A local asymptotic expansion for a local solution of the Stokes system

We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha>0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$. The results extend to Oseen systems and to the Navier-Stokes equation.


Introduction
In this paper, we study local asymptotic development of solutions of local solutions for the Stokes equation in the unit cylinder. Namely, given f = (f 1 (x, t), f 2 (x, t), . . . , f n (x, t)) we seek a polynomial in x which approximates a solution u = (u 1 (x, t), u 2 (x, t), . . . , u n (x, t)) of the system u t − △u + ∇p = f, (1.1) around a point where the solution vanishes of order d. The solution is not assumed to have a high degree of regularity and thus the Taylor expansion is not available. Replacing the force with a matrix of functions in the divergence form we also obtain development for solutions of the Navier-Stokes equations around a vanishing point as a consequence. Fabre and Lebeau in [FL1,FL2] showed that the system (1.1)-(1.2) has a unique continuation property, i.e., local solutions of (1.1)-(1.2) can not vanish to infinite order unless they vanish identically. Having a priori estimates on solutions with respect to their vanishing order is considered a crucial step in many applications. For instance, using a priori estimates on asymptotic polynomials Han [H2] improves the classical Schauder estimates in a way that the estimates of solutions and their derivatives at one point depend on the coefficient and the nonhomogeneous terms at that particular point. Also, Hardt and Simon [HS] applied an estimate of Donnelly and Fefferman for the order of vanishing of eigenfunctions to find an asymptotic bound of the (n − 1)-dimensional measure of v −1 j {0}, where v j is an eigenfunction corresponding to the j-th eigenvalue of the Laplacian on a compact Riemannian manifold.
The method we use in proving the main theorem was introduced by Q. Han, who in [H2] found an asymptotic development of a solution of a parabolic equation of an arbitrary degree (cf. also [H1] for the elliptic case). The main idea in [H2] is based on a local expansion of the corresponding fundamental solution of the global linear equation.
There are several key difficulties when trying to extend the results to the Stokes equation (1.1)-(1.2). First, due to presence of the pressure, it is not reasonable to expect that the velocity and the pressure would vanish at the same point (for instance, the unique continuation result of Fabre and Lebeau gives a unique continuation property for u and not for the pair (u, p)). Thus in our main result we do require p to vanish. The second difficulty is the lack of smoothing in the time variable in the system, which is a well-known problem for local solutions of the Stokes and Navier-Stokes systems. Indeed, taking the divergence of the evolution equation for the velocity gives which does not contain any smoothing in the time variable. The third difficulty is the nonlocal nature of the Stokes kernel, which in particular causes the Stokes kernel to decay polynomially, rather than exponentially as it is the case for the scalar equations. We note here that there have been many works on unique continuation of elliptic and parabolic equations showing that, under various assumptions on coefficients, no solution can vanish to infinite order (cf. [AE,AMRV,CRV,DF,EFV,EV,GL,JK,KT,SS1,SS2] for instance); for more complete reviews, see [K1, K2, V]. Unique continuation questions for the Stokes and Navier-Stokes systems were addressed in [CK,FL1,FL2,Ku]. The paper is organized as follows. In Section 2, we state the main results, Theorem 2.1 and 2.3, addressing the forces in standard and divergence forms respectively. We also state the two corollaries concerning the Navier-Stokes and Oseen systems. In Section 3 we recall the properties of the Stokes kernel, while the last part contains a construction of a particular solution vanishing of order d as well as the proof of Theorem 2.1.

