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Decay Estimate for some Toy-models related to the Navier-Stokes system

F. Hounkpe111Email address: [email protected] Mathematical Institute, University of Oxford, Oxford, UK
Abstract

We prove in this paper a decay estimate for scaling invariant local energy solutions for some toy-models related to the incompressible Navier-Stokes system.

1 Introduction

We consider the following models

∂tu−Δ​u−κ​∇div⁡u+u⋅∇u+u2​div⁡u=0in ​ℝ3×(0,+∞)\partial_{t}u-\Delta u-\kappa\nabla\operatorname*{\mbox{div}}u+u\cdot\nabla u+\frac{u}{2}\operatorname*{\mbox{div}}u=0\quad\mbox{in }\mathbb{R}^{3}\times(0,+\infty) (1.1)

and

∂tu−Δ​u−κ​∇div⁡u+div⁡(u⊗u+|u|2​I3)=0in ​ℝ3×(0,+∞),\partial_{t}u-\Delta u-\kappa\nabla\operatorname*{\mbox{div}}u+\operatorname*{\mbox{div}}\left(u\otimes u+\frac{|u|}{2}I_{3}\right)=0\quad\mbox{in }\mathbb{R}^{3}\times(0,+\infty), (1.2)

where I3I_{3} denotes the identity matrix in 33D and κ≥0\kappa\geq 0. Notice that if uu is a solution of either one of the previous systems, then for all λ>0\lambda>0, the vector field uλ:(x,t)↦λ​u​(λ​x,λ2​t)u^{\lambda}:(x,t)\mapsto\lambda u(\lambda x,\lambda^{2}t) is also a solution to that system. An interesting question now is whether or not one can construct a scale invariant solution uu, i.e. uλ=uu^{\lambda}=u for all λ>0\lambda>0, to system (1.1) and (1.2). This question was tackled and solved in [3] for the Cauchy problem for the 33D incompressible Navier-Stokes system

∂tv−Δ​v+v⋅∇v+∇p=0,div⁡v=0,v|t=0=u0,}\left.\begin{gathered}\partial_{t}v-\Delta v+v\cdot\nabla v+\nabla p=0,\\ \operatorname*{\mbox{div}}v=0,\\ v|_{t=0}=u_{0},\end{gathered}\right\} (1.3)

where λ​u0​(λ​x)=u0\lambda u_{0}(\lambda x)=u_{0} for all λ>0\lambda>0. To achieve this, Jia and Šverák looked for solutions of the form v​(x,t)=1t​U​(xt)v(x,t)=\frac{1}{\sqrt{t}}U(\frac{x}{\sqrt{t}}), where UU satisfies the following system

−Δ​U+U⋅∇U−x2⋅∇U−U2+∇P=0,div⁡U=0.}in ℝ3.\left.\begin{gathered}-\Delta U+U\cdot\nabla U-\frac{x}{2}\cdot\nabla U-\frac{U}{2}+\nabla P=0,\\ \operatorname*{\mbox{div}}U=0.\end{gathered}\right\}\mbox{in }\mathbb{R}^{3}. (1.4)

And they require the following asymptotics on UU

|U​(x)−u0​(x)|=o​(|x|−1)as ​|x|→∞.|U(x)-u_{0}(x)|=o(|x|^{-1})\quad\mbox{as }|x|\to\infty. (1.5)

They solve the problem (1.4)-(1.5) thanks to Leray-Schauder degree theory applied in a suitable function space. This was make possible by some quite technical decay estimates (see Theorem 4.1 in [3]). Our goal in this note is to establish a similar decay estimate for system (1.1) and (1.2). The particularity here being that the leading term in our systems is the Lamé operator for which the required regularity results we need seem not to appear in the literature. We will focus mostly on system (1.1) since the computations for system (1.2) are the same if not simpler because of the divergence structure of the non-linearity.
In section 2 and 3, we prove the intermediate results we need for the proof of our main theorem; those are ϵ−\epsilon-regularity results for a generalised version of system (1.1) and a so-called local near initial time regularity result. In section 4, we prove our main result.

1.1 Preliminaries

Before continuing our development, let us explain our notations

z=(x,t),z0=(x0,t0),B​(x0,R)={|x−x0|<R};z=(x,t),\quad z_{0}=(x_{0},t_{0}),\quad B(x_{0},R)=\{|x-x_{0}|<R\};
Q​(z0,R)=B​(x0,R)×(t0−R2,t0);Q(z_{0},R)=B(x_{0},R)\times(t_{0}-R^{2},t_{0});
B​(r)=B​(0,r),Q​(r)=Q​(0,r),B=B​(1),Q=Q​(1);B(r)=B(0,r),\quad Q(r)=Q(0,r),\quad B=B(1),\quad Q=Q(1);

For Ω\Omega an open subset of ℝn\mathbb{R}^{n} and −∞≤T1<T2≤+∞-\infty\leq T_{1}<T_{2}\leq+\infty. Set QT1,T2:=Ω×(T1,T2)Q_{T_{1},T_{2}}:=\Omega\times(T_{1},T_{2}). We will be using Lm,n​(QT1,T2):=Ln​(T1,T2;Lm​(Ω))L_{m,n}(Q_{T_{1},T_{2}}):=L_{n}(T_{1},T_{2};L_{m}(\Omega)), the Lebesgue space with the norm

‖v‖m,n,QT1,T2={(∫T1T2‖v​(⋅,t)‖Lm​(Ω)n​𝑑t)1/n,1≤n<∞ess​sup(T1,T2)‖v​(⋅,t)‖Lm​(Ω),n=∞,\|v\|_{m,n,Q_{T_{1},T_{2}}}=\begin{cases}\left(\int_{T_{1}}^{T_{2}}\|v(\cdot,t)\|^{n}_{L_{m}(\Omega)}dt\right)^{1/n},\quad&1\leq n<\infty\\ \mbox{ess}\displaystyle\sup_{(T_{1},T_{2})}\|v(\cdot,t)\|_{L_{m}(\Omega)},\quad&n=\infty,\end{cases}
Lm​(QT1,T2)=Lm,m​(QT1,T2),‖v‖m,m,QT1,T2=‖v‖m,QT1,T2;L_{m}(Q_{T_{1},T_{2}})=L_{m,m}(Q_{T_{1},T_{2}}),\quad\|v\|_{m,m,Q_{T_{1},T_{2}}}=\|v\|_{m,Q_{T_{1},T_{2}}};

Wm,n1,0​(QT1,T2)W^{1,0}_{m,n}(Q_{T_{1},T_{2}}), Wm,n2,1​(QT1,T2)W^{2,1}_{m,n}(Q_{T_{1},T_{2}}) are the Sobolev spaces with mixed norm,

Wm,n1,0​(QT1,T2)={v,∇v∈Lm,n​(QT1,T2)},W^{1,0}_{m,n}(Q_{T_{1},T_{2}})=\left\{v,\nabla v\in L_{m,n}(Q_{T_{1},T_{2}})\right\},
Wm,n2,1​(QT1,T2)={v,∇v,∇2v,∂tv∈Lm,n​(QT1,T2)},W^{2,1}_{m,n}(Q_{T_{1},T_{2}})=\left\{v,\nabla v,\nabla^{2}v,\partial_{t}v\in L_{m,n}(Q_{T_{1},T_{2}})\right\},
Wm1,0​(QT1,T2)=Wm,m1,0​(QT1,T2),Wm2,1​(QT1,T2)=Wm,m2,1​(QT1,T2).W^{1,0}_{m}(Q_{T_{1},T_{2}})=W^{1,0}_{m,m}(Q_{T_{1},T_{2}}),\quad W^{2,1}_{m}(Q_{T_{1},T_{2}})=W^{2,1}_{m,m}(Q_{T_{1},T_{2}}).

We use the following mean value notations:

(f)z0,R=1|Q​(R)|​∫Q​(z0,R)g​(z)​𝑑z,(g),R=(g)0,R,(f)_{z_{0},R}=\frac{1}{|Q(R)|}\int_{Q(z_{0},R)}g(z)dz,\quad(g)_{,R}=(g)_{0,R},

where |Ω||\Omega|, in the above, stands for the 44-dimensional Lebesgue measure of the domains Ω\Omega.
We use cc or CC to denote an absolute constant and we write C​(A,B,…)C(A,B,\ldots) when the constant depends on the parameters A,B,…A,B,\ldots
Finally, we denote by eν​Δ​te^{\nu\Delta t} the semigroup associated to the heat equation

∂tu−ν​Δ​u=0.\partial_{t}u-\nu\Delta u=0. (1.6)

To be more precise, u​(x,t)=eν​Δ​t​u0​(x)u(x,t)=e^{\nu\Delta t}u_{0}(x) denotes the unique solution of (1.6) in ℝn×(0,∞)\mathbb{R}^{n}\times(0,\infty) with initial data u0u_{0}. Moreover, if u0u_{0} belongs to a suitable function space, the following formula is available.

eν​Δ​t​u0​(x)=∫ℝnΓν​(x−y,t)​u0​(y)​𝑑y,e^{\nu\Delta t}u_{0}(x)=\int_{\mathbb{R}^{n}}\Gamma_{\nu}(x-y,t)u_{0}(y)dy,

where

Γν​(x,t)=1(4​π​ν​t)n2​exp⁡(−|x|24​ν​t).\Gamma_{\nu}(x,t)=\frac{1}{(4\pi\nu t)^{\frac{n}{2}}}\exp\left(-\frac{|x|^{2}}{4\nu t}\right).

In order to state our main result we need the following definition. Our setting is as follows: we take u0∈L2,l​o​c​(ℝ3)u_{0}\in L_{2,loc}(\mathbb{R}^{3}) such that supx0∈ℝ3∫B1​(x0)|u|2​𝑑x<∞\sup_{x_{0}\in\mathbb{R}^{3}}\int_{B_{1}(x_{0})}|u|^{2}dx<\infty.

Definition 1.1.

A vector field u∈L2,l​o​c​(ℝ3×[0,∞))u\in L_{2,loc}(\mathbb{R}^{3}\times[0,\infty)) is called a Leray solution or local energy solution to system (1.1) with initial data u0u_{0} (as above) if

  1. 1.

    For all R>0R>0, we have

    supx0∈ℝ3(ess sup0≤t<R2​∫B​(x0,R)|u​(x,t)|2​𝑑x+∫0R2∫B​(x0,R)|∇u​(x,t)|2​𝑑x​𝑑t)<∞;\sup_{x_{0}\in\mathbb{R}^{3}}\left(\operatorname*{\mbox{ess sup}}_{0\leq t<R^{2}}\int_{B(x_{0},R)}|u(x,t)|^{2}dx+\int_{0}^{R^{2}}\int_{B(x_{0},R)}|\nabla u(x,t)|^{2}dxdt\right)<\infty;
  2. 2.

    The vector field uu solves the Cauchy problem for (1.1) in the following sense:

    ∂tu−Δ​u−κ​∇div⁡u+u⋅∇u+u2​div⁡u=0in ​𝒟′​(ℝ3×(0,∞)),\partial_{t}u-\Delta u-\kappa\nabla\operatorname*{\mbox{div}}u+u\cdot\nabla u+\frac{u}{2}\operatorname*{\mbox{div}}u=0\quad\mbox{in }\mathcal{D}^{\prime}(\mathbb{R}^{3}\times(0,\infty)),

    and

    ‖u​(⋅,t)−u0‖L2​(K)→t→0+0,\|u(\cdot,t)-u_{0}\|_{L_{2}(K)}\xrightarrow[t\to 0^{+}]{}0,

    for any compact set K⊂ℝ3K\subset\mathbb{R}^{3};

  3. 3.

    For any smooth ϕ≥0\phi\geq 0 with suppϕ⊂⊂ℝ3×(0,∞)\phi\subset\subset\mathbb{R}^{3}\times(0,\infty), the following local energy inequality holds

    ∫0∞∫ℝ3(|∇u|2+κ​(div⁡u)2)​ϕ​(x,t)​𝑑x​𝑑t≤∫0∞∫ℝ3|u|22​(∂tϕ+Δ​ϕ)​𝑑x​𝑑t+∫0∞∫ℝ3(|u|22−κ​div⁡u)​u⋅∇ϕ​d​x​d​t\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\left(|\nabla u|^{2}+\kappa(\operatorname*{\mbox{div}}u)^{2}\right)\phi(x,t)dxdt\leq\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\frac{|u|^{2}}{2}(\partial_{t}\phi+\Delta\phi)dxdt\\ +\int_{0}^{\infty}\int_{\mathbb{R}^{3}}(\frac{|u|^{2}}{2}-\kappa\operatorname*{\mbox{div}}u)u\cdot\nabla\phi dxdt

An analogous definition is available for system (1.2). The proof of existence of these Leray solutions follows the same lines as the one of the so-called Lemarié-Rieusset (see e.g. [9] Chapters 32 & 33 or [13] Appendix B) and will be presented elsewhere.

1.2 Main result

Take u0=(u01,u02,u03)u_{0}=(u_{0}^{1},u_{0}^{2},u_{0}^{3}) which is a (−1)(-1)-homogeneous vector field such that u0|∂B1∈C∞​(∂B1)u_{0}|_{\partial B_{1}}\in C^{\infty}(\partial B_{1}). In this case, one can steadily show that

|∂αu0​(x)|≤C​(α,u0)|x|1+|α|,∀α∈ℕ3.|\partial^{\alpha}u_{0}(x)|\leq\frac{C(\alpha,u_{0})}{|x|^{1+|\alpha|}},\quad\forall\alpha\in\mathbb{N}^{3}.

We have the following decay estimate

Theorem 1.1.

Let u0u_{0} as above, and uu be a scale invariant Leray solution to system (1.1) or system (1.2). Then U​(⋅):=u​(⋅,1)U(\cdot):=u(\cdot,1), the solution profile at time t=1t=1, belongs to C∞​(ℝ3)C^{\infty}(\mathbb{R}^{3}) and

|∂α(U−Sκ​(1)​u0)​(x)|≤C​(α,κ,u0)(1+|x|)3+|α|,|\partial^{\alpha}(U-S_{\kappa}(1)u_{0})(x)|\leq\frac{C(\alpha,\kappa,u_{0})}{(1+|x|)^{3+|\alpha|}},

for all α∈ℕ3\alpha\in\mathbb{N}^{3} (with |α|=α1+α2+α3|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}). Here Sκ​(t)S_{\kappa}(t) denotes the semigroup associate to the time dependent Lamé system:

∂tf−Δ​f−κ​∇div⁡f=0.\partial_{t}f-\Delta f-\kappa\nabla\operatorname*{\mbox{div}}f=0.

