Asymptotics and Lower Bound for the Lifespan of Solutions to the Primitive Equations

Abstract

In a previous work we obtained a large lower bound for the lifespan of the solutions to the Primitive Equations, and proved convergence to the 3D quasi-geostrophic system for general and ill-prepared blowing-up data, when the kinematic viscosity $\nu$ is equal to the heat diffusivity $\nu '$, turning the diffusion operator $\varGamma$ into the classical Laplacian.

Obtaining the same results in the general case is much more difficult as it involves a homogeneous non-local non-radial diffusion operator $\varGamma$ whose semi-group and singular integral form kernels present sign changes. Every classical result related to non-local operators, or to Navier-Stokes system then becomes more involved here and the key ingredient will be new transport-diffusion estimates obtained in a companion paper and a precise use of the quasi-geostrophic decomposition.

Appendix

The first part is devoted to a quick presentation of the Littlewood-Paley theory. In the second section we briefly recall general definitions for vortex patches and the last section provides new properties for the operator $\varGamma$ and recalls the a priori estimates from [16].

1.1 A.1 Littlewood-Paley Theory

In this section, $C^{s}$ is the usual Hölder space, which can also be defined through the Littlewood-Paley theory if $s\notin \mathbb{N}$ (we refer to [4, 17] for a complete presentation of the theory):

$$ C^{s}=\Bigl\{ u \in {\mathcal{S}}'\bigl( \mathbb{R}^{3}\bigr), \quad \|u\|_{C^{s}} \mathop{=}^{\mathrm{def}} \sup_{q \geq -1} 2^{qs} \|\Delta _{q} u \|_{L ^{\infty }} < \infty \Bigr\} , $$

where $\Delta _{q}$ is the classical dyadic frequency localization operator defined as follows: consider a smooth radial function $\chi$ supported in the ball $B(0, \frac{4}{3})$, equal to 1 in a neighborhood of $B(0, \frac{3}{4})$ and such that $r\mapsto \chi (r.e _{1})$ is nonincreasing over $\mathbb{R}_{+}$. If we define $\varphi (\xi )=\chi (\frac{\xi }{2})-\chi (\xi )$, $\varphi$ is supported respectively in the annulus ${\mathcal{C}}(0,\frac{3}{4}, \frac{8}{3})$ (equal to 1 in a sub-annulus), and satisfy that for all $\xi \in \mathbb{R}^{3}$,

$$ \chi (\xi )+ \sum_{q\geq 0} \varphi \bigl(2^{-q} \xi \bigr)=1 \quad \mbox{and if}\quad \xi \neq 0, \quad \sum _{q\in \mathbb{Z}} \varphi \bigl(2^{-q} \xi \bigr)=1. $$

Then for all tempered distribution we define:

• $\Delta _{-1}= \mathcal{F}^{-1} (\chi (\xi ) \widehat{u}(\xi ))$ and $\forall q\leq -2$, $\Delta _{q}=0$,

• $\forall q\geq 0$, $\Delta _{q} =\mathcal{F}^{-1} (\varphi (2^{-q}\xi ) \widehat{u}(\xi ) )$ and $S_{q} u =\sum_{p< q-1} \Delta _{p} u= \chi (2^{-q}D) u$,

• $\forall q \in \mathbb{Z}$, $\dot{\Delta }_{q} =\mathcal{F}^{-1}(\varphi (2^{-q}\xi ) \widehat{u}(\xi ))$ and ${\dot{S}_{q} u =\sum_{p< q-1} \dot{\Delta }_{p} u= \chi (2^{-q}D) u}$.

Hölder spaces are particular cases of inhomogeneous Besov spaces: $C^{s}=B_{\infty , \infty }^{s}$, where

$$ B_{p,r}^{s}=\bigl\{ u \in {\mathcal{S}}'\bigl( \mathbb{R}^{3}\bigr), \ \|u\|_{B_{p,r}^{s}} \mathop{=}^{\mathrm{def}} \bigl\| \bigl(2^{qs} \|\Delta _{q} u\| _{L^{p}} \bigr)_{q \geq -1}\bigr\| _{\ell ^{r}} < \infty \bigr\} . $$

