymptotically stable in a suitable Banach space of tempered distributions. They are also asymptotically stable under arbitrary large initial perturbations of finite energy, see [7,8]. They appear in asymptotic expansions of solutions to initial-boundary value problems for the Navier-Stokes system (1), cf. [4,10,6,13,3]. Here, we also mention the article by Sverak [20] in which it is proved that the Slëzkin-Landau solution is the only stationary solution of the three dimensional Navier-Stokes system that is invariant under the natural scaling of the system.
Concerning other equations, solutions singular along curves, have been constructed by Sato and Yanagida [14,15,16,17,18] and by Takahashi and Yanagida [21,22] for the linear and non-linear heat equation.
Notation. We use the notation from [9]. Specifically, a major role in all these considerations is played by the Banach space PM a for a > 0 defined as the space of all w ∈ S (R 3 ) such that Then, for a time dependent function u : [0, T ] → PM a with fixed T ∈ (0, ∞] we define the scaling invariant norm The factor of (2π) 3/2 is added in the definition of · PM a because we are using the Fourier transform in the form Thus, the factor (2π) −3/2 appears in the well known formula F −1 (fĝ) = (2π) −3/2 f * g.

2.
The integral formulation of the problem.
Integral version of the Navier-Stokes initial value problem. For two vector fields u = (u 1 , u 2 , u 3 ) and v = (v 1 , v 2 , v 3 ), we denote by u⊗v the 3×3 matrix whose ( , k) entry is u v k . We denote the heat kernel by F (x, t) = (4πt) −3/2 exp(−|x| 2 /4t). With R j denoting the j-th Riesz transform, (that is R j f (ξ) = −i(ξ j /|ξ|)f (ξ)) we define R to be the 3 × 3 matrix of operators (R j R k ). Thus, For t ≥ 0, with a slight abuse of notation, we define the "convolution" K(t) * (u ⊗ v) to be the vector whose j-th component is given by where we have a sum of actual convolutions on the right hand side. The basic bilinear form is now defined by the formula we obtain Now, the integral version of the Navier-Stokes initial value problem can then be written in the form The matrix K = K(t) = K(x, t) with the coefficients defined by formula (6) is sometimes referred to as the Oseen tensor, see e.g. [12]. It is not too hard to prove (see e.g. [12,Sec. 4.5]) that there is a constant C > 0 such that for all x ∈ R 3 , t > 0, so that we have the estimate where, as usual, 1/p + 1/p = 1.
By direct calculation, the function and satisfies the following inhomogeneous heat equation in the sense of distributions (see [21] for other properties of this function). Thus, The Fourier transform of ϕ is given by the formula which implies the inequality We also remark that ϕ(x, 0) = 0 if x = γ(0). Navier-Stokes system with singular force. We consider now the integral equation (8) with u 0 = 0 and f (t) = κδ γ(t) e 1 , where e 1 = (1, 0, 0), κ ∈ R, and where with the function ϕ(x, t) defined by (9) and with Ψ = (ψ 1 , ψ 2 , ψ 3 ), where for j = 1, 2, 3 we define Notice that by inequality (10), we have Φ(t) PM 2 ≤ 1 and, since obviously In this notation, the integral formulation of problem (4) becomes Estimates of the bilinear form. Our next goal is to prove suitable estimates of the bilinear form B(·, ·) defined in (5). First, however, we recall an inequality from [9].
In the following lemma, we gather those estimates of B(·, ·) which will be required in the proof of the main result.
Remark 1. Note the following immediate consequence of estimate (16): We conclude this section by recalling a theorem on the existence of solutions to the singular Navier-Stokes system (4). This theorem is an immediate consequence of the Banach fixed point theorem applied to the equation u = B(u, u) + z using estimates from (12) and (14) with δ = 0.
From now on, we denote by u = u(x, t) the solution provided by Theorem 2.3 and our goal is to show that u ∈ C [0, ∞), L 2 (R 3 ) 3 .

