Time-dependent singularities in the Navier-Stokes system

We show that, for a given H\"older continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset R^3\times R^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $R^3$ which is smooth outside this curve and singular on it. This is a pointwise solution of the system outside the curve, however, as a distributional solution on $R^3\times R^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.


Introduction
The Navier-Stokes system describing a motion of an incompressible homogeneous fluid in the whole three dimensional space has the following form (1.1) ∂ t u − ∆u + (u · ∇)u + ∇p = 0, div u = 0, for (x, t) ∈ R 3 ×R + . 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. System (1.1) should be supplemented with an initial condition u| t=0 = u 0 , however, it does not play any role in the statement of the main result in this work. Our goal in this paper is to propose mathematical tools which, in particular, allow us to prove the following theorem. Theorem 1.1. For every Hölder continuous function γ : R + → R 3 with a Hölder exponent α > 3 and its one-parameter family explicit stationary solutions V c (x), Q c (x) of the following form where |x| = x 2 1 + x 2 2 + x 2 3 and c ∈ R is an arbitrary constant such that |c| > 1. These explicit stationary solutions to (1.1) seem to be discovered first by Slezkin [29] (see the translation of this work in [6,Apendix]) and described by Landau in [15] (see also [16,Sec. 23]). They play a pivotal role in this work and we are going to call them as the Slezkin-Landau solutions to system (1.1). Let us also recall that the stationary solutions (1.2) were also independently derived in [30,33] and they can be found in standard textbooks, see e.g. [1, p. 206]. To obtain such solutions, it suffices to notice that the additional axisymmetry requirement reduces the stationary Navier-Stokes system to a system of ODEs which can be solved explicitly in terms of elementary functions. Recently,Šverák [31] proved that even if we drop the requirement of axisymmetry, the Slezkin-Landau solutions (1.2) are still the only stationary solutions which are invariant under the natural scaling of system (1.1).
The Slezkin-Landau solutions appear in recent works in different contexts. 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 [8,9]. They appear in asymptotic expansions of solutions to initialboundary value problems for the Navier-Stokes system (1.1), cf. [5,12,13,20].
One can check by straightforward calculations that the functions V c 1 (x), V c 2 (x), V c 3 (x) and Q c (x) given by (1.2) satisfy system (1.1) in the pointwise sense for every x ∈ R 3 \ {(0, 0, 0)}. They are homogeneous functions of degree −1 and −2, respectively, and are smooth for x = 0. Thus, the Slezkin-Landau solution V c , Q c solves system (1.1) in a classical and pointwise sense on (R 3 × R + )\Γ 0 and is singular on the line Γ 0 = {(0, t) ∈ R 3 × R + , t ≥ 0}. On the other hand, if one treats V c , Q c as a distributional or generalized solution to (1.1) in the whole space R 3 , it corresponds to the singular external force (κδ 0 , 0, 0), where the parameter κ = 0 depends on c and δ 0 stands for the Dirac measure (details of this reasoning are recalled below in Proposition 3.8).
In this work, we generalize this idea and we construct analogous singular solutions on a sufficiently regular curve Γ ⊂ R 3 × R + . Our solutions are not explicit and they behave asymptotically as the Slezkin-Landau solution in a neighborhood of a singularity at the curve Γ. Theorem 1.1 is a particular case of Theorem 2.1 formulated in the next section.
Our results have been motivated by recent works of Yanagida and his collaborators [23,24,25,26,27,32] where solutions singular on curves have been constructed for either nonlinear or linear heat equation.
Notation. Here, R + = (0, ∞). For p ∈ [1, ∞], the usual Lebesgue space is denoted by L p (R 3 ) and the weak Marcinkiewicz L p -space -by L p,∞ (R 3 ). In the case of Banach spaces X used in this work, the norm in X is denoted by · X . Given an open set Ω, the symbol C ∞ c (Ω) denotes the set of all smooth functions which are compactly supported in Ω. S(R 3 ) is the Schwartz class of smooth and rapidly decreasing functions. Constants may change from line to line and will be denoted by C.

Main result
The result formulated in Theorem 1.1 is covered by the following more general result.