2 ϵ\epsilon-Regularity

We prove in this section an ϵ\epsilon-regularity criteria similar to that of Caffarelli-Kohn-Nirenberg (see [7]) for a generalised counterpart of system (1.1). Our setting is as follows:
Let 𝒪\mathcal{O} be an open subset of ℝ3+1\mathbb{R}^{3+1} and a,b∈Lm​(𝒪)a,b\in L_{m}(\mathcal{O}) with m>5m>5. A function uu is called suitable weak solution to

∂tu−Δ​u−κ​∇div⁡u+u⋅∇u+u2​div⁡u+a2⋅∇u+div⁡(u⊗a2)+div⁡(b⊗u)−b2​div⁡u=0,\partial_{t}u-\Delta u-\kappa\nabla\operatorname*{\mbox{div}}u+u\cdot\nabla u+\frac{u}{2}\operatorname*{\mbox{div}}u+\frac{a}{2}\cdot\nabla u+\operatorname*{\mbox{div}}(u\otimes\frac{a}{2})+\operatorname*{\mbox{div}}(b\otimes u)-\frac{b}{2}\operatorname*{\mbox{div}}u=0, (2.1)

in 𝒪\mathcal{O} if u∈L2,∞​(𝒪)∩W21,0​(𝒪)u\in L_{2,\infty}(\mathcal{O})\cap W^{1,0}_{2}(\mathcal{O}), satisfies (2.1) in the sense of distribution in 𝒪\mathcal{O} and

∂t|u|22−Δ​|u|22−κ​div⁡(u​div⁡u)+|∇u|2+κ​(div⁡u)2+div⁡((u+a)​|u|22)+u⋅div⁡(b⊗u)−12​u⋅b​div⁡u≤0\partial_{t}\frac{|u|^{2}}{2}-\Delta\frac{|u|^{2}}{2}-\kappa\operatorname*{\mbox{div}}(u\operatorname*{\mbox{div}}u)+|\nabla u|^{2}+\kappa(\operatorname*{\mbox{div}}u)^{2}+\operatorname*{\mbox{div}}\left((u+a)\frac{|u|^{2}}{2}\right)\\ +u\cdot\operatorname*{\mbox{div}}(b\otimes u)-\frac{1}{2}u\cdot b\operatorname*{\mbox{div}}u\leq 0 (2.2)

in the sense of distributions. Let us point out that the term u⋅div⁡(a⊗u)u\cdot\operatorname*{\mbox{div}}(a\otimes u) is indeed a distribution and should be understood in the following way

⟨u⋅div⁡(a⊗u),ϕ⟩=−∫𝒪ai​uj​ui,j​ϕ​𝑑x​𝑑t−∫𝒪ai​uj​ui​ϕ,j​𝑑x​𝑑t,\langle u\cdot\operatorname*{\mbox{div}}(a\otimes u),\phi\rangle=-\int_{\mathcal{O}}a_{i}u_{j}u_{i,j}\phi dxdt-\int_{\mathcal{O}}a_{i}u_{j}u_{i}\phi_{,j}dxdt,

for all ϕ∈C0∞​(𝒪)\phi\in C^{\infty}_{0}(\mathcal{O}). All the other terms in (2.2) obviously make sense because of the energy class of uu and the integrability condition on a,ba,b.
The main theorem of this section reads as follows

Theorem 2.1 (ϵ\epsilon-regularity criterion).

Let uu be a suitable weak solution to (2.1) in QQ with a,b∈Lm​(Q)a,b\in L_{m}(Q) and m>5m>5. Then there exists ϵ0=ϵ0​(κ,m)>0\epsilon_{0}=\epsilon_{0}(\kappa,m)>0 with the following properties: if

(−​∫Q|u|3​𝑑z)13+(−​∫Q|a|m​𝑑z)1m+(−​∫Q|b|m​𝑑z)1m≤ϵ0,\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|u|^{3}dz\right)^{\frac{1}{3}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b|^{m}dz\right)^{\frac{1}{m}}\leq\epsilon_{0}, (2.3)

then uu is Hölder continuous in Q​(12)¯\overline{Q(\frac{1}{2})} with exponent α=α​(m)∈(0,1)\alpha=\alpha(m)\in(0,1) and

‖u‖Cα,α2​(Q​(12)¯)≤C​(ϵ0,κ,m).\|u\|_{C^{\alpha,\frac{\alpha}{2}}(\overline{Q(\frac{1}{2})})}\leq C(\epsilon_{0},\kappa,m). (2.4)
Remark 1.

The proof of this theorem we present here is inspired from the proof of a similar result in the case of the 33D incompressible Navier-Stokes equations given in [10, 13]. The particular case a=b=0a=b=0 was tackled in [12] (see Theorem 7.1). Finally, let us point out that the case κ=0\kappa=0 and a=b=0a=b=0 was tackled in [4] using a more direct method which could also be extended to the general case (2.1) upon some minor changes.

The following auxiliary results will be needed for the proof of Theorem 2.1.

Lemma 2.2.

Set QT:=ℝ3×(0,T)Q_{T}:=\mathbb{R}^{3}\times(0,T) (T>0T>0); let f∈Ls,l​(QT)f\in L_{s,l}(Q_{T}) and F∈Ls,l​(QT;ℝ3×3)F\in L_{s,l}(Q_{T};\mathbb{R}^{3\times 3}) with 1<s,l<∞1<s,l<\infty. There exist two functions vv and ww that uniquely solve the systems

{∂tv−Δ​v−κ​∇div⁡v=fin ​QTv|t=0=0in ​ℝ3and{∂tw−Δ​w−κ​∇div⁡w=div⁡Fin ​QTw|t=0=0in ​ℝ3\left\{\begin{gathered}\partial_{t}v-\Delta v-\kappa\nabla\operatorname*{\mbox{div}}v=f\quad\mbox{in }Q_{T}\\ v|_{t=0}=0\quad\mbox{in }\mathbb{R}^{3}\end{gathered}\right.\quad\mbox{and}\quad\left\{\begin{gathered}\partial_{t}w-\Delta w-\kappa\nabla\operatorname*{\mbox{div}}w=\operatorname*{\mbox{div}}F\quad\mbox{in }Q_{T}\\ w|_{t=0}=0\quad\mbox{in }\mathbb{R}^{3}\end{gathered}\right.

such that

‖∂tv‖Ls,l​(QT)+‖∇2v‖Ls,l​(QT)≤c​(κ,s,l)​‖f‖Ls,l​(QT),‖∇w‖Ls,l​(QT)≤c​(κ,s,l)​‖F‖Ls,l​(QT);\displaystyle\|\partial_{t}v\|_{L_{s,l}(Q_{T})}+\|\nabla^{2}v\|_{L_{s,l}(Q_{T})}\leq c(\kappa,s,l)\|f\|_{L_{s,l}(Q_{T})},\quad\|\nabla w\|_{L_{s,l}(Q_{T})}\leq c(\kappa,s,l)\|F\|_{L_{s,l}(Q_{T})};
‖∇v‖Ll1​(0,T;Ls1​(ℝ3))≤c​(κ,s,s1,l,l1)​T12​(1−2l−3s+2l1+3s1)​‖f‖Ls,l​(QT)​∀T≥0,\displaystyle\|\nabla v\|_{L_{l_{1}}(0,T;L_{s_{1}}(\mathbb{R}^{3}))}\leq c(\kappa,s,s_{1},l,l_{1})T^{\frac{1}{2}\left(1-\frac{2}{l}-\frac{3}{s}+\frac{2}{l_{1}}+\frac{3}{s_{1}}\right)}\|f\|_{L_{s,l}(Q_{T})}\leavevmode\nobreak\ \forall T\geq 0,
with s≤s1≤∞,l≤l1≤∞ such that ​1−2l−3s+2l1+3s1>0;\displaystyle\mbox{with $s\leq s_{1}\leq\infty,\leavevmode\nobreak\ l\leq l_{1}\leq\infty$ such that }1-\frac{2}{l}-\frac{3}{s}+\frac{2}{l_{1}}+\frac{3}{s_{1}}>0;
‖v‖Ll2​(0,T;Ls2​(ℝ3))≤c​(κ,s,s2,l,l2)​T12​(2−2l−3s+2l2+3s2)​‖f‖Ls,l​(QT)​∀T≥0,\displaystyle\|v\|_{L_{l_{2}}(0,T;L_{s_{2}}(\mathbb{R}^{3}))}\leq c(\kappa,s,s_{2},l,l_{2})T^{\frac{1}{2}\left(2-\frac{2}{l}-\frac{3}{s}+\frac{2}{l_{2}}+\frac{3}{s_{2}}\right)}\|f\|_{L_{s,l}(Q_{T})}\leavevmode\nobreak\ \forall T\geq 0,
with s≤s2≤∞,l≤l2≤∞ such that ​2−2l−3s+2l2+3s2>0;\displaystyle\mbox{with $s\leq s_{2}\leq\infty,\leavevmode\nobreak\ l\leq l_{2}\leq\infty$ such that }2-\frac{2}{l}-\frac{3}{s}+\frac{2}{l_{2}}+\frac{3}{s_{2}}>0;
‖w‖Ll3​(0,T;Ls3​(ℝ3))≤c​(κ,s,s3,l,l3)​T12​(1−2l−3s+2l3+3s3)​‖F‖Ls,l​(QT)​∀T≥0,\displaystyle\|w\|_{L_{l_{3}}(0,T;L_{s_{3}}(\mathbb{R}^{3}))}\leq c(\kappa,s,s_{3},l,l_{3})T^{\frac{1}{2}\left(1-\frac{2}{l}-\frac{3}{s}+\frac{2}{l_{3}}+\frac{3}{s_{3}}\right)}\|F\|_{L_{s,l}(Q_{T})}\leavevmode\nobreak\ \forall T\geq 0,
with s≤s3≤∞,l≤l3≤∞ such that ​1−2l−3s+2l3+3s3>0.\displaystyle\mbox{with $s\leq s_{3}\leq\infty,\leavevmode\nobreak\ l\leq l_{3}\leq\infty$ such that }1-\frac{2}{l}-\frac{3}{s}+\frac{2}{l_{3}}+\frac{3}{s_{3}}>0.

Finally, we have Hölder continuity of vv as soon as μ:=2−2l−3s>0\mu:=2-\frac{2}{l}-\frac{3}{s}>0 and for ww when α:=1−2l−3s>0\alpha:=1-\frac{2}{l}-\frac{3}{s}>0; to be more precise, we have the following estimates:

|v​(z1)−v​(z2)|≤c​(κ,s,l)​(|x1−x2|+|t1−t2|)μ​‖f‖Ls,l​(QT)|v(z_{1})-v(z_{2})|\leq c(\kappa,s,l)\left(|x_{1}-x_{2}|+\sqrt{|t_{1}-t_{2}|}\right)^{\mu}\|f\|_{L_{s,l}(Q_{T})}

and similarly

|w​(z1)−w​(z2)|≤c​(κ,s,l)​(|x1−x2|+|t1−t2|)α​‖F‖Ls,l​(QT),|w(z_{1})-w(z_{2})|\leq c(\kappa,s,l)\left(|x_{1}-x_{2}|+\sqrt{|t_{1}-t_{2}|}\right)^{\alpha}\|F\|_{L_{s,l}(Q_{T})},

for all z1=(x1,t1),z2=(x2,t2)∈QTz_{1}=(x_{1},t_{1}),z_{2}=(x_{2},t_{2})\in Q_{T}.

Sketch of proof.

Uniqueness is straightforward. For the existence part, we present only the proof for the function ww since the function vv’s case follows the same ideas.
There exists a function qq (using the Newtonian representation for solutions of the Poisson equation together with singular integrals’ theory) such that

Δ​q=div⁡div⁡(F),\Delta q=\operatorname*{\mbox{div}}\operatorname*{\mbox{div}}(F),

and

‖q‖Ls,l​(QT)≤c​‖F‖Ls,l​(QT)\|q\|_{L_{s,l}(Q_{T})}\leq c\|F\|_{L_{s,l}(Q_{T})}

Next, we introduce the function F0:=div⁡(q​I3−F)F_{0}:=\operatorname*{\mbox{div}}(qI_{3}-F) and let us notice that div⁡F0=0\operatorname*{\mbox{div}}F_{0}=0 in 𝒟′​(QT)\mathcal{D}^{\prime}(Q_{T}). From well-known solvability results for the heat equation (see e.g. [6, 11]), we have the existence of two functions w1w^{1} and w2w^{2} such that

{∂tw1−(1+κ)​Δ​w1=∇qin ​QTw1|t=0=0in ​ℝ3and{∂tw2−Δ​w2=F0in ​QTw2|t=0=0in ​ℝ3\left\{\begin{gathered}\partial_{t}w^{1}-(1+\kappa)\Delta w^{1}=\nabla q\quad\mbox{in }Q_{T}\\ w^{1}|_{t=0}=0\quad\mbox{in }\mathbb{R}^{3}\end{gathered}\right.\quad\mbox{and}\quad\left\{\begin{gathered}\partial_{t}w^{2}-\Delta w^{2}=F_{0}\quad\mbox{in }Q_{T}\\ w^{2}|_{t=0}=0\quad\mbox{in }\mathbb{R}^{3}\end{gathered}\right. (2.5)

Moreover, the following estimate is available:

‖∇w1‖Ls,l​(QT)+‖∇w2‖Ls,l​(QT)≤c​(κ,s,l)​‖F‖Ls,l​(QT).\displaystyle\|\nabla w^{1}\|_{L_{s,l}(Q_{T})}+\|\nabla w^{2}\|_{L_{s,l}(Q_{T})}\leq c(\kappa,s,l)\|F\|_{L_{s,l}(Q_{T})}.
‖w1‖Ll3​(0,T;Ls3​(ℝ3))+‖w2‖Ll3​(0,T;Ls3​(ℝ3))≤c​(κ,s,s3,l,l3)​T12​(1−2l−3s+3l3+3s3)​‖F‖Ls,l​(QT)\displaystyle\|w^{1}\|_{L_{l_{3}}(0,T;L_{s_{3}}(\mathbb{R}^{3}))}+\|w^{2}\|_{L_{l_{3}}(0,T;L_{s_{3}}(\mathbb{R}^{3}))}\leq c(\kappa,s,s_{3},l,l_{3})T^{\frac{1}{2}\left(1-\frac{2}{l}-\frac{3}{s}+\frac{3}{l_{3}}+\frac{3}{s_{3}}\right)}\|F\|_{L_{s,l}(Q_{T})}
∀T≥0, with s≤s3≤∞,l≤l3≤∞ such that ​1−2l−3s+3l3+3s3>0.\displaystyle\forall T\geq 0,\mbox{ with $s\leq s_{3}\leq\infty,\leavevmode\nobreak\ l\leq l_{3}\leq\infty$ such that }1-\frac{2}{l}-\frac{3}{s}+\frac{3}{l_{3}}+\frac{3}{s_{3}}>0.

