In this chapter we present the solenoidal Lipschitz truncation for non-stationary problems: we show how to construct a Lipschitz truncation which preserves the divergence-free character of a given Sobolev function. As a matter of fact, it suffices to have distributional time-derivatives in the sense of divergence-free test-functions. After this, we present the A-Stokes approximation for non-stationary problems. It aims at approximating almost solutions to the non-stationary A-Stokes system by exact solutions. Thanks to the solenoidal Lipschitz truncation this can be done on the level of gradients.
Solenoidal Lipschitz truncation; Divergence-free constraint; Parabolic PDEs; Inverse curl-operator; A-Stokes approximation; Almost solutions
In this chapter we develop a non-stationary counterpart of the solenoidal Lipschitz truncation from Chapter 3. Here, the main difficulty is to handle problems connected with the distributional time derivative of the function we aim to truncate. Let us be a little bit more precise. Let Q0=I0×B0⊂R×R3 be a space time cylinder and σ∈(1,∞). Let u∈Lσ(I0,W1,σdiv(B0)) and G∈Lσ(Q0) satisfy
∫Q0∂tu⋅ξdxdt=∫Q0G:∇ξdxdtfor allξ∈C∞0,div(Q0).
The main purpose of the solenoidal Lipschitz truncation is to avoid the appearance of the pressure function. Hence we start in (6.0.1) with an equation on the level of divergence-free test-functions. Unfortunately, this is not enough information on the time derivative for a Poincaré-type inequality as in (5.2.6). Hence the approach from [65] as explained in Section 5.2 will not give L∞-estimates for the gradient of the truncation, cf. the proof of Lemma 5.2.3. Our aim is to construct a truncation which preserves the properties from [65] and is, in addition, divergence-free.
We will show that there is a truncation uλ of u with roughly the following properties (see Theorem 6.1.25 for a precise formulation).
(a) ∇uλ∈L∞(Q0) with ‖∇uλ‖∞⩽cλ and divuλ=0.
(b) uλ=u a.e. outside a suitable set Oλ.
(c) There holds
|〈∂tu,uλ−u〉|+‖χOαλ∇uλ‖pp⩽cλp|Oλ|⩽δ(λ),
with δ(λ)→0 if λ→∞.
In the following we sketch the construction on a heuristic level. In fact, the rigorous approach which we shall present in the next section requires a series of localization arguments, so it is quite technical. Let us start with a function
u∈L∞(I0;L2(B0))∩Lp(I0;W1,p0,div(B0))
with ∂tu=divH in D′div(B0), where H∈Lσ(B) for some σ>1. We define
w:=curl−1u∈L∞(I0;W1,2(B0))∩Lp(I0;W2,pdiv(B0)).
It follows that ∂tΔw=curldivH in D′(B0). Also we can obtain an information about the time derivative of w as a distribution acting on all test-functions. However, we do not have control about a possible harmonic part of w. Hence we decompose w into a harmonic and anti-harmonic part. To do this we define, pointwise in time,
w(t)=z(t)+h(t),
where z(t)∈ΔW2,p0(B0) and Δh(t)=0. This decomposition is based on a singular integral operator which is continuous on Lp-spaces such that
z,w∈L∞(I0;W1,2(B0))∩Lp(I0;W2,p(B0)).
Moreover, we have
∂tΔz=∂tw=curldivH
in D′(B0). As z is anti-harmonic by construction this yields
‖∂tz‖σ⩽c‖H‖σ.
In fact, ∂tz is a measurable function. Now, we truncate z to zλ with an approach similar to (5.2.8). This truncation satisfies with ‖∇2zλ‖∞⩽cλ as well as zλ=z in Oλ, where Oλ=Oλ(M(∇2z);M(∂tz)). Finally, we set
uλ:=curlzλ+curlh.
Obviously, we have divuλ=0. Due to (6.0.2) and the properties of harmonic functions we have h∈L∞(I0;Wk,2(B0)) for any k∈N (at least locally in space). Hence uλ has the same regularity as curlzλ. In particular, ∇uλ is bounded.