Notation and the main result on the asymptotic expansion
In this paper, we consider a solution (u, p) of the Stokes system (1.1)-(1.2) in an open set containing (0, 0) (which can always be assumed using translation). For any (x, t) ∈ R n × R and r > 0 we denote the parabolic cylinder label by (x, t) with radius r > 0 by The corresponding parabolic norm for (x, t) ∈ R n × R is given by Denote by W m,1 q (Q 1 ) the Sobolev space of L q (Q 1 ) functions whose all the x-derivatives up to m-th order and t-derivative of first order belong to L q (Q 1 ).
Theorem 2.1. Let q > 1 + n/2. Suppose that f j ∈ L q (Q 1 ), for j = 1, 2, . . . , n satisfy for some constants γ > 0 and α ∈ (0, 1), where d ≥ 2 is an integer. Then for any solution u = (u 1 , . . . , u n ) ∈ W 2,1 q (Q 1 ) of (1.1)-(1.2) there exists P = (P 1 d,t , . . . , P n d,t ), whose each component P j d,t is a polynomial in x of degree less than or equal to d, such that where C is a positive constant depending on n, q, d, and α. Moreover, P satisfies the Stokes system where R is the corresponding pressure.
Remark 2.2. The pressure term is found explicitly in the proof of the main theorem and is given by for (x, t) ∈ Q 1 .
As we are interested in obtaining estimates in Q 1 , we assume without loss of generality that In the case when the function on the right side of (1.1) is in the divergence form, the Stokes system reads as for some function g = (g jk ) n j,k=1 ∈ W 1,0 q (Q 1 ). Here also we may assume without loss of generality that Then we have the following variant of Theorem 2.1.
for some constants γ > 0 and α ∈ (0, 1). Then for any solution u = (u 1 , u 2 , . . . , u n ) ∈ W 2,1 q of (2.6)-(2.7) there exists P = (P 1 d,t , . . . , P n d,t ) whose each component P j d,t is a polynomial in x of degree less than or equal to d such that for any (x, t) ∈ Q 1/2 , where C is a positive constant depending on n, q, d, α. Also, P satisfies the Stokes system where R is the corresponding pressure.
Having a force in divergence form on the right side of (2.7) allows us to apply the above results to the solutions of the Navier-Stokes equations. (2.14) Also, assume that u vanishes of the order at least d ≥ 2. Then there exists P = (P 1 d,t , . . . , P n d,t ) whose each component P j d,t is a polynomial in x of degree less than or equal to d such that for j = 1, . . . , n, where C is a positive constant depending on n, d, q, and u. Moreover, P satisfies the Stokes system where R a suitable pressure term depending on u and p.
We note that u is not assumed to be smooth in the space or time variable. Therefore, the inequality in (2.15) can not be obtained by expanding the solution in the Taylor series. The result in Theorem 2.1 can also be applied to the Oseen system considered in [FL1].

20)
for j = 1, . . . , n, where α ∈ (0, 1) and C is a positive constant depending on n, d, q, α, and u. Moreover, P satisfies the Stokes system where R a suitable pressure term depending on u and p.

The basic results
We start by recalling pointwise estimates on the derivatives of solutions to the homogeneous heat equation The fundamental solution is given by Γ(x, t) = (4πt) −n/2 exp −|x| 2 /4t for t > 0 and Γ(x, t) = 0 for t ≤ 0. Recall that the derivatives are bounded as First recall that for any solution u of (3.1) we have where C depends on |µ| + 2l [Li].
For completeness, we briefly recall the derivation of the fundamental solution to the Stokes system (1.1)-(1.2). Let u(0, ·) = u 0 be the initial condition. By uniqueness of solutions, we have for k = 1, . . . , n. Using the Fourier transform of both sides in (1.3), we get where R j g = ξ j i|ξ|ĝ 5 denotes the j-th Riesz transform, using the Fourier transformf (ξ) = f (x)e −iξ·x dx. Thus (3.4) can be written as For each j, k = 1, . . . , n, the function K jk solves the heat equation, i.e., Also, where C depends on |µ| and l [FJR, L, S].

Proof of the Main Theorem
In the next lemma, we construct a solution of the system (1.1)-(1.2) which vanishes with a certain prescribed degree.
Proof of Lemma 4.1. We start by setting Then we have . . . , n (4.5) and Now, we consider the Taylor expansion of K jk (x − y, t − s) around (0, 0). Let |(y, s)| < 1 be such that s = 0. Denote by K m jk the m-th order terms, i.e., It is easy to check that K m jk solves the heat equation for each j, k, i.e., K m jk (x, y; t, s)f j (y, s) dy ds, k = 1, . . . , n. (4.8) Each v k is a polynomial of degree less than or equal to d and it satisfies Moreover, we have ∇ · v = 0 (4.10) Indeed, we may write where e k is the standard k-th unit vector in R n . Note that Using ∂ k K jk = 0 for j = 1, . . . , n, we get ∇ · v = 0. Now, set K m jk (x, y; t, s) f j (y, s) dy ds, k = 1, . . . , n (4.11) and note that we have ∂ t u k − △u k + ∂ k p = f k , k = 1, . . . , n. (4.12) We now check the condition (1.2). Since where we used ∂ k K jk = 0 for j = 1, . . . , n, we get ∇ · u = 0. Next, we claim that (4.13) Fixing |(x, t)| ≤ 1/2, we split the integral on the far right side of (4.11) into three parts K m jk (x, y; t, s)f j (y, s) dy ds K m jk (x, y; t, s) f j (y, s) dy ds.
Proof of Theorem 2.1. The proof of this result follows that of Theorem 2.1 and it is thus omitted.