The homogeneous Besov spaces are defined as follows:

$$ \dot{B}_{p,r}^{s}=\bigl\{ u \in {\mathcal{S}}'\bigl(\mathbb{R}^{3}\bigr), \mbox{with } \mathop{\mbox{lim}}_{q\rightarrow -\infty }\dot{S}_{q} u=0 \mbox{ and } \|u\|_{\dot{B}_{p,r}^{s}} \mathop{=}^{\mathrm{def}} \bigl\| \bigl(2^{qs} \|\dot{\Delta }_{q}u\|_{L^{p}} \bigr)_{q \in \mathbb{Z}}\bigr\| _{\ell ^{r}} < \infty \bigr\} . $$

When the regularity index $s$ is negative, another way to express the Besov norm involves the operator $S_{q}$ instead of $\Delta _{q}$ (for more details we refer to [4] Proposition 2.76 and to 2.31 for the homogeneous case):

Proposition 10

There exists a constant $C>0$ such that for all $s<0$, $p,r\in [1,\infty ]$ and $u$, then $u\in B_{p,r}^{s}$ if and only if

$$ \bigl(2^{qs} \|S_{q} u\|_{L^{p}} \bigr)_{q \geq -1} \in \ell ^{r}. $$

Moreover, we have

$$ \bigl\| \bigl(2^{qs} \|S_{q} u\|_{L^{p}} \bigr)_{q}\bigr\| _{\ell ^{r}} \sim \|u\| _{B_{p,r}^{s}}. $$

(A.96)

Remark 18

Due to the supports, we easily obtain that

$$ \Delta _{j} \Delta _{l} =\dot{\Delta }_{j} \dot{\Delta }_{l} =0 \quad \mbox{if } |j-l|\geq 2. $$

(A.97)

• For any functions $f$, $g$, and any $\alpha \in \{-1,0,1\}$, the product $\Delta _{q} f. \Delta _{q+\alpha } g$ has its frequencies in a ball of size $2^{q}$.

• For any functions $f$, $g$, the product $S_{q-1} f. \Delta _{q} g$ has its frequencies in an annulus of size $2^{q}$.

We will end this section with the Bony decomposition, which comes from the fact that for all distributions $u$, $v$, we can write the product as follows:

$$ uv=\biggl(\sum_{q\geq -1} \Delta _{q} u \biggr) \biggl(\sum_{l\geq -1} \Delta _{l} v \biggr). $$

In fact, a more efficient way to write this product is the following Bony decomposition, where we basically set three parts according to the fact that the frequency $q$ of $u$ is smaller, comparable or bigger than the frequency $l$ of $v$:

$$ uv= T_{u} v+ T_{v} u+ R(u, v), $$

(A.98)

where

• $T$ is the paraproduct: $T_{u} v= \sum_{p\leq q-2} \Delta _{p} u \Delta _{q} v= \sum_{q} S_{q-1} u \Delta _{q} v$,

• $R$ is the remainder: $R(u, v)= \sum_{|p-q| \leq 1} \Delta _{p} u \Delta _{q} v$.

A similar decomposition can be defined with the homogeneous Littlewood-Paley operators.

1.2 A.2 Vortex Patches

We refer to [17] for a full description of the persistence of the vortex patches structure in the case of the Euler system, to [25] and [35] for the case of the Navier-Stokes system, and to [28] for the case of the inviscid Primitive Equations ($\nu =\nu '=0$). In the present paper, we take here the same definitions of vortex patches and tangential regularity as in [28]: a potential vortex patch will be defined with respect to the scalar potential vorticity instead of the vorticity (rotational of the velocity). The potential vorticity is called a vortex patch if it is the characteristic function of a regular open set:

Definition 1

We say that $\varOmega _{0}$ is a vortex patch of class $C^{s}$ if, for some $s\in ]0,1[$,

$$ \varOmega _{0}= \varOmega _{0, i} \textbf{1}_{D} +\varOmega _{0, e} \textbf{1} _{\mathbb{R}^{3}-D}, $$

where $\varOmega _{0, i} \in C^{s}(\overline{D})$, $\varOmega _{0, e} \in C ^{s}(\mathbb{R}^{3}-D)$ and $D$ is an open bounded domain of class $C^{s+1}$.