Remark 2.3. Property (iii) of Theorem 2.1 means that u(x, t) and p(x, t) have to be singular at the point x = γ(t) for every t > 0 and this singularity is comparable with the singularity of the functions V c (x − γ(t)) and of Q c (x − γ(t)), respectively. This is due to the fact that V c = V c (x) is a homogeneous function of degree −1, so, it does not belong to L q loc (R 3 ) for each q ≥ 3. Similarly, Q c = Q c (x) is homogeneous of degree −2 and it does not belong to L q loc (R 3 ) for each q ≥ 3 2 .
Remark 2.4. As we have emphasized in Remark 2.3, the stationary solution V c = V c (x) defined in (1.2) is singular with singularity of the kind O(1/|x|) as |x| → 0. This is the critical singularity, because as it was shown in [4,10], every pointwise stationary solution is also a solution in the sense of distributions in the whole B R . In other words, such a singularity at the origin is removable. Analogous results on removable singularities of time-dependent weak solutions to the Navier-Stokes equations has been proved by Kozono [14]. Recent results on removable (time-dependent) singularities to semilinear parabolic equations can be found in [7,11,32].
The velocity vector field and the pressure (u, p) in Theorem 2.1 are obtained as solutions to the following initial value problem for the incompressible Navier-Stokes system with a singular force where κ ∈ R,ē 1 = (1, 0, 0) and δ γ(t) is the Dirac measure on R 3 concentrated at the point x = γ(t). Solutions to the singular problem (2.1) with sufficiently small |κ| has been constructed in [2] in a suitable space of tempered distributions. In this work, using the approach introduced in [2], we show that those solutions are, in fact, functions with all the properties stated in Theorem 2.1.
Remark 2.5. For simplicity of the exposition, we supplement problem (2.1) with the zero initial datum, however, a completely analogous result may be proved in the case of a sufficiently small initial datum u| t=0 ∈ PM 2 (see the next section). Such a nontrivial initial condition will just give us another solution of problem (2.1) with properties stated in Theorem 2.1.
Remark 2.6. Applying results from the recent work [9], we obtain immediately that solutions to problem (2.1) are asymptotically stable under arbitrary large initial perturbations from L 2 (R 3 ).
In the next section, we recall mathematical tools which allows us to study problem (2.1). In Section 4, we explain our idea of proving Theorem 2.1 by using it in the case of the heat equation u t = ∆u. Theorem 2.1 is proved in Section 5.
3. Estimates in spaces PM a 3.1. Preliminary properties. First, we precise our assumption on the curve γ.
Following [2], we introduce a family Banach spaces in which we solve singular problem (2.1). For every fixed a ≥ 0, we set In this paper, we mainly deal with vector fields u = (u 1 , u 2 , u 3 ) hence, by the very definition u PM a = max u 1 PM a , u 2 PM a , u 3 PM a . Below, we construct a solution to problem (2.1) satisfying u ∈ C w [0, ∞), PM 2 which means that u ∈ L ∞ [0, ∞); PM 2 and the function u(t), ϕ is continuous with respect to t ≥ 0 for every test function ϕ ∈ S(R 3 ). Here, ·, · denotes the usual pairing between S ′ (R 3 ) and S(R 3 ).
To study regularity properties of u, also following [2], we introduce the Banach space for each a ≥ 2 and T ∈ (0, ∞]. The space Y a T is normed by the quantity v Y a T = ||| v ||| 2,T + ||| v ||| a,T and of course, Y 2 ∞ = C w [0, ∞), PM 2 with this definition. This is a usual procedure to eliminate the pressure p = p(x, t) from problem (2.1) by applying the Leray projector P on solenoidal vector fields, which is given formally by the formula P = I − ∇(∆) −1 div . It is well-known that this is a pseudodifferential operator corresponding to the matrix P(ξ) with the components where δ jk = 0 for j = k and δ jk = 1 for j = k. In particular, | P(ξ)| ≤ 2 for all ξ = 0. Using the Leray projector P, at least formally, we may rewrite problem (2.1) as the following integral equation where S(t) is the heat semigroup given as the convolution with the Gauss-Weierstrass kernel G(x, t) = (4πt) −3/2 exp(−|x| 2 /(4t)) and the bilinear form To give a precise meaning of equations (3.4)-(3.5), we reformulate them using the Fourier transform in the following way.