The latter estimate comes from well-known properties of the volume heat potential but the former is a bit more subtle; we refer to [5] (Theorem 1.1) for a proof of this statement. Next, using Campanato’s characterisation for Hölder continuous functions, one gets without too much difficulty (and with the help of Poincaré’s inequality on balls) that

|w1​(z1)−w1​(z2)|+|w2​(z1)−w2​(z2)|≤c​(κ,s,l)​(|x1−x2|+|t1−t2|)α​‖F‖Ls,l​(QT),|w^{1}(z_{1})-w^{1}(z_{2})|+|w^{2}(z_{1})-w^{2}(z_{2})|\leq c(\kappa,s,l)\left(|x_{1}-x_{2}|+\sqrt{|t_{1}-t_{2}|}\right)^{\alpha}\|F\|_{L_{s,l}(Q_{T})},

for all z1=(x1,t1),z2=(x2,t2)∈QTz_{1}=(x_{1},t_{1}),z_{2}=(x_{2},t_{2})\in Q_{T} as long as α>0\alpha>0.
Finally, notice that curl⁡w1=div⁡w2=0\operatorname*{\mbox{curl}}w^{1}=\operatorname*{\mbox{div}}w^{2}=0 in QTQ_{T} and Δ​w1=∇div⁡w1\Delta w^{1}=\nabla\operatorname*{\mbox{div}}w^{1} (at least in the sense of distributions); and we are done by setting w:=w1+w2w:=w^{1}+w^{2}. ∎

Remark 2.

A suitable scaling argument in (2.5) allow us to see how the constants in the above estimates depend on κ\kappa.

Next, we have the following local regularity result for the time-dependent Lamé system.

Lemma 2.3 (Local regularity).

Let u∈L2​(Q)u\in L_{2}(Q) such that

∂tu−Δ​u−κ​∇div⁡u=0​ in ​𝒟′​(Q).\partial_{t}u-\Delta u-\kappa\nabla\operatorname*{\mbox{div}}u=0\mbox{ in }\mathcal{D}^{\prime}(Q).

Then, for any k=0,1,2,…k=0,1,2,\ldots and any 0<ϱ<10<\varrho<1, there exists C=C​(κ,ϱ,k)>0C=C(\kappa,\varrho,k)>0 such that

sup(x,t)∈Q​(ϱ)|∇ku​(x,t)|≤C​(−​∫Q|u|2​𝑑z)12.\sup_{(x,t)\in Q(\varrho)}|\nabla^{k}u(x,t)|\leq C\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|u|^{2}dz\right)^{\frac{1}{2}}.
Proof.

We see without too much difficulty that

∫Q​((1+ϱ)/2)|∇u|2​𝑑z≤c​(κ,ϱ)​∫Q|u|2​𝑑z,\int_{Q((1+\varrho)/2)}|\nabla u|^{2}dz\leq c(\kappa,\varrho)\int_{Q}|u|^{2}dz, (2.6)

(for an arbitrary 0<ϱ<10<\varrho<1) and

∂tcurl⁡u−Δ​curl⁡u=0and∂tdiv⁡u−(1+κ)​Δ​div⁡u=0,\partial_{t}\operatorname*{\mbox{curl}}u-\Delta\operatorname*{\mbox{curl}}u=0\quad\mbox{and}\quad\partial_{t}\operatorname*{\mbox{div}}u-(1+\kappa)\Delta\operatorname*{\mbox{div}}u=0,

in the sense of distributions. Next, from local well-known regularity results for the heat equation (see e.g. [6, 11])

sup(x,t)∈Q​((1+3​ϱ)/4)|∇kcurl⁡u​(x,t)|≤C​(ϱ,k)​(∫Q​((1+ϱ)/2)|curl⁡u|2​𝑑z)12\displaystyle\sup_{(x,t)\in Q((1+3\varrho)/4)}|\nabla^{k}\operatorname*{\mbox{curl}}u(x,t)|\leq C(\varrho,k)\left(\int_{Q((1+\varrho)/2)}|\operatorname*{\mbox{curl}}u|^{2}dz\right)^{\frac{1}{2}}
sup(x,t)∈Q​((1+3​ϱ)/4)|∇kdiv⁡u​(x,t)|≤C​(κ,ϱ,k)​(∫Q​((1+ϱ)/2)|div⁡u|2​𝑑z)12\displaystyle\sup_{(x,t)\in Q((1+3\varrho)/4)}|\nabla^{k}\operatorname*{\mbox{div}}u(x,t)|\leq C(\kappa,\varrho,k)\left(\int_{Q((1+\varrho)/2)}|\operatorname*{\mbox{div}}u|^{2}dz\right)^{\frac{1}{2}}

for any k=0,1,2​…k=0,1,2\ldots Thus, from (2.6), we have

sup(x,t)∈Q​((1+3​ϱ)/4)(|∇kcurl⁡u​(x,t)|+|∇kdiv⁡u​(x,t)|)≤c​(κ,ϱ,k)​(∫Q|u|2​𝑑z)12,\sup_{(x,t)\in Q((1+3\varrho)/4)}(|\nabla^{k}\operatorname*{\mbox{curl}}u(x,t)|+|\nabla^{k}\operatorname*{\mbox{div}}u(x,t)|)\leq c(\kappa,\varrho,k)\left(\int_{Q}|u|^{2}dz\right)^{\frac{1}{2}}, (2.7)

for any k=0,1,2​…k=0,1,2\ldots Finally, using the identity

−Δ​u=curl⁡(curl⁡u)−∇div⁡u​ in ​𝒟′​(Q),-\Delta u=\operatorname*{\mbox{curl}}(\operatorname*{\mbox{curl}}u)-\nabla\operatorname*{\mbox{div}}u\mbox{ in }\mathcal{D}^{\prime}(Q),

together with the stationary analogue of the first estimate in Theorem 2.4.9 of [6] (for instance) and taking into account (2.7), we have that the lemma is proved. ∎

Our next lemma gives us an estimate for a generalised time dependent Lamé system.

Lemma 2.4.

Let a,b∈Lm​(Q;ℝ3)a,b\in L_{m}(Q;\mathbb{R}^{3}) and F∈Lm​(Q;ℝ3×3)F\in L_{m}(Q;\mathbb{R}^{3\times 3}) such that

‖a‖Lm​(Q)+‖b‖Lm​(Q)+‖F‖Lm​(Q)≤M,\|a\|_{L_{m}(Q)}+\|b\|_{L_{m}(Q)}+\|F\|_{L_{m}(Q)}\leq M, (2.8)

for some arbitrary M>0M>0 and m>5m>5. Let also λ∈ℝ3\lambda\in\mathbb{R}^{3} with |λ|≤M|\lambda|\leq M and u∈L2,∞​(Q)∩W1,0​(Q)u\in L_{2,\infty}(Q)\cap W^{1,0}(Q) such that

ess sup−1<t<0​∫B|u​(x,t)|2​𝑑x+∫−10∫B|∇u|2​𝑑x​𝑑t≤M2.\operatorname*{\mbox{ess sup}}_{-1<t<0}\int_{B}|u(x,t)|^{2}dx+\int_{-1}^{0}\int_{B}|\nabla u|^{2}dxdt\leq M^{2}. (2.9)

Assume uu satisfies

∂tu−Δ​u−κ​∇div⁡u+λ⋅∇u+λ2​div⁡u+a2⋅∇u+div⁡(u⊗a2)+div⁡(b⊗u)−b2​div⁡u=div⁡F,\partial_{t}u-\Delta u-\kappa\nabla\operatorname*{\mbox{div}}u+\lambda\cdot\nabla u+\frac{\lambda}{2}\operatorname*{\mbox{div}}u+\frac{a}{2}\cdot\nabla u+\operatorname*{\mbox{div}}(u\otimes\frac{a}{2})+\operatorname*{\mbox{div}}(b\otimes u)-\frac{b}{2}\operatorname*{\mbox{div}}u=\operatorname*{\mbox{div}}F,

in the sense of distributions. Then uu is Hölder continuous in Q​(12)Q(\frac{1}{2}) with exponent α=α​(m)∈(0,1)\alpha=\alpha(m)\in(0,1) and

‖u‖Cα,α2​(Q​(12)¯)≤C​(κ,m,M).\|u\|_{C^{\alpha,\frac{\alpha}{2}}(\overline{Q(\frac{1}{2})})}\leq C(\kappa,m,M).
Proof.

Let 0<ϱ<r≤10<\varrho<r\leq 1 and 0≤φϱ,r∈C0∞​(B)0\leq\varphi_{\varrho,r}\in C^{\infty}_{0}(B) such that φϱ,r≡1\varphi_{\varrho,r}\equiv 1 in B​((r+ϱ)/2)B((r+\varrho)/2) and φϱ,r≡0\varphi_{\varrho,r}\equiv 0 in B​(r)∖B​((3​r+ϱ)/4)B(r)\setminus B((3r+\varrho)/4). Next, let us set

Fϱ,r:=φϱ,r​(F−u⊗λ−λ2⊗u−u⊗a2−b⊗u),fϱ,r:=(b2​div⁡u−a2⋅∇u)​φϱ,r;F^{\varrho,r}:=\varphi_{\varrho,r}\left(F-u\otimes\lambda-\frac{\lambda}{2}\otimes u-u\otimes\frac{a}{2}-b\otimes u\right),\leavevmode\nobreak\ f^{\varrho,r}:=\left(\frac{b}{2}\operatorname*{\mbox{div}}u-\frac{a}{2}\cdot\nabla u\right)\varphi_{\varrho,r};

If u∈Lq​(Q​(r))u\in L_{q}(Q(r)) and ∇u∈Lp​(Q​(r))\nabla u\in L_{p}(Q(r)) (p,q≥2p,q\geq 2) with

1m+1q<1p+12​(1m−15)and1m+1p−25<1q+12​(1m−15)\frac{1}{m}+\frac{1}{q}<\frac{1}{p}+\frac{1}{2}\left(\frac{1}{m}-\frac{1}{5}\right)\quad\mbox{and}\quad\frac{1}{m}+\frac{1}{p}-\frac{2}{5}<\frac{1}{q}+\frac{1}{2}\left(\frac{1}{m}-\frac{1}{5}\right) (2.10)

(e.g. (2.10) holds true for q=10/3q=10/3 and p=2p=2) then

Fϱ,r∈Lm​qm+q​(Q​(r))​ and ​fϱ,r∈Lm​pm+p​(Q​(r)).F^{\varrho,r}\in L_{\frac{mq}{m+q}}(Q(r))\mbox{ and }f^{\varrho,r}\in L_{\frac{mp}{m+p}}(Q(r)).

From Lemma 2.2, we have the existence of two functions vϱ,rv^{\varrho,r} and wϱ,rw^{\varrho,r} such that

{∂tvϱ,r−Δ​vϱ,r−κ​∇div⁡vϱ,r=fϱ,rin ​ℝ3×(−1,0)vϱ,r|t=−1=0in ​ℝ3\left\{\begin{gathered}\partial_{t}v^{\varrho,r}-\Delta v^{\varrho,r}-\kappa\nabla\operatorname*{\mbox{div}}v^{\varrho,r}=f^{\varrho,r}\quad\mbox{in }\mathbb{R}^{3}\times(-1,0)\\ v^{\varrho,r}|_{t=-1}=0\quad\mbox{in }\mathbb{R}^{3}\end{gathered}\right.

and

{∂twϱ,r−Δ​wϱ,r−κ​∇div⁡wϱ,r=div⁡Fϱ,rin ​ℝ3×(−1,0)wϱ,r|t=−1=0in ​ℝ3\left\{\begin{gathered}\partial_{t}w^{\varrho,r}-\Delta w^{\varrho,r}-\kappa\nabla\operatorname*{\mbox{div}}w^{\varrho,r}=\operatorname*{\mbox{div}}F^{\varrho,r}\quad\mbox{in }\mathbb{R}^{3}\times(-1,0)\\ w^{\varrho,r}|_{t=-1}=0\quad\mbox{in }\mathbb{R}^{3}\end{gathered}\right.

Moreover the following estimates are available

‖wϱ,r‖Lq^​(ℝ3×(−1,0))+‖∇wϱ,r‖Lm​qm+q​(ℝ3×(−1,0))≤c​(m,p,q)​‖Fϱ,r‖Lm​qm+q​(Q​(r)),\|w^{\varrho,r}\|_{L_{\hat{q}}(\mathbb{R}^{3}\times(-1,0))}+\|\nabla w^{\varrho,r}\|_{L_{\frac{mq}{m+q}}(\mathbb{R}^{3}\times(-1,0))}\leq c(m,p,q)\|F^{\varrho,r}\|_{L_{\frac{mq}{m+q}}(Q(r))},

with

1q^>1q+1m−15,\frac{1}{\hat{q}}>\frac{1}{q}+\frac{1}{m}-\frac{1}{5},

and

‖vϱ,r‖Lq~​(ℝ3×(−1,0))+‖∇vϱ,r‖Lp~​(ℝ3×(−1,0))≤c​(m,p,q)​‖fϱ,r‖Lm​pm+p​(Q​(r)),\|v^{\varrho,r}\|_{L_{\tilde{q}}(\mathbb{R}^{3}\times(-1,0))}+\|\nabla v^{\varrho,r}\|_{L_{\tilde{p}}(\mathbb{R}^{3}\times(-1,0))}\leq c(m,p,q)\|f^{\varrho,r}\|_{L_{\frac{mp}{m+p}}(Q(r))},

with

1p~>1p+1m−15and1q~>1p+1m−25.\frac{1}{\tilde{p}}>\frac{1}{p}+\frac{1}{m}-\frac{1}{5}\quad\mbox{and}\quad\frac{1}{\tilde{q}}>\frac{1}{p}+\frac{1}{m}-\frac{2}{5}.