In our results we will use estimates involving the tangential regularity with respect to a set $X$ of vectorfields:

Definition 2

If $X=(X_{\lambda })_{\lambda =1,\ldots,N}$ is a finite family of vectorfields we will say that this family is admissible if and only if (× is the usual vector product in $\mathbb{R}^{3}$):

$$ [X]^{-1}\mathop{=}^{\mathrm{def}} \biggl(\frac{2}{N(N-1)} \sum _{\lambda < \lambda '} |X_{\lambda } \times X_{\lambda '}|^{2} \biggr)^{-\frac{1}{4}} < \infty . $$

If $s\in ]0,1[$ and $X$ is an admissible family of vectorfields $C^{s}$ we define the space:

$$ C^{s}(X)=\bigl\{ w \in L^{\infty } \mbox{ such that} \ X_{\lambda }(x, D)w \mathop{=}^{\mathrm{def}} \mbox{div} (w \otimes X _{\lambda })\in C^{s-1} \bigr\} $$

and as corresponding norm we take:

$$ \|w\|_{C^{s}(X)}\mathop{=}^{\mathrm{def}}\|w\|_{L^{\infty }}+ \bigl\| [X]^{-1} \bigr\| _{L^{\infty }} +\sum_{\lambda =1}^{N} \bigl( \|X_{\lambda }\|_{C^{s}}+ \bigl\| X_{\lambda }(x, D)w\bigr\| _{C^{s-1}} \bigr). $$

(A.99)

Remark 19

We took here the same definition as in [28] for $X_{\lambda }(x, D)w$ which is a slightly simplified formulation of the definitions from [17], or [35].

1.3 A.3 Definition and Additional Properties for the Non-local Operator $\varGamma$

1.3.1 A.3.1 Product Estimates

We refer to [16] for a study of operator $\varGamma$ where we give various formulations. This operator is defined as

$$ \varGamma = \Delta \Delta _{F}^{-1} \bigl(\nu \partial _{1}^{2} +\nu \partial _{2}^{2}+ \nu ' F^{2} \partial _{3}^{2}\bigr). $$

First we decompose $\varGamma$ into its local and non-local parts:

$$ \varGamma =\varGamma _{L}+\bigl(\nu -\nu '\bigr) F^{2}\bigl(1-F^{2}\bigr) \varLambda ^{2}, $$

where we denote:

$$ \textstyle\begin{cases} \varGamma _{L}= \nu \partial _{1}^{2} +\nu \partial _{2}^{2}+ ( (1-F ^{2})\nu +F^{2} \nu ' ) \partial _{3}^{2}, \\ \varLambda = \partial _{3}^{2} (-\Delta _{F})^{-\frac{1}{2}}. \end{cases} $$

(A.100)

We also refer to [16] for the following expressions of $\varLambda$ as singular or convergent integrals, directly related to the alternative expression of homogeneous Besov norms involving finite differences:

$$\begin{aligned} \varLambda f (x) =& \lim_{\varepsilon \rightarrow 0}\int _{|y|\geq \varepsilon } K(y) \bigl(f(x-y)-f(x) \bigr) dy \\ =& \frac{1}{2}\int _{\mathbb{R}^{3}} K(y) \bigl(f(x-y)+f(x+y)-2f(x) \bigr) dy, \end{aligned}$$

(A.101)

where the kernel $K$ is defined for all $y\in \mathbb{R}^{3}$ by ($C$ is a universal constant):

$$ K(y)= -\frac{2C}{F^{3}}\frac{y_{1}^{2}+ y_{2}^{2} -\frac{3}{F^{2}}y _{3}^{2}}{ (y_{1}^{2}+ y_{2}^{2} +\frac{1}{F^{2}}y_{3}^{2} )^{3}}. $$

(A.102)

The main feature of [16] was an estimate of the commutator of $\varLambda$ with a lagrangian change of variable (crucial to obtain the a priori estimates), but we also obtained the following result that allows to consider commutators like $\varGamma (fg)-f\varGamma g$:

Proposition 11

([16] Sect. 3.3.2)

For any smooth functions $f$, $g$ we can write:

$$ \varLambda (fg)= f\varLambda g +g\varLambda f +M(f,g), $$

where the bilinear operator $M$ is defined for all $x\in \mathbb{R} ^{3}$ by:

$$ M(f,g) (x) = \int _{\mathbb{R}^{3}} K(y) \bigl(f(x-y)-f(x) \bigr) \bigl(g(x-y)-g(x) \bigr) dy. $$

(A.103)