In Section 6.2 we develop the A-Stokes approximation for non-stationary problems, see [29]. This is, on the one hand, a non-stationary variant of the A-Stokes approximation from Section 3.3. On the other hand it is a fluid-mechanical counterpart of the A-caloric approximation from [68] which is concerned with the A-heat equation.
In this section we examine solenoidal functions, whose time derivative is only a distribution acting on solenoidal test-functions. Let u∈Lσ(I0,W1,σdiv(B0)) be such that (6.0.1) holds for some G∈Lσ(Q0). So the time derivative is only well defined via the duality with solenoidal test functions. The goal of this section is to construct a solenoidal truncation uλ of u which preserves the properties of the truncation in [65].
First we extend our function u in a suitable way to the whole space and then apply the inverse curl operator. Let γ∈C∞0(B0) with χ12B0⩽γ⩽χB0, where B0 is a ball. Let C0 denote the annulus B0∖12B0. Then according to Theorem 2.1.6 (with A(t)=B(t)=tq) there exists a Bogovskiĭ operator BogC0:C∞0,⊥(C0)→C∞0(C0) which is bounded from Lq⊥(C0)→W1,q0(C0) for all q∈(1,∞), and such that divBogC0=Id. Define
˜u:=γu−BogC0(div(γu))=γu−BogC0(∇γ⋅u).
Then div˜u=0 on I0×B0 and ˜u(t)∈W1,σ0(B0), so we can extend ˜u by zero in space to ˜u∈Lσ(I0,W1,σdiv(R3)). Since ˜u=u on I0×12B0, we have
∫Q0∂t˜u⋅ξdxdt=∫Q0G:∇ξdxdtfor allξ∈C∞0,div(12Q0).
Now, we define, pointwise in time,
w:=curl−1(˜u)=curl−1(γu−BogC0(∇γ⋅u)).
Overall, we get the following lemma.
Let us derive from (6.0.1) the equation for w. For ψ∈C∞0(12Q0) we have
∫Q0∂tu⋅curlψdxdt=∫Q0G:∇curlψdxdt.
We use u=curlw and partial integration to show that
∫Q0∂tw⋅curlcurlψdxdt=∫Q0G:∇curlψdxdt.
Now, because
∫Q0w⋅∂t∇divψdxdt=∫Q0divw∂tdivψdxdt=0
and curlcurlψ=−Δψ+∇divψ we obtain
∫Q0w⋅∂tΔψdxdt=−∫Q0G:∇curlψdxdt
for every ψ∈C∞0(12Q0). We can rewrite this as
∫Q0w⋅∂tΔψdxdt=−∫Q0H:∇2ψdxdt,
with |G|∼|H| pointwise. In particular, in the sense of distributions we have
∂tΔw=−curldivG=−divdivH.
So in passing from u to w we got a system valid for all test functions ψ∈C∞0(Q0). However, we only have control of ∂tΔw, so that the time derivative of the harmonic part of w cannot be seen. Hence, a parabolic Poincaré inequality for w still does not hold; i.e. ∂tw is not controlled! In order to remove this harmonic invariance we will replace w by some function z such that ∂tΔw=∂tΔz. This will imply that ∂tz can be controlled by H. To define z conveniently we need some auxiliary results.
For a ball B′⊂R3 and a function f∈Ls(B′) we define Δ−2B′Δf as the weak solution F∈W2,s0(B′) of
∫B′ΔFΔφdx=∫B′fΔφdxfor allφ∈C∞0(B′).
Then f−Δ(Δ−2B′Δf) is harmonic on B′.
According to [117] and Lemma 2.1 of [140] we have the following variational estimate.
This implies the following two corollaries.
For V∈Ls(B′) we define Δ−2B′divdivV as the weak solution F∈W2,s0(B′) of
∫B′ΔFΔφdx=∫B′V∇2φdxfor allφ∈C∞0(B′).
Similar to Corollary 6.1.1 we get the following result.
The next lemma shows the wanted control of the time derivative.
Defining z(t):=z12Q0(t)=ΔΔ−212B0Δw(t) for t∈12I0, we then have
∫Q0z⋅∂tΔψdxdt=∫Q0w⋅∂tΔψdxdt=−∫Q0H:∇2ψdxdt,
for all ψ∈C∞0(12Q0). Since the function Δ−212B0w(t)∈W2,s0(12B0), we can extend it by zero to a function in W2,s(R3). In this sense it is natural to extend z(t) by zero to a function in Ls(R3).