We construct a solution to integral equation (3.6) using the Banach fixed point theorem which, in the case of the incompressible Navier-Stokes system, is often reformulated in the following way.
Lemma 3.1. Let (X, · ) be an abstract Banach space, L : X → X be a linear bounded operator such that for a constant λ ∈ [0, 1), we have L(x) ≤ λ x for all x ∈ X, and B : X × X → X be a bilinear mapping such that for every x 1 , x 2 ∈ X for some constant η > 0. Then, for every y ∈ X satisfying 4η y < (1 − λ) 2 , the equation has a solution x ∈ X. In particular, this solution satisfies x ≤ 2 y 1−λ , and it is the only one among all solutions satisfying x < 1−λ 2η .
We skip the proof of this lemma because it is elementary, and it simply consists in applying the Banach contraction principle to equation (3.7) in the ball {x ∈ X : x ≤ ε} with arbitrary ε < 1−λ 2η . 3.2. Auxiliary estimates. Due to Lemma 3.1, to show the existence of solutions to equation (3.6), we need estimates of the bilinear form B(·, ·) defined in (3.5).
We skip the proof of Lemma 3.2, because it was proved in [2, Proposition 7.1]. Notice that in [2], inequality (3.8) was shown for T = ∞, however, after a minor modification that proof works for each finite T as well.
Proof. This inequality is an immediate consequence of the following estimate To study a regularity of solutions to equation (3.6), we need some imbedding properties of spaces PM a . We formulate then in the following three lemmas.
Then there exists a positive constant . Proof. In the particular case of a = 2, this Lemma was proved in [2,Lemma 7.4]. For the completeness of the exposition, we show the general case. Using a standard approximation procedure one may assume that v is smooth and rapidly decreasing. Since q ≥ 2, by the Hausdorff-Young inequality with 1/p + 1/q = 1 and p ∈ [1,2] and the definition of the PM a -norm we obtain for all R > 0 and C independent of v and R. In these calculations, we require ap < 3 which is equivalent to q > 3/(3 − a). Moreover, we have to assume that bp > 3 which leads to the inequality q < 3/(3 − b). Now, we optimize inequality (3.10) with respect to R to get formula (3.9).
In the case of the integral over the low frequency domain |ξ| ≤ R, we estimate as follows For high frequencies |ξ| > R, by the definition of space PM β , we have we obtain estimate (3.10). Next, we show that tempered distributions from PM 2 are in fact functions from L 3,∞ (R 3 ).
Lemma 3.6. There exists a constant C such that Proof. The imbedding PM 2 ⊂ L 3,∞ (R 3 ) has been noticed in [9,Remark 3.2]. Here, we present its proof for the completeness of the exposition. First, we recall (see for example [21]) the Hausdorff-Young inequality in the Lorentz space for p ≥ 2 and p ′ = p p−1 as well as the Hölder inequality in the Lorentz space Thus, by the Plancherel theorem, the definition of the norm in PM 2 , and inequalities (3.12)-(3.13), for every test functions ϕ ∈ S(R 3 ), we obtain is a dense subset, we obtain that the tempered distribution u ∈ PM 2 defines a bounded linear functional on L 3,1 (R 3 ) with the norm estimated by C u PM 2 . Since L Finally, we prove a simple inequality in weak L p -spaces.
Lemma 3.7. There exists a constant C > 0 such that for every u ∈ L p,∞ (R 3 ) 3 with Proof. We use the well-known fact that the norm in the Marcinkiewicz space L p,∞ (R 3 ) is comparable with the quantity sup λ≥0 λ {x ∈ R 3 : |u(x)| ≥ λ} 1 p , where |A| denotes the Lebesgue measure of a set A ⊂ R 3 . Thus, by a direct calculation, we obtain the following inequalities which complete the proof of Lemma 3.7.

Slezkin-Landau solution.