Moreover there exists a constant $C_{F}$ such that for all $f$, $g$:

$$ \bigl\| M(f,g)\bigr\| _{L^{p}} \leq C_{F} \sqrt{\|f\|_{L^{p}} \| \nabla f\|_{L ^{p}} \|g\|_{L^{\infty }} \|\nabla g\|_{L^{\infty }}}. $$

(A.104)

In this article we will need more precise estimates where we can in particular, even if $M(f,g)=M(g,f)$, make the derivatives pound differently on $f$ or $g$, which is the object of the following result:

Proposition 12

There exists a constant $C_{F}$ such that for all $f$, $g$ and all $p,p_{1}, P_{2},r,\overline{r}\in [1,\infty ]$ and $\eta$ satisfying:

$$ \textstyle\begin{cases} \displaystyle {\frac{1}{p}= \frac{1}{p_{1}} +\frac{1}{p_{2}}, \quad 1= \frac{1}{r} +\frac{1}{ \overline{r}},} \\ \displaystyle {2-\eta -\frac{3}{r} \in ]0,1[,} \end{cases} $$

then we have

$$ \bigl\| M(f,g)\bigr\| _{L^{p}} \leq C_{F} \|f\|_{\displaystyle {\dot{B}_{p_{1},r} ^{2-\eta -\frac{3}{r}}}}\|g \|_{\displaystyle {\dot{B}_{p_{2}, \overline{r}}^{2+\eta -\frac{3}{\overline{r}}}}}. $$

(A.105)

Remark 20

In the present article we will use the previous proposition in the case $p=p_{1}=p_{2}=\infty$, $r=\infty$, $\overline{r}=1$ and for $\eta =2-(s+\gamma )$ where $s\in ]0,1[$ and $\sigma >0$ is such that $s+\sigma <1$. In this case we end up with:

$$ \bigl\| M(f,g)\bigr\| _{L^{\infty }} \leq C_{F} \|f\|_{\dot{B}_{\infty ,\infty } ^{s+\sigma }}\|g \|_{\dot{B}_{\infty ,1}^{1-(s+\sigma )}}. $$

(A.106)

Proof

Let $f$, $g$ be smooth functions. From the expression of the kernel, there exists a constant $C_{F}>0$ such that, thanks to the relations between the parameters:

$$\begin{aligned} \bigl\| M(f,g)\bigr\| _{L_{x}^{p}} &\leq C_{F}\int _{\mathbb{R}^{3}} \frac{\|f(.-y)-f(.) \|_{L_{x}^{p_{1}}}}{|y|^{2-\eta }} \cdot \frac{\|g(.-y)-g(.)\|_{L_{x} ^{p_{2}}}}{|y|^{2+\eta }} dy \\ &\leq C_{F} \biggl( \int _{\mathbb{R}^{3}} \frac{\|f(.-y)-f(.)\|_{L _{x}^{p_{1}}}^{r}}{|y|^{(2-\eta )r}} dy \biggr) ^{\frac{1}{r}} \biggl( \int _{\mathbb{R}^{3}} \frac{\|g(.-y)-g(.)\|_{L_{x}^{p_{2}}}^{ \overline{r}}}{|y|^{(2+\eta )\overline{r}}} dy \biggr) ^{\frac{1}{ \overline{r}}}. \end{aligned}$$

(A.107)

If we denote $s_{1}=2-\eta -\frac{3}{r}$ and $s_{2}=2+\eta -\frac{3}{ \overline{r}}$ then $(2-\eta )r= s_{1} r+3$ and $(2+\eta ) \overline{r} =s_{2} \overline{r} +3$. Moreover $s_{1}+s_{2}=1$ so if $1-\frac{3}{r}<\eta <2-\frac{3}{r}$ both regularity exponents are in $]0,1[$ and we can use the following result

Theorem 2

([4], 2.36)

Let $s \in ]0,1[$ and $p,r\in [1,\infty]$. There exists a constant $C$ such that for any $u\in {\mathcal{C}} _{h}'$,

$$ C^{-1} \|u\|_{\dot{B}_{p,r}^{s}}\leq \biggl\| \frac{\|\tau _{-y}u -u\|_{L^{p}}}{|y|^{s}}\biggr\| _{L^{r} (\mathbb{R}^{d}; \frac{dy}{|y|^{d}})} \leq C \|u\|_{\dot{B}_{p,r}^{s}}. $$

This concludes the proof.  □

1.3.2 A.3.2 A Priori Estimates

For the comfort of the reader we state here the a priori estimates obtained in [16]. We consider the following transport diffusion system:

$$ \textstyle\begin{cases} \partial _{t} u +v.\nabla u -\varGamma u =F^{e}, \\ u_{|t=0}=u_{0} \end{cases} $$

(A.108)

Let us introduce $\mu _{visc}=\frac{\max (\nu , \nu ')}{\min (\nu , \nu ')}$.