Now, we choose q1≥2q_{1}\geq 2 and p1≥2p_{1}\geq 2 such that

1q1=1q+12​(1m−15)and1p1=1p+12​(1m−15);\frac{1}{q_{1}}=\frac{1}{q}+\frac{1}{2}\left(\frac{1}{m}-\frac{1}{5}\right)\quad\mbox{and}\quad\frac{1}{p_{1}}=\frac{1}{p}+\frac{1}{2}\left(\frac{1}{m}-\frac{1}{5}\right); (2.11)

because of (2.10), we have that

‖wϱ,r‖Lq1​(Q)+‖∇wϱ,r‖Lp1​(Q)≤c​(κ,m,p,q)​‖Fϱ,r‖Lm​qm+q​(Q​(r)),‖vϱ,r‖Lq1​(Q)+‖∇vϱ,r‖Lp1​(Q)≤c​(κ,m,p,q)​‖fϱ,r‖Lm​pm+p​(Q​(r));\displaystyle\begin{split}\|w^{\varrho,r}\|_{L_{q_{1}}(Q)}+\|\nabla w^{\varrho,r}\|_{L_{p_{1}}(Q)}\leq c(\kappa,m,p,q)\|F^{\varrho,r}\|_{L_{\frac{mq}{m+q}}(Q(r))},\\ \|v^{\varrho,r}\|_{L_{q_{1}}(Q)}+\|\nabla v^{\varrho,r}\|_{L_{p_{1}}(Q)}\leq c(\kappa,m,p,q)\|f^{\varrho,r}\|_{L_{\frac{mp}{m+p}}(Q(r))};\end{split} (2.12)

moreover (2.10) is true for qq and pp replaced respectively by q1q_{1} and p1p_{1}.
Finally, let us notice that

∂t(u−vϱ,r−wϱ,r)−Δ​(u−vϱ,r−wϱ,r)−κ​∇div⁡(u−vϱ,r−wϱ,r)=0\partial_{t}(u-v^{\varrho,r}-w^{\varrho,r})-\Delta(u-v^{\varrho,r}-w^{\varrho,r})-\kappa\nabla\operatorname*{\mbox{div}}(u-v^{\varrho,r}-w^{\varrho,r})=0 (2.13)

in Q​((r+ϱ)/2)Q((r+\varrho)/2). which together with Lemma 2.3 lead to u∈Lq1​(Q​(ϱ))u\in L_{q_{1}}(Q(\varrho)) and ∇u∈Lp1​(Q​(ϱ))\nabla u\in L_{p_{1}}(Q(\varrho)) where q1>qq_{1}>q and p1>pp_{1}>p. The goal now is to iterate this process. We set q0=10/3q_{0}=10/3, p0=2p_{0}=2 and we define a sequence (qk)(q_{k}) and (pk)(p_{k}) via the following recursive formula

1qk=1qk−1+12​(1m−15)and1pk=1pk−1+12​(1m−15)\frac{1}{q_{k}}=\frac{1}{q_{k-1}}+\frac{1}{2}\left(\frac{1}{m}-\frac{1}{5}\right)\quad\mbox{and}\quad\frac{1}{p_{k}}=\frac{1}{p_{k-1}}+\frac{1}{2}\left(\frac{1}{m}-\frac{1}{5}\right)

and we find u∈Lqk​(Q​(2−1+2−(1+k)))u\in L_{q_{k}}(Q(2^{-1}+2^{-(1+k)})) and ∇u∈Lpk​(Q​(2−1+2−(1+k)))\nabla u\in L_{p_{k}}(Q(2^{-1}+2^{-(1+k)})) for all k≥0k\geq 0. Now, for a large enough k0=k0​(m)k_{0}=k_{0}(m), we have that

m​qk0m+qk0>5​ and ​m​pk0m+pk0>52\frac{mq_{k_{0}}}{m+q_{k_{0}}}>5\mbox{ and }\frac{mp_{k_{0}}}{m+p_{k_{0}}}>\frac{5}{2}

thus for ϱ=2−1+2−(2+k0)\varrho=2^{-1}+2^{-(2+k_{0})} and r=2−1+2−(1+k0)r=2^{-1}+2^{-(1+k_{0})}, we find that there exists α=α​(m)∈(0,1)\alpha=\alpha(m)\in(0,1) such that vϱ,r,wϱ,r∈Cα,α2​(Q​(2−1+2−(2+k0)))v^{\varrho,r},w^{\varrho,r}\in C^{\alpha,\frac{\alpha}{2}}(Q(2^{-1}+2^{-(2+k_{0})})). Using one more time (2.13) and Lemma 2.3, we get that u∈Cα,α2​(Q​(1/2)¯)u\in C^{\alpha,\frac{\alpha}{2}}(\overline{Q(1/2)}). A careful track of the constants in the above process yields also the estimate claimed in the Lemma, we omit the details here. Thus the lemma is proved. ∎

We can return now to the proof of Theorem 2.1. We first prove a so-called "oscillation lemma", which roughly speaking asserts that if uu is of small oscillation in QQ, then the oscillation is even smaller in QθQ_{\theta} for 0<θ<10<\theta<1 (see e.g. [13, 3] where an analogue of this result was proved for the incompressible Navier-Stokes equations).

In order to state our lemma and for what follows, some additional notions are needed; we introduce

Y​(z0,R;u):=(−​∫Q​(z0,R)|u−(u)z0,R|3​𝑑z)13,Yθ​(u):=Y​(0,θ;u)Y(z_{0},R;u):=\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},R)}|u-(u)_{z_{0},R}|^{3}dz\right)^{\frac{1}{3}},\leavevmode\nobreak\ Y_{\theta}(u):=Y(0,\theta;u)
Lemma 2.5.

Given any numbers θ∈(0,1/3)\theta\in(0,1/3), m>5m>5 and M>0M>0, there are three constants α=α​(m)∈(0,1)\alpha=\alpha(m)\in(0,1), ϵ=ϵ​(θ,m,M)>0\epsilon=\epsilon(\theta,m,M)>0 and C1=C1​(κ,m,M)>0C_{1}=C_{1}(\kappa,m,M)>0 such that for any ‖a‖Lm​(Q)≤M,‖b‖Lm​(Q)≤c0\|a\|_{L_{m}(Q)}\leq M,\leavevmode\nobreak\ \|b\|_{L_{m}(Q)}\leq c_{0}, c0>0c_{0}>0 being a small absolute constant (to be specified later), and any suitable weak solution uu to (2.1) in QQ, satisfying the additional conditions

|(u),1|≤M,Y1​(u)+|(u),1|​(−​∫Q|a|m​𝑑z)1m+|(u),1|​(−​∫Q|b|m​𝑑z)1m<ϵ,|(u)_{,1}|\leq M,\quad Y_{1}(u)+|(u)_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a|^{m}dz\right)^{\frac{1}{m}}+|(u)_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b|^{m}dz\right)^{\frac{1}{m}}<\epsilon,

the following estimate is valid:

Yθ​(u)≤C1​θα​[Y1​(u)+|(u),1|​(−​∫Q|a|m​𝑑z)1m+|(u),1|​(−​∫Q|b|m​𝑑z)1m].Y_{\theta}(u)\leq C_{1}\theta^{\alpha}\left[Y_{1}(u)+|(u)_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a|^{m}dz\right)^{\frac{1}{m}}+|(u)_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b|^{m}dz\right)^{\frac{1}{m}}\right].
Proof.

Assume that the statement is false. This means that there exist numbers θ∈(0,1/3)\theta\in(0,1/3), m>5m>5 and M>0M>0 and sequences aka^{k}, bkb^{k} and a sequence of suitable weak solutions uku^{k} to (2.1) (with aa and bb replaced respectively by aka^{k} and bkb^{k}) such that

|(uk)1|≤M,‖ak‖Lm​(Q)≤M,‖bk‖Lm​(Q)≤c0,Y1​(uk)+|(uk),1|​(−∫Q|ak|m​𝑑z)1m+|(uk),1|​(−∫Q|bk|m​𝑑z)1m=ϵk→0+​ as ​k→∞,Yθ​(uk)>C1​θα​ϵk,\begin{gathered}|(u^{k})_{1}|\leq M,\quad\|a^{k}\|_{L_{m}(Q)}\leq M,\leavevmode\nobreak\ \|b^{k}\|_{L_{m}(Q)}\leq c_{0},\\ Y_{1}(u^{k})+|(u^{k})_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a^{k}|^{m}dz\right)^{\frac{1}{m}}+|(u^{k})_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b^{k}|^{m}dz\right)^{\frac{1}{m}}=\epsilon_{k}\to 0^{+}\mbox{ as }k\to\infty,\\ Y_{\theta}(u^{k})>C_{1}\theta^{\alpha}\epsilon_{k},\end{gathered}

for all k∈ℕk\in\mathbb{N}. Next, we introduce the following functions

vk:=uk−(uk),1ϵk,−Fk:=(uk),1⊗ak2​ϵk+bk⊗(uk),1ϵk;v^{k}:=\frac{u^{k}-(u^{k})_{,1}}{\epsilon_{k}},\quad-F^{k}:=\frac{(u^{k})_{,1}\otimes a^{k}}{2\epsilon_{k}}+\frac{b^{k}\otimes(u^{k})_{,1}}{\epsilon_{k}};

we have

(−​∫Q|vk|3​𝑑z)13+(−​∫Q|Fk|m​𝑑z)1m≤4and ​Yθ​(vk)>C1​θα.\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|v^{k}|^{3}dz\right)^{\frac{1}{3}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|F^{k}|^{m}dz\right)^{\frac{1}{m}}\leq 4\quad\mbox{and }Y_{\theta}(v^{k})>C_{1}\theta^{\alpha}. (2.14)

Moreover,

∂tvk−Δ​vk−κ​∇div⁡vk+ϵk​(vk⋅∇vk+vk2​div⁡vk)+(uk),1⋅∇vk+(uk),12​div⁡vk+ak2⋅∇vk+div⁡(vk⊗ak2)+div⁡(bk⊗vk)−bk2​div⁡vk=div⁡(Fk),\partial_{t}v^{k}-\Delta v^{k}-\kappa\nabla\operatorname*{\mbox{div}}v^{k}+\epsilon_{k}\left(v^{k}\cdot\nabla v^{k}+\frac{v^{k}}{2}\operatorname*{\mbox{div}}v^{k}\right)+(u^{k})_{,1}\cdot\nabla v^{k}+\frac{(u^{k})_{,1}}{2}\operatorname*{\mbox{div}}v^{k}\\ +\frac{a^{k}}{2}\cdot\nabla v^{k}+\operatorname*{\mbox{div}}(v^{k}\otimes\frac{a^{k}}{2})+\operatorname*{\mbox{div}}(b^{k}\otimes v^{k})-\frac{b^{k}}{2}\operatorname*{\mbox{div}}v^{k}=\operatorname*{\mbox{div}}(F^{k}), (2.15)

in the sense of distributions on QQ and

∂t|vk|22−Δ​|vk|22−κ​div⁡(vk​div⁡vk)+|∇vk|2+κ​(div⁡vk)2+div⁡((ϵk​vk+ak+(uk),1)​|vk|22)+vk⋅div⁡((uk),1⊗vk2+bk⊗vk−Fk)−12​vk⋅bk​div⁡vk≤0,\partial_{t}\frac{|v^{k}|^{2}}{2}-\Delta\frac{|v^{k}|^{2}}{2}-\kappa\operatorname*{\mbox{div}}(v^{k}\operatorname*{\mbox{div}}v^{k})+|\nabla v^{k}|^{2}+\kappa(\operatorname*{\mbox{div}}v^{k})^{2}+\operatorname*{\mbox{div}}\left((\epsilon_{k}v^{k}+a^{k}+(u^{k})_{,1})\frac{|v^{k}|^{2}}{2}\right)\\ +v^{k}\cdot\operatorname*{\mbox{div}}((u^{k})_{,1}\otimes\frac{v^{k}}{2}+b^{k}\otimes v^{k}-F^{k})-\frac{1}{2}v^{k}\cdot b^{k}\operatorname*{\mbox{div}}v^{k}\leq 0, (2.16)

in the sense of distributions on QQ. From the previous inequation, we get that

12​∫B|vk​(x,t)|2​ϕ​(x,t)​𝑑x+∫−1t∫B|∇vk|2​ϕ​𝑑z+κ​∫−1t∫B(div⁡vk)2​ϕ​𝑑z≤∫−1∫B|vk|22​(∂tϕ+Δ​ϕ)​𝑑z+∫−1t∫B|vk|22​(ϵk​vk+ak+(uk),1)⋅∇ϕ​d​z+∫−1t∫Bϕ​∇vk:((uk),1⊗vk2+bk⊗vk−Fk)​d​z+∫−1t∫Bvk⊗∇ϕ:((uk),1⊗vk2+bk⊗vk−Fk)​d​z−κ​∫−1t∫Bvk⋅∇ϕ​div⁡vk​d​z+12​∫−1∫Bϕ​vk⋅bk​div⁡vk​d​z,\frac{1}{2}\int_{B}|v^{k}(x,t)|^{2}\phi(x,t)dx+\int_{-1}^{t}\int_{B}|\nabla v^{k}|^{2}\phi dz+\kappa\int_{-1}^{t}\int_{B}(\operatorname*{\mbox{div}}v^{k})^{2}\phi dz\\ \leq\int_{-1}\int_{B}\frac{|v^{k}|^{2}}{2}(\partial_{t}\phi+\Delta\phi)dz+\int_{-1}^{t}\int_{B}\frac{|v^{k}|^{2}}{2}(\epsilon_{k}v^{k}+a^{k}+(u^{k})_{,1})\cdot\nabla\phi dz\\ +\int_{-1}^{t}\int_{B}\phi\nabla v^{k}:((u^{k})_{,1}\otimes\frac{v^{k}}{2}+b^{k}\otimes v^{k}-F^{k})dz+\int_{-1}^{t}\int_{B}v^{k}\otimes\nabla\phi:((u^{k})_{,1}\otimes\frac{v^{k}}{2}+b^{k}\otimes v^{k}-F^{k})dz\\ -\kappa\int_{-1}^{t}\int_{B}v^{k}\cdot\nabla\phi\operatorname*{\mbox{div}}v^{k}dz+\frac{1}{2}\int_{-1}\int_{B}\phi v^{k}\cdot b^{k}\operatorname*{\mbox{div}}v^{k}dz,

for any 0≤ϕ∈C0∞​(B×(−1,1))0\leq\phi\in C^{\infty}_{0}(B\times(-1,1)). Next, we define

Ek​(r):=12​ess sup−r2<t<0​∫B​(r)|vk​(x,t)|2​𝑑x+∫−r2∫B​(r)[|∇vk​(x,t)|2+κ​(div⁡vk​(x,t))2]​𝑑z,E_{k}(r):=\frac{1}{2}\operatorname*{\mbox{ess sup}}_{-r^{2}<t<0}\int_{B(r)}|v^{k}(x,t)|^{2}dx+\int_{-r^{2}}\int_{B(r)}[|\nabla v^{k}(x,t)|^{2}+\kappa(\operatorname*{\mbox{div}}v^{k}(x,t))^{2}]dz,

for any 0<r<10<r<1. Our goal now is to get an uniform estimate (in kk) for EkE_{k}. To this end we start by recalling the following well-known multiplicative inequality