Note that Lemma 6.1.3 enables us to control ∂tz by H in Ls(12Q0).
For λ,α>0 and σ>1 we define
Oαλ:={Mασ(χ13Q0|∇2z|)>λ}∪{αMασ(χ13Q0|∂tz|)>λ}.
Later we will choose α=λ2−p and σ smaller than the integrability exponent of ∂tz. We want to redefine z on Oαλ. The first step is to cover Oαλ by well selected cubes. By the lower-semi-continuity property of the maximal functions the set Oαλ is open. We assume in the following that Oαλ is non-empty. (In the case that Oαλ is empty, we do not need to truncate at all.) We cover Oαλ by an α-parabolic Whitney covering {Qi} with partition of unity in accordance with Lemmas 5.2.1 and 5.2.2.
Due to property (PW3) we have that 16Qj∩(Rd+1∖Oαλ)≠∅. Thus, the definition of Oαλ implies that
(⨍16Qj|∇2z|σχ13Q0dxdt)1σ⩽λ,
α(⨍16Qj|∂tz|σχ13Q0dxdt)1σ⩽λ.
Let us show that the assumption λp|Oαλ|⩽c0 from Lemma 6.1.5 is satisfied in our situation. To do this we assume from now on that
α:=λ2−p
and that σ<min{p,p′}.
In the following we choose λ0 such that the conclusion of Lemma 6.1.5 is valid and assume λ⩾λ0. Without loss of generality we can assume further that
λ0⩾(⨍13Q0|∇2z|σdxdt)1σ+r−20(⨍13Q0|z|σdxdt)1σ.
We define
I:={i:Qi∩14Q0≠∅}.
Then Lemma 6.1.5 implies that Qi⊂13Q0 (and Qj⊂13Q0 for j∈Ai) for all i∈I. For each i∈I we define local approximation zi for z on Qi by
zi:=Π0IiΠ1Bi(z),
where Π1Bi(z) is the first order averaged Taylor polynomial [37,63] with respect to space and Π0Ii is the zero order averaged Taylor polynomial in time. Note that this definition implies the Poincaré-type inequality.
We can now define our truncation zαλ for λ⩾λ0 on 14Q0 by
zαλ:=z−∑i∈Iφi(z−zi).
It suffices to sum over i with Qi∩14Q0≠∅.
Since the φi are locally finite, this sum is pointwise well-defined. We will see later that the sum converges also in other topologies. Using ∑i∈Iφi=1 on 14Q0, we can also write zαλ in the form
zαλ={zin14Q0∖Oαλ,∑i∈Iφiziin14Q0∩Oαλ.
In the following we describe some properties of the truncation (e.g. ∇2zαλ∈L∞(14Q0)).
Next, we prove the stability of the truncation.
The truncation zαλ has better regularity properties than z. Indeed, ∇z is Lipschitz.
The next lemma will control the time error we get when we apply the truncation as a test function.
The following corollary is useful in the application of the solenoidal Lipschitz truncation.
The next corollary follows by combining Lemma 6.1.9, Lemma 6.1.10, Theorem 6.1.24 (g) (with α=1) and the continuity of curl−1 with a scaling procedure.
By A we denote a symmetric, elliptic tensor, i.e.
c0|τ|2⩽A(τ,τ)⩽c1|τ|2for allτ∈Rd×d.
We set |A|:=c1/c0. Let B⊂Rd be a ball and J=(t0,t1) a bounded interval. We set Q=J×B. For a function w∈L1(Q) with ∂tw∈Lq′(J;W−1,q′(B)) we introduce the unique function Hw∈Lq′0(Q) with
∫Qw⋅∂tφdxdt=∫QHw:∇φdxdt
for all φ∈C∞0,div(Q). We begin with a variational inequality for the non-stationary A-Stokes system.
Let us now state the A-Stokes approximation. In the following let B be a ball with radius r and J an interval with length 2r2. Let ˜Q denote either Q=J×B or 2Q. We use similar notations for ˜J and ˜B.
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