To conclude this section, we recall properties of the Slezkin-Landau solution given by formula (1.2).
and Q c be defined by (1.2). For every test function ϕ ∈ C ∞ c (R 3 ) 3 the following equalities hold true: (3.14)  Proof. It follows from the explicit formula for V c that where the functions |x| ) 2 and their derivatives are bounded on R 3 uniformly in |c| ≥ 2. A similar reasoning should be applied in the case of V c 2 (x) and V c 3 (x). Thus, by direct calculations, for every multiindex α = (α 1 , α 2 , α 3 ) and D α = ∂ α 1 where constants C(α) are independent of c ∈ R such that |c| ≥ 2.
Next, using the Littlewood-Paley theory, one can write the following decomposition for where for each q ∈ Z the symbol∆ q = ϕ(2 −q D) denotes the homogeneous Littlewood- and satisfying q∈Z ϕ(2 −q x) = 1 for each x ∈ R 3 \{0}. We refer the reader to [3] and to references therein for more results on the Littlewood-Paley decomposition. Now, we deal with the terms I and II, separately.
In the case where q ≤ 1 |ξ| , by the Hausdorff-Young inequality, the support property of ϕ, and by estimate (3.18) with α = (0, 0, 0), term I can be bounded as follows Next, for q > 1 |ξ| , the term II can be written as |ξ| −|α| q> 1 |ξ| |ξ| |α|∆ q V c (ξ) for each |α| ≥ 0. Moreover, by the Hausdorff-Young inequality and the Leibniz formula, we easily find that Now, we fix α such that |α| = 3. Inequality (3.19) together with the support property of ϕ and estimate (3.18) allows us to conclude that Both estimates of I and II yields the required inequality (3.17).

Singular solutions to the heat equation
In order to illustrate our method of constructing singular solutions to the Navier-Stokes system (1.1), we apply it first to the linear heat equation We are going to construct a pointwise solution to Hence, −∆U = δ 0 in R 3 , where δ 0 denotes the Dirac measure. Following this idea, we construct a solution to the inhomogeneous heat equation with the singular force δ γ(t) for every t > 0.
Supplementing this equation with the initial condition u(ξ, 0) = 0 for all ξ ∈ R 3 , we obtain the following formula for its solution Proof of Theorem 4.1. The proof consists in showing that the tempered distribution u(t) defined in the Fourier variables by formula (4.4) is, in fact, a function u = u(x, t) with the properties stated in (i)-(iii) of Theorem 4.1.
Step 1. First, let us show that u ∈ C w [0, ∞); PM 2 ; see the definition of this space in Section 3. Indeed, we notice that Hence, by the definition of the norm in PM 2 , we have Now, for every test function ϕ ∈ S(R 3 ), by the definition of the Fourier transform of a tempered distribution, we have By formula (4.4), u(ξ, t) is a continuous function of t ≥ 0 for each fixed ξ ∈ R 3 . Moreover, by (4.5), we have the inequality u(ξ, t) ϕ(ξ) ≤ 1 |ξ| 2 ϕ(ξ) , where the right hand side is integrable over R 3 . Thus, the continuity of the right-hand side of identity (4.6) with respect to t ≥ 0 is an immediate consequence of the Lebesgue dominated convergence theorem.
Step 3. Now, we prove that ω 0 ∈ Y a T for each a ∈ [2, 1 + 2α] and T > 0, where the Banach space Y a T is defined in (3.2). By Step 2, we have ω 0 ∈ C w [0, ∞); PM 2 . Hence, it remains to estimate ω 0 (t) in the PM a -norm, and we use here the representation of ω 0 by the right-hand side of equation (4.8).
First, we notice that for each a ≥ 2 there exists a positive constant C = C(a) such that ess sup for all t > 0.
To deal with the second term on the right-hand side of (4.8), we fix T > 0 and consider s, t ∈ [0, T ]. Thus, by the Hölder continuity of γ(t), there is a constant   Consequently, using estimate (4.9) for |ξ| ≤ 1 and estimate (4.10) for |ξ| ≥ 1, we obtain Here, the right-hand side is finite for every a ∈ [2, 1 + 2α], and this interval is non-empty if α > 1 2 . Thus, we have proved that ω 0 ∈ Y a T for each a ∈ [2, 1 + 2α] and T > 0.