‖vk‖L103​(Q​(r))2≤C​Ek​(r),\|v^{k}\|^{2}_{L_{\frac{10}{3}}(Q(r))}\leq CE_{k}(r),

with C>0C>0 being an universal constant. Then, for any 1/2<r1<r2≤11/2<r_{1}<r_{2}\leq 1, if we choose appropriately the test function ϕ\phi in the above local energy inequality together with the help of Hölder’s inequality and the estimates on vk,ak,bkv^{k},\leavevmode\nobreak\ a^{k},\leavevmode\nobreak\ b^{k} and FkF^{k}, we have that:

Ek​(r1)\displaystyle E_{k}(r_{1}) ≤C(r2−r1)2+Cr2−r1{ϵk∫Q|vk|3dz+(∫Q|vk|3dz)23(∫Q|ak|3dz)13\displaystyle\leq\frac{C}{(r_{2}-r_{1})^{2}}+\frac{C}{r_{2}-r_{1}}\left\{\epsilon_{k}\int_{Q}|v^{k}|^{3}dz+\left(\int_{Q}|v^{k}|^{3}dz\right)^{\frac{2}{3}}\left(\int_{Q}|a^{k}|^{3}dz\right)^{\frac{1}{3}}\right.
+|(vk),1|∫Q|vk|2dz}+Ek(r2)12{|(uk),1|(∫Q|vk|2dz)12\displaystyle\mathrel{\phantom{=}}\left.+|(v^{k})_{,1}|\int_{Q}|v^{k}|^{2}dz\right\}+E_{k}(r_{2})^{\frac{1}{2}}\left\{|(u^{k})_{,1}|\left(\int_{Q}|v^{k}|^{2}dz\right)^{\frac{1}{2}}\right.
+(∫Q|bk|5dz)15(∫Q​(r2)|vk|103dz)310+(∫Q|Fk|2dz)12}\displaystyle\mathrel{\phantom{=}}\left.+\left(\int_{Q}|b^{k}|^{5}dz\right)^{\frac{1}{5}}\left(\int_{Q(r_{2})}|v^{k}|^{\frac{10}{3}}dz\right)^{\frac{3}{10}}+\left(\int_{Q}|F^{k}|^{2}dz\right)^{\frac{1}{2}}\right\}
+Cr2−r1{|(uk),1|∫Q|vk|2dz+(∫Q|vk|3dz)23(∫Q|ak|3dz)13\displaystyle\mathrel{\phantom{=}}+\frac{C}{r_{2}-r_{1}}\left\{|(u^{k})_{,1}|\int_{Q}|v^{k}|^{2}dz+\left(\int_{Q}|v^{k}|^{3}dz\right)^{\frac{2}{3}}\left(\int_{Q}|a^{k}|^{3}dz\right)^{\frac{1}{3}}\right.
(∫Q|vk|2dz)12(∫Q|Fk|2dz)12}+Ek(r2)12{C​(κ)(r2−r1)2∫Q|vk|2dz}12\displaystyle\mathrel{\phantom{=}}\left.\left(\int_{Q}|v^{k}|^{2}dz\right)^{\frac{1}{2}}\left(\int_{Q}|F^{k}|^{2}dz\right)^{\frac{1}{2}}\right\}+E_{k}(r_{2})^{\frac{1}{2}}\left\{\frac{C(\kappa)}{(r_{2}-r_{1})^{2}}\int_{Q}|v^{k}|^{2}dz\right\}^{\frac{1}{2}}
≤C​(κ,M)(r2−r1)2+C​(M)​Ek​(r2)12+(14+C​‖bk‖Lm​(Q))​Ek​(r2)\displaystyle\leq\frac{C(\kappa,M)}{(r_{2}-r_{1})^{2}}+C(M)E_{k}(r_{2})^{\frac{1}{2}}+(\frac{1}{4}+C\|b^{k}\|_{L_{m}(Q)})E_{k}(r_{2})
≤C​(κ,M)(r2−r1)2+(12+C​‖bk‖Lm​(Q))​Ek​(r2).\displaystyle\leq\frac{C(\kappa,M)}{(r_{2}-r_{1})^{2}}+(\frac{1}{2}+C\|b^{k}\|_{L_{m}(Q)})E_{k}(r_{2}).

Note that ‖b‖Lm​(Q)≤c0\|b\|_{L_{m}(Q)}\leq c_{0} with c0c_{0} small. Therefore, if we choose c0c_{0} such that C​c0<1/2Cc_{0}<1/2, we can iterate the above estimate (see e.g. [1] Lemma 5.2) and conclude that

Ek​(3/4)≤C​(κ,M),∀k∈ℕ.E_{k}(3/4)\leq C(\kappa,M),\leavevmode\nobreak\ \forall k\in\mathbb{N}. (2.17)

This together with the fact that vkv^{k} satisfies (2.15) yield

‖∂tvk‖L4/3​(−(3/4)2,0;H−1​(B​(3/4)))≤C​(κ,M),∀k∈ℕ,\|\partial_{t}v^{k}\|_{L_{4/3}(-(3/4)^{2},0;H^{-1}(B(3/4)))}\leq C(\kappa,M),\leavevmode\nobreak\ \forall k\in\mathbb{N}, (2.18)

where H−1​(B​(3/4))H^{-1}(B(3/4)) stands here for the dual of the Sobolev space H01​(B​(3/4))H^{1}_{0}(B(3/4)). Now, from Aubin-Lions and Banach-Alaoglu compactness results, we can choose subsequences of vk,ak,bk,(uk),1v^{k},\leavevmode\nobreak\ a^{k},\leavevmode\nobreak\ b^{k},\leavevmode\nobreak\ (u^{k})_{,1} and FkF^{k} (which we still denote vk,ak,bk,(uk),1v^{k},\leavevmode\nobreak\ a^{k},\leavevmode\nobreak\ b^{k},\leavevmode\nobreak\ (u^{k})_{,1} and FkF^{k}) such that for some λ∈ℝ,v∈L3​(Q​(3/4)),a,b,F∈Lm​(Q​(3/4))\lambda\in\mathbb{R},\leavevmode\nobreak\ v\in L_{3}(Q(3/4)),\leavevmode\nobreak\ a,b,F\in L_{m}(Q(3/4)), we have

vk→v​ strongly in ​L3​(Q​(3/4)),(uk)→λak⇀a,bk⇀b,Fk⇀Fweakly in ​Lm​(Q​(3/4)).\begin{gathered}v^{k}\to v\mbox{ strongly in }L_{3}(Q(3/4)),\leavevmode\nobreak\ (u^{k})\to\lambda\\ a^{k}\rightharpoonup a,\leavevmode\nobreak\ b^{k}\rightharpoonup b,\leavevmode\nobreak\ F^{k}\rightharpoonup F\quad\mbox{weakly in }L_{m}(Q(3/4)).\end{gathered}

Moreover, we have that

(∫Q​(3/4)(|a|+|b|+|F|)m​𝑑z)1m≤C​(M),|λ|≤M,ess sup−(3/4)2<t<0​∫B​(3/4)|v​(x,t)|2​𝑑x+∫Q​(3/4)|∇v​(x,t)|2​𝑑z≤C​(κ,M).\begin{gathered}\left(\int_{Q(3/4)}(|a|+|b|+|F|)^{m}dz\right)^{\frac{1}{m}}\leq C(M),\quad|\lambda|\leq M,\\ \operatorname*{\mbox{ess sup}}_{-(3/4)^{2}<t<0}\int_{B(3/4)}|v(x,t)|^{2}dx+\int_{Q(3/4)}|\nabla v(x,t)|^{2}dz\leq C(\kappa,M).\end{gathered} (2.19)

Finally, from (2.15), we see that

∂tv−Δ​v−κ​∇div⁡v+λ⋅∇v+λ2​div⁡v+a2⋅∇v+div⁡(v⊗a2)+div⁡(b⊗v)−b2​div⁡v=div⁡F,\partial_{t}v-\Delta v-\kappa\nabla\operatorname*{\mbox{div}}v+\lambda\cdot\nabla v+\frac{\lambda}{2}\operatorname*{\mbox{div}}v+\frac{a}{2}\cdot\nabla v+\operatorname*{\mbox{div}}(v\otimes\frac{a}{2})+\operatorname*{\mbox{div}}(b\otimes v)-\frac{b}{2}\operatorname*{\mbox{div}}v=\operatorname*{\mbox{div}}F,

in the sense of distributions in Q​(3/4)Q(3/4). Thus, from Lemma 2.4, we have that

‖v‖Cα,α2​(B​(1/2)¯)≤C​(κ,m,M),\|v\|_{C^{\alpha,\frac{\alpha}{2}}(\overline{B(1/2)})}\leq C(\kappa,m,M),

for some α=α​(m)∈(0,1)\alpha=\alpha(m)\in(0,1). This implies that

Yθ​(v)≤C​(κ,m,M)​θα;Y_{\theta}(v)\leq C(\kappa,m,M)\theta^{\alpha}; (2.20)

but because of (2.14) and the strong L3L_{3}-convergence of vkv^{k}, we also have that Yθ​(v)≥C1​θαY_{\theta}(v)\geq C_{1}\theta^{\alpha}. This together with (2.20) give us

C1≤C​(κ,m,M).C_{1}\leq C(\kappa,m,M).

If from the beginning, C1C_{1} is chosen so that C1>2​C​(κ,m,M)C_{1}>2C(\kappa,m,M), we arrive at a contradiction and the Lemma is proved. ∎

Lemma 2.5 admits the following iterations.

Lemma 2.6.

Given numbers M>0M>0 and m>5m>5, we choose θ∈(0,1/3)\theta\in(0,1/3) so that

C1​θα−β<1andθ<c1C_{1}\theta^{\alpha-\beta}<1\quad\mbox{and}\quad\theta<c_{1} (2.21)

where c1=c1​(m)>0c_{1}=c_{1}(m)>0 is a small number to be specified later, C1,αC_{1},\alpha are as in Lemma 2.5 and β:=α/2\beta:=\alpha/2. Then, there exists ϵ∗=ϵ∗​(κ,θ,m,M)<ϵ\epsilon_{*}=\epsilon_{*}(\kappa,\theta,m,M)<\epsilon sufficiently small, such that for any ‖a‖Lm​(Q)≤M,‖b‖Lm​(Q)≤c0\|a\|_{L_{m}(Q)}\leq M,\leavevmode\nobreak\ \|b\|_{L_{m}(Q)}\leq c_{0} (ϵ=ϵ​(θ,m,M)>0\epsilon=\epsilon(\theta,m,M)>0 and c0>0c_{0}>0 being also as in Lemma 2.5), and any suitable weak solution uu to (2.1) in QQ, satisfying the additional conditions

|(u),1|≤M/2,Y1​(u)+M​(−​∫Q|a|m​𝑑z)1m+M​(−​∫Q|b|m​𝑑z)1m<ϵ∗,|(u)_{,1}|\leq M/2,\quad Y_{1}(u)+M\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a|^{m}dz\right)^{\frac{1}{m}}+M\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b|^{m}dz\right)^{\frac{1}{m}}<\epsilon_{*},

the following holds: ∀k=1,2,…,\forall k=1,2,\ldots, we have

|(u),θk−1|≤M,|(u)_{,\theta^{k-1}}|\leq M,
Yθk−1​(u)+θk−1​|(u),θk−1|​[(−​∫Q​(θk−1)|a|m​𝑑z)1m+(−​∫Q​(θk−1)|b|m​𝑑z)1m]<ϵ∗,Y_{\theta^{k-1}}(u)+\theta^{k-1}|(u)_{,\theta^{k-1}}|\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k-1})}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k-1})}|b|^{m}dz\right)^{\frac{1}{m}}\right]<\epsilon_{*},
Yθk​(u)≤θβ​{Yθk−1​(u)+θk−1​|(u),θk−1|​[(−​∫Q​(θk−1)|a|m​𝑑z)1m+(−​∫Q​(θk−1)|b|m​𝑑z)1m]}.Y_{\theta^{k}}(u)\leq\theta^{\beta}\left\{Y_{\theta^{k-1}}(u)+\theta^{k-1}|(u)_{,\theta^{k-1}}|\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k-1})}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k-1})}|b|^{m}dz\right)^{\frac{1}{m}}\right]\right\}.
Proof.

We prove the lemma by induction; the case k=1k=1 is true thanks to Lemma 2.5. Now, suppose the conclusion is true for k≤k0k\leq k_{0} (k0≥1k_{0}\geq 1) and let us show that it remains the case for k=k0+1k=k_{0}+1.
For all k≤k0k\leq k_{0}, we have

Yθk​(u)\displaystyle Y_{\theta^{k}}(u) ≤θβ​{Yθk−1​(u)+θk−1​|(u),θk−1|​[(−∫Q​(θk−1)|a|m​𝑑z)1m+(−∫Q​(θk−1)|b|m​𝑑z)1m]}\displaystyle\leq\theta^{\beta}\left\{Y_{\theta^{k-1}}(u)+\theta^{k-1}|(u)_{,\theta^{k-1}}|\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k-1})}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k-1})}|b|^{m}dz\right)^{\frac{1}{m}}\right]\right\}
≤θβ​{Yθk−1​(u)+θ(k−1)​(1−5/m)​M​[(−∫Q|a|m​𝑑z)1m+(−∫Q|b|m​𝑑z)1m]}\displaystyle\leq\theta^{\beta}\left\{Y_{\theta^{k-1}}(u)+\theta^{(k-1)(1-5/m)}M\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b|^{m}dz\right)^{\frac{1}{m}}\right]\right\}
≤θβ​Yθk−1​(u)+θk​β1​ϵ∗,\displaystyle\leq\theta^{\beta}Y_{\theta^{k-1}}(u)+\theta^{k\beta_{1}}\epsilon_{*},

with β1=min⁡{β,1−5/m}\beta_{1}=\min\{\beta,1-5/m\}. By iterating the last inequality, we get

Yθk​(u)≤θk​β​Y1​(u)+k​θk​β1​ϵ∗,∀k≤k0.Y_{\theta^{k}}(u)\leq\theta^{k\beta}Y_{1}(u)+k\theta^{k\beta_{1}}\epsilon_{*},\quad\forall k\leq k_{0}.

