Abstract 3
1. Introduction 4
1.1. Problem statement and main results 4
1.2. Known result 7
1.3. Plan of the proof 7
2. Integral identities 9
3. The Stokes problem with a drift 10
3.1. Existence of unique weak solution 10
3.2. Smoothness of solutions to the Stokes problem 13
3.3. Solution of the Stokes problem as a mild solution 14
4. Upper bounds 15
5. Pikard’s method 19
5.1. Pikard’s method for the solutions of the Cauchy problem 19
5.2. Extension of the solutions obtained by Pikard’s method 20
6. Duality method 21
7. Decreasing in time for sufficiently small singularity 23
8. Proof of the main result. Concluding remarks 24
8.1. Proof of the Liouville-type theorem 24
8.2. Concluding remarks 25
References 26
1.1. Problem statement and main results.
and t1 G R, t2 G R U {Ix} such that t1 < t2.
space of all Banach-valued measurable functions
t2
HUHLp(ti,t2;X) = j Hu(t)HX dt < +~.
t1
Also, denote by L^(t1, t2; X) the Banach space of all Banach-valued measurable functions u : [t1,t2] ^ X such that
l|u||L~(ti,t2;x) = ess sup{||u(t)||x | t G (ti,t2)} < +x.
For simplicity of notation we omit the spatial variable of function by matching them as u(-,t) ^ u(t) G Lp(t1, t2; X), a.e. in (t1, t2), 1 < p < +x.
Let Q be a hypercube in R", and |Q| is the volume of Q, i.e., its Lebesgue measure. Denote by BMO(R3) the Banach space of all locally integrable functions whose mean oscillation is bounded, namely, the following norm if finite:
II/ ||BMO(R3) = Яф |Q У f (x) — [f]Q| dx < x-,
q
|Q Jq f (x) dx
open problems of local regularity of weak solutions to the
Consider so-called suitable weak solution
dtw — Aw + (w • V)w + Vg = 0, div w = 0
in the unit parabolic space-time ball Q = [—1, 0] x Br (0) c R x Rn. Here, Br (x) stands for the ball in R" of radius r centred at the point x G Rn. For a definition of suitable weak solutions, we refer to the paper. Let us assume that function w satisfies the additional restriction:
c
|w(x,t)| Si < r~i, y(x,t) G Q,
|x| I —t
for some constant c > 0. In that case we say that w has a singularity of type I.
The question is to understand whether or not the origin z = (x, t) = (0,0) G Rn+1 is a regular point of w, i.e., there exists 5 > 0 such that w is essentially bounded in the parabolic ball Q(S) = [—52,0] x Bd(0).
In this thesis we consider only the case of a three-dimensional space R", so n = 3. Denote Q+ = R3 x (0, +x). We say that the functions u and p are a mild bounded ancient solution of the backward Navier-Stokes equations, if
u G C~(Q+) П L^(Q+), p G C™(Q+) П L^(0, x; BMO(R3)),
and these functions u and p obey following equations
/кГсР. [—dtu — Au + (u • V)u + Vp = 0 .
(NS) : < n in Q+.
div u = 0
It has been shown that if the origin z = 0 is a singular point of w, then there exists the non-trivial mild bounded ancient solution u and p such that |u(0,0)| = 1 and
Since u is a smooth function let us fix such constant M > 0 that
IIuIIl«,(Q+) + Vu l q. < M- (I-3)
If w has type I singularity satisfying (1.1) then the mild bounded ancient solution u corresponding to w also satisfies the condition (1.4):
3c* > 0: u(x,t) < , C* r, V(x,t) G Q+. (1.4)
x + Vt
The duality method has been first developed and exploited by G. Seregin. This method allows to prove Liouville type theorems not only for scalar equations, but also for systems. Soon after that M. Schonbeck and G. Seregin considered the application of this method to the above-mentioned problem, see e.g. In particular, the following dual problem, namely, the Stokes system with a drift u, has been considered:
dtv — Av — (u • V)v + Vq = -div F, div v = 0, (1.5)
in Q+ and v(x, 0) = 0 for all x G R3. It has been supposed that a tensor-valued field F G C0°(R3) is smooth and compactly supported in Q+. In addition, it has been assumed that F is skew symmetric and therefore
divdiv F = 0.
The following identity takes place:
— J Vu : F dxdt J u • div F dxdt = — ?lim J u(x, T) • v(x, T) dx. (1.6) Q+ Q+ R3
If the solution v to the dual problem has a certain decay, the limit on the right hand side vanishes. This means that the skew symmetric part of Vu vanishes in Q+. Then one can easily show that u must be a function of time only. Since u is a divergence free field, u is a bounded harmonic function. But also u is a bounded mild ancient solution to the Navier-Stokes equation and thus must be a constant. However, the condition (1.4) means that u is identically equal to zero. Thus, if we had a statement about the decay of v, we would immediately obtain a Liouville-type statement.
Therefore, since we know that u(0,0) = 1, this finally would prove that z = 0 is not a singular point of w and it, in turns, says that the origin is a regular point of w . So, the problem of local regularity of weak solutions to the Navier-Stokes equations stated in the beginning is solved under the additional assumption: u has a singularity of type I.
...
In this section, we describe how the Theorem 7.1 could be proved for any constant c* > 0, that is, without assuming that c* < e0 for some e0 > 0. Unfortunately, we show that this theorem can not be proved under such assumptions. What is interesting here is that we come to exactly the same estimate as M. Schonbeck and G. Seregin came to in paper [5]. Moreover, we come to some conclusions about the duality method.
Let p, q ^ 1 be such that p + q = 1.
The duality method is applicable under different assumptions and is indeed a method rather than a special case of solving the problem of local regularity of a weak solution of the Navier-Stokes equations. As we mentioned in Section 1.1, our main goal is to investigate the limits of applicability of the duality method developed by G. Seregin. We examined a modification fo duality method expecting that we could obtain results similar to those obtained by M. Schonbeck and G. Seregin. Since all of our results, both positive and negative, are exactly the same as those of M. Schonbeck and G. Seregin, we finally conclude that the duality method requires the search for further applications in the theory of the Navier-Stokes equations.