Singular solutions to the Navier-Stokes system
In this section, we construct a vector field u = (u 1 , u 2 , u 3 ) and a pressure p = p(x, t) with properties stated in Theorem 2.1 as a solution to the singular initial value problem (2.1). First, we recall from [2] a result on the existence of a family of tempered distributions u = u(t) which satisfies problem (2.1).
Proof. This is an immediate consequence of the continuous imbedding PM 2 ⊂ L 3,∞ (R 3 ) proved in Lemma 3.6.
In the next step, we compare u(x, t) with the shifted Slezkin-Landau solution V c γ (x, t) = V c (x − γ(t)). We begin with an integral representation of V c analogous to that one for u(ξ, t) in (3.6).
Proof. Since V c (x) is a distribution solution of the Navier-Stokes system with the singular force κδ 0ē1 , in the Fourier variables, we have To derive the Fourier integral representation (5.3), we notice that . Hence, multiplying equation (5.5) by e iγ(t)·ξ , we obtain the relation 1 , for all ξ ∈ R 3 and t ≥ 0, which is equivalent by a direct calculation to the following formula Let us modify the sum of the last two terms on the right-hand side of the above identity using the relation V c γ (ξ, t) = e iγ(t)·ξ V c (ξ) in the following way Proof. Here, the reasoning is completely analogous to the one in the proof of Theorem 5.1. We apply Lemma 3.1 to equation (5.8) with the linear operator L ω = B(V c γ , ω)+B(ω, V c γ ). It follows from estimate (3.8) with a = 2 and T = ∞ that Now, we apply Lemma 3.1 with λ = 2η 2 V c PM 2 and η = η 2 in the following way. Notice that, by Lemma 3.9, we have λ ≤ 2η 2 K |c| < 1 6 for all |c| ≥ c 0 > 12Kη 2 . Next, by a direct calculation, if c 0 > 12Kη 2 , we have Thus, by a direct calculation, if c 0 > 12Kη 2 we have for all |c| ≥ c 0 , which is possible due to Lemma 3.9. Finally, using inequality (3.8) with a = 2 and T = ∞ and applying Lemma 3.1, we obtain a solution ω ∈ Y 2 ∞ of equation (5.7). The following corollary is a direct consequence of the uniqueness of solution to the considered equations.
Corollary 5.5. Let u be a solution to problem (2.1) constructed in Theorem 5.1. Then, choosing c 0 sufficiently large, we obtain that the solution ω in Theorem 5.4 is of the form ω = u − V c γ for all |c| ≥ c 0 . Proof. We use the notation from the proof of Theorem 5.4. By Lemma 3.1, estimate (5.9), and inequality (3.17), the solution ω of equation (5.7) satisfies Thus, for sufficiently large |c|, we have ω = u − V c γ by the uniqueness of the solution u to equation (5.1) established in Theorem 5.1 and the uniqueness of ω from Theorem 5.4.
In the next step, we prove that the solution ω ∈ Y 2 ∞ of equation (5.8) satisfies ω ∈ Y a T with suitable a > 2.
In the proof of this theorem, we need the following property of y 0 defined by formula (5.4).
Lemma 5.7. Let γ : [0, ∞) → R 3 be Hölder continuous with an exponent α ∈ ( 1 2 , 1]. Then, for each T > 0 and each a ∈ [2, 1 + 2α], we have y 0 ∈ Y a T . Proof. First, we notice that y 0 (t) given by (5.4) has the form of the tempered distribution ω(t) defined in the case of the heat equation by formula (4.8) with 1 |ξ| 2 replaced by V c (ξ). Since | V c (ξ)| ≤ C |ξ| 2 , to complete the proof of this lemma, it suffices to repeat the reasoning form Step 3 of the proof of Theorem 4.1.
Since c > 2(8η a + η 2 )K, we have V c PM 2 < 1 4ηa by Lemma 3.9. Moreover, it follows from Theorem 5.4 that sup t>0 ω(t) PM 2 ≤ 4K c−2η 2 K < 1 4ηa . Thus, T is a contraction in the norm ||| · ||| a,T for sufficiently large c which implies that the sequence {ω n } converges toward ω ∈ Y a T . We are in a position to complete the proof of the main result from this work.
Proof of Theorem 2.1.