Thus,

|(u),θk0|\displaystyle|(u)_{,\theta^{k_{0}}}| ≤∑k=1k0|(u),θk−(u),θk−1|+|(u),1|\displaystyle\leq\sum_{k=1}^{k_{0}}|(u)_{,\theta^{k}}-(u)_{,\theta^{k-1}}|+|(u)_{,1}|
≤θ−5/3​∑k=1k0Yθk−1​(u)+|(u),1|\displaystyle\leq\theta^{-5/3}\sum_{k=1}^{k_{0}}Y_{\theta^{k-1}}(u)+|(u)_{,1}|
≤θ−5/3​∑k=1k0(θ(k−1)​β+(k−1)​θ(k−1)​β1)​ϵ∗+M/2\displaystyle\leq\theta^{-5/3}\sum_{k=1}^{k_{0}}\left(\theta^{(k-1)\beta}+(k-1)\theta^{(k-1)\beta_{1}}\right)\epsilon_{*}+M/2
≤θ−5/3​((1−θβ)−1+∑k=0∞k​θk​β1)⏟C​(θ,m)​ϵ∗+M/2.\displaystyle\leq\underbrace{\theta^{-5/3}\left((1-\theta^{\beta})^{-1}+\sum_{k=0}^{\infty}k\theta^{k\beta_{1}}\right)}_{C(\theta,m)}\epsilon_{*}+M/2.

By choosing ϵ∗≤M​(2​C​(θ,m))−1\epsilon_{*}\leq M(2C(\theta,m))^{-1}, we find that

|(u),θk0|≤M.|(u)_{,\theta^{k_{0}}}|\leq M.

Moreover,

Yθk0​(u)+θk0​|(u),θk0|​[(−∫Q​(θk0)|a|m​𝑑z)1m+(−∫Q​(θk0)|b|m​𝑑z)1m]\displaystyle Y_{\theta^{k_{0}}}(u)+\theta^{k_{0}}|(u)_{,\theta^{k_{0}}}|\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k_{0}})}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k_{0}})}|b|^{m}dz\right)^{\frac{1}{m}}\right] ≤θβ​ϵ∗+θ(1−5/m)​k0​ϵ∗\displaystyle\leq\theta^{\beta}\epsilon_{*}+\theta^{(1-5/m)k_{0}}\epsilon_{*}
≤θβ​ϵ∗+θ(1−5/m)​ϵ∗\displaystyle\leq\theta^{\beta}\epsilon_{*}+\theta^{(1-5/m)}\epsilon_{*}
<ϵ∗,\displaystyle<\epsilon_{*},

if we choose θ<c1​(m)\theta<c_{1}(m) to be small enough. Now, set

u0​(y,s)=θk0​u​(θk0​y,θ2​k0​s),a0​(y,s)=θk0​a​(θk0​y,θ2​k0​s),b0​(y,s)=θk0​b​(θk0​y,θ2​k0​s),\begin{gathered}u^{0}(y,s)=\theta^{k_{0}}u(\theta^{k_{0}}y,\theta^{2k_{0}}s),\\ a^{0}(y,s)=\theta^{k_{0}}a(\theta^{k_{0}}y,\theta^{2k_{0}}s),\quad b^{0}(y,s)=\theta^{k_{0}}b(\theta^{k_{0}}y,\theta^{2k_{0}}s),\end{gathered}

with (y,s)∈Q(y,s)\in Q. We show steadily that u0u^{0} is also a suitable weak solution to (2.1) with aa and bb replaced respectively by a0a^{0} and b0b^{0}; moreover the conditions stated in Lemma 2.5 are satisfied for these new functions. Consequently, we have (thanks to Lemma 2.5)

Yθ​(u0)≤θβ​[Y1​(u0)+|(u0),1|​(−​∫Q|a0|m​𝑑z)1m+|(u0),1|​(−​∫Q|b0|m​𝑑z)1m],Y_{\theta}(u^{0})\leq\theta^{\beta}\left[Y_{1}(u^{0})+|(u^{0})_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a^{0}|^{m}dz\right)^{\frac{1}{m}}+|(u^{0})_{,1}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b^{0}|^{m}dz\right)^{\frac{1}{m}}\right],

that is

Yθk0+1​(u)≤θβ​{Yθk0​(u)+θk0​|(u),θk0|​[(−​∫Q​(θk0)|a|m​𝑑z)1m+(−​∫Q​(θk0)|b|m​𝑑z)1m]}.Y_{\theta^{k_{0}+1}}(u)\leq\theta^{\beta}\left\{Y_{\theta^{k_{0}}}(u)+\theta^{k_{0}}|(u)_{,\theta^{k_{0}}}|\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k_{0}})}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(\theta^{k_{0}})}|b|^{m}dz\right)^{\frac{1}{m}}\right]\right\}.

This concludes the induction and the proof of the lemma. ∎

By translation and dilatation, we obtain the following corollary.

Corollary 2.6.1.

Let uu be a suitable weak solution to (2.1) in Q​(z0,R)Q(z_{0},R) with

R1−5/m​‖a‖Lm​(Q​(z0,R))≤MandR1−5/m​‖b‖Lm​(Q​(z0,R))≤c0R^{1-5/m}\|a\|_{L_{m}(Q(z_{0},R))}\leq M\quad\mbox{and}\quad R^{1-5/m}\|b\|_{L_{m}(Q(z_{0},R))}\leq c_{0}

with c0c_{0} as in Lemma 2.5. Take θ,β\theta,\beta and ϵ∗\epsilon_{*} as in the previous lemma. If we have in addition that

R​|(u)z0,R|≤M/2​ and ​R​Y​(z0,R;u)+R​M​[(−​∫Q​(z0,R)|a|m​𝑑z)1m+(−​∫Q​(z0,R)|b|m​𝑑z)1m]<ϵ∗,R|(u)_{z_{0},R}|\leq M/2\mbox{ and }RY(z_{0},R;u)+RM\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},R)}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},R)}|b|^{m}dz\right)^{\frac{1}{m}}\right]<\epsilon_{*},

then for all k≥1k\geq 1

R​|(u)Q​(z0,θk−1​R)|≤M,R|(u)_{Q(z_{0},\theta^{k-1}R)}|\leq M,

and

Y(z0,θkR;u)≤θβ[RY(z0,θk−1R;u)+Rθk−1|(u)z0,θk−1​R|(−∫Q​(z0,θk−1​R)|a|mdz)1m+Rθk−1|(u)z0,θk−1​R|(−∫Q​(z0,θk−1​R)|b|mdz)1m].Y(z_{0},\theta^{k}R;u)\leq\theta^{\beta}\left[RY(z_{0},\theta^{k-1}R;u)+R\theta^{k-1}|(u)_{z_{0},\theta^{k-1}R}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},\theta^{k-1}R)}|a|^{m}dz\right)^{\frac{1}{m}}\right.\\ +\left.R\theta^{k-1}|(u)_{z_{0},\theta^{k-1}R}|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},\theta^{k-1}R)}|b|^{m}dz\right)^{\frac{1}{m}}\right].

We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1.

Let M=2020M=2020 and choose θ\theta according to Lemma 2.6. Define

A:=(−​∫Q|u|3​𝑑z)13+(−​∫Q|a|m​𝑑z)1m+(−​∫Q|b|m​𝑑z)1m.A:=\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|u|^{3}dz\right)^{\frac{1}{3}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|b|^{m}dz\right)^{\frac{1}{m}}.

Observe that

Q​(z0,1/4)⊂Q​ if ​z0∈Q​(1/2),Q(z_{0},1/4)\subset Q\mbox{ if }z_{0}\in Q(1/2),
14​|(u)z0,1/4|≤c​A,(14)1−5/m​(‖a‖Lm​(Q​(z0,1/4))+‖b‖Lm​(Q​(z0,1/4)))≤c​(m)​A,\frac{1}{4}|(u)_{z_{0},1/4}|\leq cA,\quad\left(\frac{1}{4}\right)^{1-5/m}(\|a\|_{L_{m}(Q(z_{0},1/4))}+\|b\|_{L_{m}(Q(z_{0},1/4))})\leq c(m)A,

and

14​Y​(z0,1/4;u)+14×M​[(−​∫Q​(z0,1/4)|a|m​𝑑z)1m+(−​∫Q​(z0,1/4)|b|m​𝑑z)1m]≤c​(m)​A;\frac{1}{4}Y(z_{0},1/4;u)+\frac{1}{4}\times M\left[\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},1/4)}|a|^{m}dz\right)^{\frac{1}{m}}+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q(z_{0},1/4)}|b|^{m}dz\right)^{\frac{1}{m}}\right]\leq c(m)A;

Now, we choose

ϵ0<min⁡{2020/c,2020/c​(m),c0/c​(m),ϵ∗/c​(m)}\epsilon_{0}<\min\{2020/c,2020/c(m),c_{0}/c(m),\epsilon_{*}/c(m)\}

with c0c_{0} and ϵ∗\epsilon_{*} as in Lemma 2.6. Consequently, by applying Corollary 2.6.1, we are able to prove (using a similar iteration process we used in the proof of Lemma 2.6) that

Y​(z0,θk/4;u)≤C​(κ,θ,m,M)​θk​β2,Y(z_{0},\theta^{k}/4;u)\leq C(\kappa,\theta,m,M)\theta^{k\beta_{2}},

for all z0∈Q​(1/2)z_{0}\in Q(1/2), k≥1k\geq 1 and with β2=1/2​(β+β1)\beta_{2}=1/2(\beta+\beta_{1}). Hölder continuity of uu in Q​(1/2¯)Q(\overline{1/2}) follows from Campanato’s type condition. The theorem is proved. ∎

Theorem 2.1 can be strengthen in the following manner, by removing the smallness condition on aa and bb.

Theorem 2.7 (Improved ϵ\epsilon-regularity criterion).

Let uu be a suitable weak solution to (2.1) in QQ with ‖a‖Lm​(Q)+‖b‖Lm​(Q)≤M\|a\|_{L_{m}(Q)}+\|b\|_{L_{m}(Q)}\leq M for some M>0M>0 and m>5m>5. Then there exists ϵ1=ϵ1​(κ,m,M)>0\epsilon_{1}=\epsilon_{1}(\kappa,m,M)>0 with the following properties: if

(−​∫Q|u|3​𝑑z)13≤ϵ1,\left(\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-4.25003pt}}\!\int_{Q}|u|^{3}dz\right)^{\frac{1}{3}}\leq\epsilon_{1}, (2.22)

then uu is Hölder continuous in Q​(12)¯\overline{Q(\frac{1}{2})} with exponent α=α​(m)∈(0,1)\alpha=\alpha(m)\in(0,1) and

‖u‖Cα,α2​(Q​(12)¯)≤C​(ϵ1,κ,m).\|u\|_{C^{\alpha,\frac{\alpha}{2}}(\overline{Q(\frac{1}{2})})}\leq C(\epsilon_{1},\kappa,m). (2.23)

We skip the proof of Theorem 2.7 since it is essentially a repetition of the proof of an analogous result for the incompressible Navier-Stokes system in [3] (see Theorem 2.2 in there).

3 Local in space near initial time smoothness of Leray solutions

In this section, we use the ϵ\epsilon-regularity theorem(s) proved in the previous section to study the local in space near the initial time smoothness of the Leray solutions (or energy solutions) to our model (1.1).
For our future analysis, the following a priori estimate will be needed. It was first proved in [9] for the incompressible Navier-Stokes system; here we follow the proof given in [2] (see Lemma 2.2) which is much simpler than the original one.

Lemma 3.1 (A priori estimate for Leray solutions).

Let u0∈L2,l​o​c​(ℝ3)u_{0}\in L_{2,loc}(\mathbb{R}^{3}) such that for some R>0R>0, α​(R):=supx0∈ℝ3∫B​(x0,R)|u0​(x)|2​𝑑x<∞\alpha(R):=\sup_{x_{0}\in\mathbb{R}^{3}}\int_{B(x_{0},R)}|u_{0}(x)|^{2}dx<\infty and let uu be a Leray solution to system (1.1) with initial data u0u_{0}. Then, there exists some small absolute number μ>0\mu>0 such that for 0<λ<μ​min⁡{1,α​(R)−2​R2,(1+κ)−1}0<\lambda<\mu\min\{1,\alpha(R)^{-2}R^{2},(1+\kappa)^{-1}\}, we have

supx0∈ℝ3ess sup0<t<λ​R2​∫B​(x0,R)|u​(x,t)|22​𝑑x+supx0∈ℝ3∫0λ​R2∫B​(x0,R)(|∇u​(x,t)|2+κ​(div⁡u​(x,t))2)​𝑑x​𝑑t≤C0​α​(R),\sup_{x_{0}\in\mathbb{R}^{3}}\operatorname*{\mbox{ess sup}}_{0<t<\lambda R^{2}}\int_{B(x_{0},R)}\frac{|u(x,t)|^{2}}{2}dx\\ +\sup_{x_{0}\in\mathbb{R}^{3}}\int_{0}^{\lambda R^{2}}\int_{B(x_{0},R)}\left(|\nabla u(x,t)|^{2}+\kappa(\operatorname*{\mbox{div}}u(x,t))^{2}\right)dxdt\leq C_{0}\alpha(R),

with C0C_{0} an absolute large constant.

Proof.

Let 0≤ϕ∈C0∞​(B)0\leq\phi\in C^{\infty}_{0}(B) such that ϕ≡1\phi\equiv 1 in B​(1/2)B(1/2) and ϕ≡0∈B∖B​(3/4)\phi\equiv 0\in B\setminus B(3/4); let x0∈ℝ3x_{0}\in\mathbb{R}^{3}, R>0R>0 and set ϕx0,2​R​(x)=ϕ​((x−x0)/2​R)\phi_{x_{0},2R}(x)=\phi((x-x_{0})/2R). We have, from the local energy inequality verified by uu, that:

∫B​(x0,2​R)|u​(x,t)|22​ϕx0,2​R​(x)​𝑑x+∫0t∫B​(x0,2​R)(|∇u|2+κ​(div⁡u)2)​ϕx0,2​R​𝑑x​𝑑s≤∫B​(x0,2​R)|u0​(x)|22​ϕx0,2​R​(x)​𝑑x+∫0t∫B​(x0,2​R)|u|22​Δ​ϕx0,2​R​𝑑x​𝑑s+∫0t∫B​(x0,2​R)(|u|22−κ​div⁡u)​u⋅∇ϕx0,2​R​d​x​d​s.\int_{B(x_{0},2R)}\frac{|u(x,t)|^{2}}{2}\phi_{x_{0},2R}(x)dx\\ +\int_{0}^{t}\int_{B(x_{0},2R)}\left(|\nabla u|^{2}+\kappa(\operatorname*{\mbox{div}}u)^{2}\right)\phi_{x_{0},2R}dxds\leq\int_{B(x_{0},2R)}\frac{|u_{0}(x)|^{2}}{2}\phi_{x_{0},2R}(x)dx\\ +\int_{0}^{t}\int_{B(x_{0},2R)}\frac{|u|^{2}}{2}\Delta\phi_{x_{0},2R}dxds+\int_{0}^{t}\int_{B(x_{0},2R)}(\frac{|u|^{2}}{2}-\kappa\operatorname*{\mbox{div}}u)u\cdot\nabla\phi_{x_{0},2R}dxds.