Asymptotics by the Slëzkin-Landau solution.
In this section, we compare our solution u given by Theorem 2.3 with the Slëzkin-Landau solution made time dependent by translating the origin to γ(t) for t ≥ 0. Thus we introduce the function h(x) = 1/(c|x| − x 1 ) which allows us to write the following equalities

Using these formulas and writing h in the form
it is easy to prove that for each multi-index α = (α 1 , α 2 , α 3 ) there exists a constant C α , depending on α but not on c (recall that |c| > 1), such that and α = (α 1 , α 2 , α 3 ) with |α| = α 1 + α 2 + α 3 ≤ 3. These are pointwise estimates of the Slëzkin-Landau solutions which extends those in [7, Lemma 3.1] and the following lemma is a consequence of them. It appears in [9, Lemma 3.10], however, for the sake of completeness, we provide a proof that is somewhat simpler than the one in [9].
Lemma 3.1. There exists a constant K independent of c such that Proof. It suffices to prove that if a function W ∈ C 3 (R 3 \{0}) satisfies the estimate Let ξ ∈ R 3 , ξ = 0; it will remain fixed for most of the proof. Let Next for j = 1, 2, 3, Let C = max |α|≤3 C α ; then for k = 0, 1, 2, since D α χ = 0 for |x| ≤ 1 and for |x| ≥ 2 if α = 0, where K is a constant that does not depend on c or ξ, not necessarily the same from expression to expression. Finally Putting it all together, we proved for j = 1, 2, 3 the inequality

GRZEGORZ KARCH, MARIA E. SCHONBEK AND TOMAS P. SCHONBEK
Combining it with the estimate on W 1 , we complete the proof.
which satisfies, for each fixed t > 0, the system for all t ≥ 0.
where Proof. Computing the divergence of both sides of the equation −∆V c + V c · ∇V c + ∇Q = κδ 0 e 1 and using div V c = 0 we get the following equation understood in the sense of distributions where R j are the Riesz transforms. Moreover, it follows that with the Kronecker delta δ j1 . Applying the heat semigroup e (t−s)∆ to both sides of equation for ∂Q ∂xj (translated by γ(s)) and integrating with respect to s from 0 to t we get (after a slight rearrangement) the following equation In the equality above, the left hand side is the j-th component of B(V c γ , V c γ )(t) and the terms on the right hand side correspond to W c (t), −κΨ(t), and κΦ(t), respectively.

Remark 2.
Returning to the expression for W c , we see that Comparing equation (27) with the one in (13), we can write Now, we improve estimates from [9, Theorem 5.3 and Corollary 2].
for all t ≥ 0.
Proof. By equation (29), . Estimating by (14) we get, after some rearranging that |||w||| 2,∞ satisfies the following quadratic inequality It follows that w PM 2 cannot be between the roots of the corresponding quadratic equation; that is, we either have assuming, as we will, that |c| is large enough so that 1 − 2η 0 V c PM 2 > 0.
However, if |c| is sufficiently large we will have and then so the first possibility prevails. To simplify the notation, let us set so that |||w||| 2,∞ ≤ ν. Returning to the first chain of equalities in this proof, we can now estimate , from which we can solve to get for all t ≥ 0 the required inequality (30).
Proof. We have Taking δ = 2α − 1, we get The lemma follows with µ 0 = C V c PM 2 .
We can now state the estimate we had in mind.
Proof. We begin with a simple estimate that will be needed in our reasoning. Assume 0 < δ < 1. There exists a constant C (depending on δ) such that for all ξ ∈ R 3 , t > 0. In fact, using the estimate |ξ| 2 e −t|ξ| 2 /2 ≤ Ct −1 for every t > 0, we get Thus, inequality (33) follows. We recall that, by Theorem 2.3 and remarks following it, the function u = u(x, t), as a solution of the fixed point problem (13), is the limit in PM 2 -norm of the sequence defined by Setting w n = u n − V c γ and using the same type of decomposition used at the beginning of the proof of Theorem 3.3, we have that We proceed by induction. Consider 0 < t ≤ 1, first, and assume the inequality w n (t) PM 2+δ ≤ µt −δ/2 for some n ≥ 1, µ > 0. In view of inequalities (18) and (33), We use similar estimates for B(w n , V c γ ) and B(w n , w n ) obtaining The expression V c PM 2 + |||w n ||| 2,∞ can be made arbitrarily small by selecting |c| large enough to obtain 2µ 0 V c PM 2 + |||w n ||| 2,∞ ≤ µ 0 .
Remark 3. If we restrict our attention to the solution in an interval 0 ≤ t ≤ T with 0 < T < ∞, we can define C(T ) = max µ 0 , µ 0 T δ/2 and then we can state the result of Theorem 3.5 in the flowing form. There exists a constant C(T ), depending on T such that w(t) PM 2+δ ≤ C(T )t −δ/2 , 0 < t ≤ T.

4.
Estimates of the L 2 (R 3 )-norm. Now, we are in a position to show the main result of this work asserting L 2 -estimates of solutions to the singular Navier-Stokes initial value problem (4). We begin with a simple estimate that will be needed below. We are in a position to prove the main result of this work.