Now, set

AR​(λ)=supx0∈ℝ3ess sup0<t<λ​R2​∫B​(x0,R)|u​(x,t)|22​𝑑x+supx0∈ℝ3∫0λ​R2∫B​(x0,R)(|∇u​(x,t)|2+κ​(div⁡u​(x,t))2)​𝑑x​𝑑t(λ>0);A_{R}(\lambda)=\sup_{x_{0}\in\mathbb{R}^{3}}\operatorname*{\mbox{ess sup}}_{0<t<\lambda R^{2}}\int_{B(x_{0},R)}\frac{|u(x,t)|^{2}}{2}dx\\ +\sup_{x_{0}\in\mathbb{R}^{3}}\int_{0}^{\lambda R^{2}}\int_{B(x_{0},R)}\left(|\nabla u(x,t)|^{2}+\kappa(\operatorname*{\mbox{div}}u(x,t))^{2}\right)dxdt\quad(\lambda>0);

For a.e. t∈(0,λ​R2)t\in(0,\lambda R^{2}), we have

∫B​(x0,R)|u​(x,t)|22​𝑑x+∫0t∫B​(x0,R)|∇u|2​𝑑x​𝑑s+∫0t∫B​(x0,2​R)κ(divu)2ϕx0,2​Rdxds≤C(α(R)+λAR(λ)+1R∫0λ​R2∫B​(x0,2​R)|u|3dxds+∫0λ​R2∫B​(x0,2​R)κ|divu|ϕx0,2​R12|u||∇ϕx0,2​R12|dxds).\int_{B(x_{0},R)}\frac{|u(x,t)|^{2}}{2}dx+\int_{0}^{t}\int_{B(x_{0},R)}|\nabla u|^{2}dxds\\ +\int_{0}^{t}\int_{B(x_{0},2R)}\kappa(\operatorname*{\mbox{div}}u)^{2}\phi_{x_{0},2R}dxds\leq C\left(\alpha(R)+\lambda A_{R}(\lambda)+\frac{1}{R}\int_{0}^{\lambda R^{2}}\int_{B(x_{0},2R)}|u|^{3}dxds\right.\\ \left.+\int_{0}^{\lambda R^{2}}\int_{B(x_{0},2R)}\kappa|\operatorname*{\mbox{div}}u|\phi_{x_{0},2R}^{\frac{1}{2}}|u||\nabla\phi_{x_{0},2R}^{\frac{1}{2}}|dxds\right).

Next, by known multiplicative inequality, we have

∫0λ​R2∫B​(x0,2​R)|u|3​𝑑x​𝑑s≤C​λ14​R12​AR​(λ)32(if ​λ≤1),\int_{0}^{\lambda R^{2}}\int_{B(x_{0},2R)}|u|^{3}dxds\leq C\lambda^{\frac{1}{4}}R^{\frac{1}{2}}A_{R}(\lambda)^{\frac{3}{2}}\quad(\mbox{if }\lambda\leq 1),

and thanks to Young’s inequality that

∫0λ​R2∫B​(x0,2​R)κ​|div⁡u|​ϕx0,2​R12​|u|​|∇ϕx0,2​R12|​𝑑x​𝑑s≤12​∫0t∫B​(x0,2​R)κ​(div⁡u)2​ϕx0,2​R​𝑑x​𝑑s+C​κ​λ​AR​(λ).\int_{0}^{\lambda R^{2}}\int_{B(x_{0},2R)}\kappa|\operatorname*{\mbox{div}}u|\phi_{x_{0},2R}^{\frac{1}{2}}|u||\nabla\phi_{x_{0},2R}^{\frac{1}{2}}|dxds\leq\frac{1}{2}\int_{0}^{t}\int_{B(x_{0},2R)}\kappa(\operatorname*{\mbox{div}}u)^{2}\phi_{x_{0},2R}dxds\\ +C\kappa\lambda A_{R}(\lambda).

Consequently, we get that

∫B​(x0,R)|u​(x,t)|22​𝑑x+∫0t∫B​(x0,R)|∇u|2​𝑑x​𝑑s+12​∫0t∫B​(x0,2​R)κ​(div⁡u)2​ϕx0,2​R​𝑑x​𝑑s≤C​[α​(R)+(1+κ)​λ​AR​(λ)+λ14​R−12​AR​(λ)32],\int_{B(x_{0},R)}\frac{|u(x,t)|^{2}}{2}dx+\int_{0}^{t}\int_{B(x_{0},R)}|\nabla u|^{2}dxds\\ +\frac{1}{2}\int_{0}^{t}\int_{B(x_{0},2R)}\kappa(\operatorname*{\mbox{div}}u)^{2}\phi_{x_{0},2R}dxds\leq C\left[\alpha(R)+(1+\kappa)\lambda A_{R}(\lambda)+\lambda^{\frac{1}{4}}R^{-\frac{1}{2}}A_{R}(\lambda)^{\frac{3}{2}}\right],

for a.e. t∈(0,λ​R2)t\in(0,\lambda R^{2}) and all x0∈R3x_{0}\in R^{3}. Therefore

AR​(λ)≤C​[α​(R)+(1+κ)​λ​AR​(λ)+λ14​R−12​AR​(λ)32];A_{R}(\lambda)\leq C\left[\alpha(R)+(1+\kappa)\lambda A_{R}(\lambda)+\lambda^{\frac{1}{4}}R^{-\frac{1}{2}}A_{R}(\lambda)^{\frac{3}{2}}\right];

By choosing λ≤min⁡{1,(2​C​(1+κ))−1}\lambda\leq\min\{1,(2C(1+\kappa))^{-1}\}, we find that

AR​(λ)≤2​C​(α​(R)+λ14​R−12​AR​(λ)32),A_{R}(\lambda)\leq 2C\left(\alpha(R)+\lambda^{\frac{1}{4}}R^{-\frac{1}{2}}A_{R}(\lambda)^{\frac{3}{2}}\right),

and from there the conclusion follows by standard continuation arguments. ∎

Now we can prove the first important result of this section.

Theorem 3.2.

Let u0∈L2,l​o​c​(ℝ3)u_{0}\in L_{2,loc}(\mathbb{R}^{3}) such that α:=supx0∈ℝ3∫B|u0​(x)|2​𝑑x<∞\alpha:=\sup_{x_{0}\in\mathbb{R}^{3}}\int_{B}|u_{0}(x)|^{2}dx<\infty. Suppose in addition that M:=‖u0‖Lm​(B)<∞M:=\|u_{0}\|_{L_{m}(B)}<\infty with m>3m>3; Let us decompose u0=u01+u02u_{0}=u^{1}_{0}+u^{2}_{0} with u01|B​(4/3)=u0u_{0}^{1}|_{B(4/3)}=u_{0}, suppu01⊂⊂B(2)u_{0}^{1}\subset\subset B(2) and ‖u01‖Lm​(ℝ3)≤M\|u_{0}^{1}\|_{L_{m}(\mathbb{R}^{3})}\leq M. Now, let aa be the locally in time defined mild solution to system (1.1) with initial data u01u_{0}^{1}. Then, there exists a time T=T​(α,κ,m,M)>0T=T(\alpha,\kappa,m,M)>0 such that any Leray solution uu to (1.1) satisfies:

‖u−a‖Cγ,γ2​(B​(1/2)×[0,T]¯)≤C​(α,κ,m,M),\|u-a\|_{C^{\gamma,\frac{\gamma}{2}}(\overline{B(1/2)\times[0,T]})}\leq C(\alpha,\kappa,m,M),

for some γ=γ​(m)∈(0,1)\gamma=\gamma(m)\in(0,1).

Proof.

Let us start by discussing the decomposition of u0u_{0} in the statement of the theorem; introduce the cut-off function 0≤φ∈C0∞​(B)0\leq\varphi\in C^{\infty}_{0}(B) such that φ≡1\varphi\equiv 1 in B​(4/3)B(4/3) and φ≡0\varphi\equiv 0 in B​(2)∖B​(3/2)B(2)\setminus B(3/2). We have the required splitting if we set u01:=u0​φu_{0}^{1}:=u_{0}\varphi and u02:=u0​(1−φ)u_{0}^{2}:=u_{0}(1-\varphi).
Next, by assumption aa solves the Cauchy problem for system (1.1) with initial data u01u_{0}^{1} in ℝ3×[0,T1]\mathbb{R}^{3}\times[0,T_{1}], where T1=T1​(κ,m,M)>0T_{1}=T_{1}(\kappa,m,M)>0. Such construction can be done by mimicking the one done for the incompressible Navier-Stokes equations in the Appendix of [8] (see Theorem 7.4). We present the details elsewhere. Their arguments also allow us to get that

‖a‖C​([0,T1];L2​(ℝ3))+‖∇a‖L2​(ℝ3×(0,T1))≤C​(m,M),‖a‖L∞​(0,T1;Lm​(ℝ3))+‖a‖L5​m3​(ℝ3×(0,T1))≤C​(κ,m,M).\begin{gathered}\|a\|_{C([0,T_{1}];L_{2}(\mathbb{R}^{3}))}+\|\nabla a\|_{L_{2}(\mathbb{R}^{3}\times(0,T_{1}))}\leq C(m,M),\\ \|a\|_{L_{\infty}(0,T_{1};L_{m}(\mathbb{R}^{3}))}+\|a\|_{L_{\frac{5m}{3}}(\mathbb{R}^{3}\times(0,T_{1}))}\leq C(\kappa,m,M).\end{gathered} (3.1)

Now, we set v:=u−av:=u-a and we observe that

∂tv−Δ​v−κ​∇div⁡v+v⋅∇v+v2​div⁡v+a2⋅∇v+div⁡(v⊗a2)+div⁡(a⊗v)−a2​div⁡v=0,\partial_{t}v-\Delta v-\kappa\nabla\operatorname*{\mbox{div}}v+v\cdot\nabla v+\frac{v}{2}\operatorname*{\mbox{div}}v+\frac{a}{2}\cdot\nabla v+\operatorname*{\mbox{div}}(v\otimes\frac{a}{2})+\operatorname*{\mbox{div}}(a\otimes v)-\frac{a}{2}\operatorname*{\mbox{div}}v=0,

in the sense of distributions in ℝ3×(0,T1)\mathbb{R}^{3}\times(0,T_{1}). Moreover, because uu and aa satisfy a local energy inequality, we see that

∂t|v|22−Δ​|v|22−κ​div⁡(v​div⁡v)+|∇v|2+κ​(div⁡v)2+div⁡((v+a)​|v|22)+v⋅div⁡(a⊗v)−12​v⋅a​div⁡v≤0in ​𝒟′​(ℝ3×(0,T1)).\partial_{t}\frac{|v|^{2}}{2}-\Delta\frac{|v|^{2}}{2}-\kappa\operatorname*{\mbox{div}}(v\operatorname*{\mbox{div}}v)+|\nabla v|^{2}+\kappa(\operatorname*{\mbox{div}}v)^{2}+\operatorname*{\mbox{div}}\left((v+a)\frac{|v|^{2}}{2}\right)\\ +v\cdot\operatorname*{\mbox{div}}(a\otimes v)-\frac{1}{2}v\cdot a\operatorname*{\mbox{div}}v\leq 0\quad\mbox{in }\mathcal{D}^{\prime}(\mathbb{R}^{3}\times(0,T_{1})). (3.2)

In other words, vv is a suitable weak solution to (2.1) with b=ab=a. Note also that limt→0+‖v​(⋅,t)−u02‖L2​(B​(x0,1))→0\lim_{t\to 0^{+}}\|v(\cdot,t)-u^{2}_{0}\|_{L_{2}(B(x_{0},1))}\to 0 for all x0∈ℝ3x_{0}\in\mathbb{R}^{3}, u02|B​(4/3)=0u^{2}_{0}|_{B(4/3)=0} (thus we have as a byproduct limt→0+‖v​(⋅,t)‖L2​(B​(4/3))=0\lim_{t\to 0^{+}}\|v(\cdot,t)\|_{L_{2}(B(4/3))}=0) and from Lemma 3.1, there exists 0<T2=T2​(α,κ,m,M)<T10<T_{2}=T_{2}(\alpha,\kappa,m,M)<T_{1} such that

ess sup0<t<T2⁡12​∫B​(2)|v​(x,t)|2​𝑑x+∫0T2∫B​(2)|∇v​(x,t)|2​𝑑x​𝑑t≤C​(α,m,M).\operatorname*{\mbox{ess sup}}_{0<t<T_{2}}\frac{1}{2}\int_{B(2)}|v(x,t)|^{2}dx+\int_{0}^{T_{2}}\int_{B(2)}|\nabla v(x,t)|^{2}dxdt\leq C(\alpha,m,M). (3.3)

From the local energy (3.2) for vv and the fact that limt→0+‖v​(⋅,t)‖L2​(B​(4/3))=0\lim_{t\to 0^{+}}\|v(\cdot,t)\|_{L_{2}(B(4/3))}=0, we obtain

12​∫B​(4/3)|v​(x,t)|2​ϕ​(x)​𝑑x+∫0t∫B​(4/3)|∇v|2​ϕ​𝑑z+∫0t∫B​(4/3)κ​(div⁡v)2​ϕ​𝑑z≤∫0t∫B​(4/3)|v|22​Δ​ϕ​𝑑z+∫0t∫B​(4/3)|v|22​(v+a)⋅∇ϕ​d​z+∫0t∫B​(4/3)a⊗v:∇v​ϕ​d​z+∫0t∫B​(4/3)a⊗v:v⊗∇ϕ​d​z−κ​∫0t∫B​(4/3)v⋅∇ϕ​div⁡v​d​z+12​∫0t∫B​(4/3)ϕ​v⋅a​div⁡v​d​z,\frac{1}{2}\int_{B(4/3)}|v(x,t)|^{2}\phi(x)dx+\int_{0}^{t}\int_{B(4/3)}|\nabla v|^{2}\phi dz+\int_{0}^{t}\int_{B(4/3)}\kappa(\operatorname*{\mbox{div}}v)^{2}\phi dz\\ \leq\int_{0}^{t}\int_{B(4/3)}\frac{|v|^{2}}{2}\Delta\phi dz+\int_{0}^{t}\int_{B(4/3)}\frac{|v|^{2}}{2}(v+a)\cdot\nabla\phi dz\\ +\int_{0}^{t}\int_{B(4/3)}a\otimes v:\nabla v\phi dz+\int_{0}^{t}\int_{B(4/3)}a\otimes v:v\otimes\nabla\phi dz\\ -\kappa\int_{0}^{t}\int_{B(4/3)}v\cdot\nabla\phi\operatorname*{\mbox{div}}vdz+\frac{1}{2}\int_{0}^{t}\int_{B(4/3)}\phi v\cdot a\operatorname*{\mbox{div}}vdz, (3.4)

for a.e. t∈(0,T2)t\in(0,T_{2}) and 0≤ϕ∈C0∞​(B​(4/3))0\leq\phi\in C^{\infty}_{0}(B(4/3)) such ϕ≡1\phi\equiv 1 in BB. The previous estimate and a repetitive use of Hölder inequality (with estimate (3.3) at hand) yield:

∫B|v​(x,t)|22​𝑑x+∫0t∫B|∇v|2​𝑑z+∫0t∫Bκ​(div⁡v)2​𝑑z≤C​(α,κ,m,M)​(t+t110+t2​m−35​m+tm−35​m),\int_{B}\frac{|v(x,t)|^{2}}{2}dx+\int_{0}^{t}\int_{B}|\nabla v|^{2}dz+\int_{0}^{t}\int_{B}\kappa(\operatorname*{\mbox{div}}v)^{2}dz\\ \leq C(\alpha,\kappa,m,M)(t+t^{\frac{1}{10}}+t^{\frac{2m-3}{5m}}+t^{\frac{m-3}{5m}}), (3.5)

for a.e. t∈(0,T2)t\in(0,T_{2}).
Now, fix t0∈(0,T2)t_{0}\in(0,T_{2}) to be specified later. Then extend vv to B×(−1+t0,t0)B\times(-1+t_{0},t_{0}) by setting v≡0v\equiv 0 in B×(−1+t0,0)B\times(-1+t_{0},0). Extend also aa to B×(−1+t0,t0)B\times(-1+t_{0},t_{0}) by setting a≡0a\equiv 0 for t<0t<0. Clearly the extended function vv is a suitable weak solution to (2.1), with the extended aa, in B×(−1+t0,t0)B\times(-1+t_{0},t_{0}). Indeed, the fact that limt→0+‖v​(⋅,t)‖L2​(B)=0\lim_{t\to 0^{+}}\|v(\cdot,t)\|_{L_{2}(B)}=0 insure that ∂tv\partial_{t}v and ∂t|v|22\partial_{t}\frac{|v|^{2}}{2} will not cause any problem across {t=0}\{t=0\}. Finally, because of (3.5), if we choose t0=t0​(α,κ,m,M)<T2t_{0}=t_{0}(\alpha,\kappa,m,M)<T_{2} sufficiently small, we can apply Theorem 2.7 and conclude that vv is Hölder continuous in B​(1/2)×[0,t0]B(1/2)\times[0,t_{0}], for some γ=γ​(m)∈(0,1)\gamma=\gamma(m)\in(0,1). This concludes the proof of the theorem. ∎

Theorem 3.2 allows us to prove the following.

Theorem 3.3 (Local Hölder regularity of Leray solutions).

Let u0∈L2,l​o​c​(ℝ3)u_{0}\in L_{2,loc}(\mathbb{R}^{3}) such that α:=supx0∈ℝ3∫B|u0​(x)|2​𝑑x<∞\alpha:=\sup_{x_{0}\in\mathbb{R}^{3}}\int_{B}|u_{0}(x)|^{2}dx<\infty. Suppose in addition that M:=‖u0‖Cγ,γ2​(B​(2))<∞M:=\|u_{0}\|_{C^{\gamma,\frac{\gamma}{2}}(B(2))}<\infty. Then, there exists T=T​(α,γ,κ,M)>0T=T(\alpha,\gamma,\kappa,M)>0 such that any Leray weak solution uu to (1.1) satisfies:

‖u‖Cγ,γ2​(B​(1/4)¯×[0,T])≤C​(α,γ,κ,M).\|u\|_{C^{\gamma,\frac{\gamma}{2}}(\overline{B(1/4)}\times[0,T])}\leq C(\alpha,\gamma,\kappa,M).
Sketch of proof.

With the same notation as in Theorem 3.2, we have that suppu01⊂⊂B(2)u^{1}_{0}\subset\subset B(2) and ‖u01‖Cγ​(ℝ3)≤C​M\|u_{0}^{1}\|_{C^{\gamma}(\mathbb{R}^{3})}\leq CM. Consequently, u−au-a is Hölder continuous with some exponent β∈(0,γ)\beta\in(0,\gamma) in B​(1/2)¯×[0,T1]\overline{B(1/2)}\times[0,T_{1}] where T1=T1​(α,κ,M)>0T_{1}=T_{1}(\alpha,\kappa,M)>0. Since the initial data u01u_{0}^{1} for aa is in Cγ​(ℝ3)C^{\gamma}(\mathbb{R}^{3}), it is not difficult to show that a∈Cγ,γ2​(ℝ3×[0,T1])a\in C^{\gamma,\frac{\gamma}{2}}(\mathbb{R}^{3}\times[0,T_{1}]). Therefore, uu is Hölder continuous with exponent β\beta in B​(1/2)¯×[0,T1]\overline{B(1/2)}\times[0,T_{1}]. From this point, a standard bootstrap argument with repetitive use of Lemma 2.2 yields the required Hölder continuity of uu; and a careful track of the constants gives us the estimate in the theorem. This concludes the proof. ∎

4 Proof of Theorem 1.1

We are ready now to proof the main result of this paper.

Proof of Theorem 1.1.

From Lemma 3.1, we have that

sup0<t<T112​∫B|u​(x,t)|2​𝑑x+∫0T1∫B|∇u​(x,t)|2​𝑑x​𝑑t≤C​(κ,‖u0‖C​(∂B)),\sup_{0<t<T_{1}}\frac{1}{2}\int_{B}|u(x,t)|^{2}dx+\int_{0}^{T_{1}}\int_{B}|\nabla u(x,t)|^{2}dxdt\leq C(\kappa,\|u_{0}\|_{C(\partial B)}), (4.1)

with T1=T1​(κ,‖u0‖C​(∂B))T_{1}=T_{1}(\kappa,\|u_{0}\|_{C(\partial B)}).
Since uu is scale invariant i.e. λ​u​(λ​x,λ2​t)=u​(x,t)\lambda u(\lambda x,\lambda^{2}t)=u(x,t), we have that

u​(x,t)=1t​U​(xt),t>0,u(x,t)=\frac{1}{\sqrt{t}}U\left(\frac{x}{\sqrt{t}}\right),\quad t>0,

where U​(⋅):=u​(⋅,1)U(\cdot):=u(\cdot,1). Thus, we deduce from (4.1) that

t∗​∫B​(1/t∗)|U​(y)|2​𝑑y+t∗​∫B​(1/t∗)|∇U​(y)|2​𝑑y≤C​(κ,‖u0‖C​(∂B)),\sqrt{t^{*}}\int_{B(1/\sqrt{t^{*}})}|U(y)|^{2}dy+\sqrt{t^{*}}\int_{B(1/\sqrt{t^{*}})}|\nabla U(y)|^{2}dy\leq C(\kappa,\|u_{0}\|_{C(\partial B)}), (4.2)

for all t∗∈(0,T1)t^{*}\in(0,T_{1}).
On the other hand, for all |x0|=8|x_{0}|=8, we have u0∈C∞​(B​(x0,4))u_{0}\in C^{\infty}(B(x_{0},4)). Therefore, by Theorem 3.3 (and some simple bootstrapping arguments), we have that there exists T2=T2​(κ,u0)>0T_{2}=T_{2}(\kappa,u_{0})>0 such that

‖∂t∂αu‖L∞​(B​(x0,1/8)¯×[0,T2])≤C​(α,κ,u0),\|\partial_{t}\partial^{\alpha}u\|_{L_{\infty}(\overline{B(x_{0},1/8)}\times[0,T_{2}])}\leq C(\alpha,\kappa,u_{0}), (4.3)

for all Leray solution uu to system (1.1) with u0u_{0} initial data.
Since for all λ>0\lambda>0, uλ:(x,t)↦λ​u​(λ​x,λ2​t)u^{\lambda}:(x,t)\mapsto\lambda u(\lambda x,\lambda^{2}t) is also a Leray solution to (1.1) with initial data u0u_{0}, then (4.3) holds also for uλu^{\lambda} and we obtain that

|λ1+|α|​∂αu​(λ​x0,λ2​t)−∂αu0​(x0)|≤C​(α,κ,u0)​t.|\lambda^{1+|\alpha|}\partial^{\alpha}u(\lambda x_{0},\lambda^{2}t)-\partial^{\alpha}u_{0}(x_{0})|\leq C(\alpha,\kappa,u_{0})t.

Setting y=x0/ty=x_{0}/\sqrt{t}, and by using the homogeneity of ∂αu0\partial^{\alpha}u_{0}, we get that:

|∂α(U−u0)​(y)|≤C​(α,κ,u0)|y|3+|α|,∀|y|>8T2.|\partial^{\alpha}(U-u_{0})(y)|\leq\frac{C(\alpha,\kappa,u_{0})}{|y|^{3+|\alpha|}},\quad\forall|y|>\frac{8}{\sqrt{T_{2}}}. (4.4)

Now, we choose t∗=t∗​(κ,u0)t^{*}=t^{*}(\kappa,u_{0}) in (4.2) sufficiently small so that

∫B​(16T2)(|U​(y)|2+|∇U​(y)|2​d​y)≤C​(κ,u0);\int_{B(\frac{16}{\sqrt{T_{2}}})}\left(|U(y)|^{2}+|\nabla U(y)|^{2}dy\right)\leq C(\kappa,u_{0}); (4.5)

and because uu satisfies (1.1), it’s not difficult to see that

−Δ​U−κ​∇div⁡U+U⋅∇U+U2​div⁡U−x2⋅∇U−U2=0in ​ℝ3.-\Delta U-\kappa\nabla\operatorname*{\mbox{div}}U+U\cdot\nabla U+\frac{U}{2}\operatorname*{\mbox{div}}U-\frac{x}{2}\cdot\nabla U-\frac{U}{2}=0\quad\mbox{in }\mathbb{R}^{3}. (4.6)

Thus, from Elliptic estimates (alongside ideas we used in the proof of Lemma 2.2), we find that

‖U‖Ck​(B​(9/T2)¯)≤C​(k,u0)(k=0,1,2​…)\|U\|_{C^{k}(\overline{B(9/\sqrt{T_{2}})})}\leq C(k,u_{0})\quad(k=0,1,2\ldots) (4.7)

Finally, let us explain how we define the semigroup Sκ​(t)S_{\kappa}(t) and derive the required estimates to close the proof of the theorem.
The classical Calderon-Zygmund combined with real interpolation methods allow us to get the existence of a unique function (up to a constant) q0q_{0} such that Δ​q0=div⁡u0\Delta q_{0}=\operatorname*{\mbox{div}}u_{0} and

‖q0‖B​M​O​(ℝ3)+‖∇q0‖L3,∞​(ℝ3)≤c​‖u0‖C​(∂B).\|q_{0}\|_{BMO(\mathbb{R}^{3})}+\|\nabla q_{0}\|_{L_{3,\infty}(\mathbb{R}^{3})}\leq c\|u_{0}\|_{C(\partial B)}. (4.8)

Set u0(1):=∇q0u_{0}^{(1)}:=\nabla q_{0} and notice that, because of the uniqueness of q0q_{0} and the scaling symmetry of u0u_{0}, we have that u0(1)u_{0}^{(1)} is also (−1)(-1)-homogeneous. Next, by elliptic estimates we see that u0(1)∈C∞​(∂B)u_{0}^{(1)}\in C^{\infty}(\partial B) and we have (thanks to (4.8))

|∂αu0(1)​(x)|≤C​(α,u0)|x|1+|α|.|\partial^{\alpha}u_{0}^{(1)}(x)|\leq\frac{C(\alpha,u_{0})}{|x|^{1+|\alpha|}}.

Introduce now u0(0):=u0−u0(1)u_{0}^{(0)}:=u_{0}-u^{(1)}_{0} and notice that div⁡u0(0)=0\operatorname*{\mbox{div}}u^{(0)}_{0}=0 (we should also point out that curl⁡u0(1)=0\operatorname*{\mbox{curl}}u_{0}^{(1)}=0 by definition). We set

Sκ​(t)​u0=eΔ​t​u0(0)+e(1+κ)​Δ​t​u0(1);S_{\kappa}(t)u_{0}=e^{\Delta t}u_{0}^{(0)}+e^{(1+\kappa)\Delta t}u_{0}^{(1)}; (4.9)

It is clear from this definition that Sκ​(t)S_{\kappa}(t) is a semigroup. Moreover, using similar arguments as the ones we use in the proof of Lemma 2.2, we see that v​(x,t):=Sκ​(t)​u0​(x)v(x,t):=S_{\kappa}(t)u_{0}(x) solves the Lamé system

{∂tv−Δ​v−κ​∇div⁡v=0in ​ℝ3×(0,∞)v|t=0=u0in ​ℝ3.\left\{\begin{gathered}\partial_{t}v-\Delta v-\kappa\nabla\operatorname*{\mbox{div}}v=0\quad\mbox{in }\mathbb{R}^{3}\times(0,\infty)\\ v|_{t=0}=u_{0}\quad\mbox{in }\mathbb{R}^{3}.\end{gathered}\right. (4.10)

Also, we have λ​v​(λ​x,λ2​t)=v​(x,t)\lambda v(\lambda x,\lambda^{2}t)=v(x,t) for all λ>0\lambda>0. So, if we set V:=Sκ​(1)​u0V:=S_{\kappa}(1)u_{0}, we have that

v​(x,t)=1t​V​(xt)∀(x,t)∈ℝ3×(0,∞).v(x,t)=\frac{1}{\sqrt{t}}V\left(\frac{x}{\sqrt{t}}\right)\quad\forall(x,t)\in\mathbb{R}^{3}\times(0,\infty).

and finally

|∂α(Sκ​(1)​u0−u0)|≤|∂α(eΔ​u0(0)−u0(0))|+|∂α(e(1+κ)​Δ​u0(1)−u0(1))|≤C​(α,κ,u0)(1+|x|)3+|α||\partial^{\alpha}(S_{\kappa}(1)u_{0}-u_{0})|\leq|\partial^{\alpha}(e^{\Delta}u_{0}^{(0)}-u_{0}^{(0)})|+|\partial^{\alpha}(e^{(1+\kappa)\Delta}u_{0}^{(1)}-u_{0}^{(1)})|\leq\frac{C(\alpha,\kappa,u_{0})}{(1+|x|)^{3+|\alpha|}}

by well-known properties of the heat equation. Same machinery for system (1.2). And this concludes the proof. ∎

Acknowledgement

This work was supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]. The author would like to thank Gregory Seregin for the insightful discussions during the completion of this paper.

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