Proposition 13

($L^{p}$-estimates) Assume that $u$ solves (A.108) on $[0,T]$ with $u_{0} \in L^{p}$ and that $\|v\|_{L_{T}^{\infty }L^{6}} \leq C'$ (for some constant $C'$) with $\operatorname{div}v=0$. Then there exists a constant $D$ (depending on $F$, $\mu _{visc}$ and $C'$) such that for all $t\in [0,T]$,

$$ \|u\|_{L_{t}^{\infty }L^{p}} \leq D^{t} \biggl(\|u_{0} \|_{L^{p}}+ \int _{0} ^{t} \|F^{e}(\tau ) \|_{L^{p}} d\tau \biggr). $$

(A.109)

Theorem 3

(Smoothing effect)

Assume that $u$ solves (A.108) on $[T_{1}, T_{2}]$ with $v$ satisfying $\operatorname{div}v=0$ and $\|v\|_{L^{\infty }({[T_{1},T_{2}]}, L^{6})} \leq C'$, $u(T_{1}) \in L^{p}$, $F^{e}\in L_{loc}^{1} L^{p}$ (for $p\in [1,\infty ]$). There exist two constants $C$ and $C_{F}$ such that if $T_{2}-T_{1}>0$ is so small that:

• 1. $2CC'(T_{2}-T_{1})^{\frac{1}{4}} \leq \nu _{0}^{\frac{3}{4}}$,

• 2. $e^{C\int _{T_{1}}^{T_{2}} \|\nabla v (\tau )\|_{L^{\infty }}}-1 \leq \frac{1}{C_{F} \mu _{visc}}$.

Then, for all $r\in [1,\infty ]$, there exists a constant $C_{r,F}>0$ such that for all $t\in [T_{1}, T_{2}]$,

$$ (\nu _{0}r)^{\frac{1}{r}} \|u\|_{\widetilde{L}^{r}([T_{1},t], B_{p, \infty }^{\frac{2}{r}})} \leq C_{r, F} \biggl( \bigl\| u(T_{1})\bigr\| _{L^{p}} + \int _{T_{1}}^{t} \bigl\| F^{e}(\tau )\bigr\| _{L^{p}} d\tau \biggr) . $$

(A.110)

Remark 21

In the particular case $r=1$, $p=\infty$ we obtain that:

$$ \nu _{0}\|u\|_{\widetilde{L}^{1}([T_{1},t], C_{*}^{2})} \leq C_{F} \biggl(\bigl\| u(T_{1})\bigr\| _{L^{\infty }} + \int _{T_{1}}^{t} \bigl\| F^{e}(\tau )\bigr\| _{L^{\infty }} d\tau \biggr) . $$

(A.111)

Theorem 4

(a priori estimates)

Let $s\in ]-1,1[$. Assume that $u$ solves (A.108) on $[T_{1}, T_{2}]$ with $v$ satisfying $\operatorname{div}v=0$ and $\|v\|_{L^{\infty }({[T_{1},T_{2}]}, L^{6})} \leq C'$, $u(T_{1})\in B_{p,\infty }^{s}$. Assume in addition that the external force term can be decomposed into $F^{e}+G^{e}$, with $F^{e}\in \widetilde{L}^{1}([T_{1},T_{2}], B_{p,\infty }^{s})$ (for $p\in [1,\infty ]$) and $G^{e}\in \widetilde{L}^{\infty }([T_{1},T _{2}], B_{p,\infty }^{s+\frac{2}{r}-2})$ for $r\in [1,\infty ]$ with $s+\frac{2}{r}\in ]-1,1[$. There exist two constants $C_{s}$ and $C_{F}$ such that if $T_{2}-T_{1}>0$ is so small that:

• 1. $2CC'(T_{2}-T_{1})^{\frac{1}{4}} \leq \nu _{0}^{\frac{3}{4}}$,

• 2. $e^{C\int _{T_{1}}^{T_{2}} \|\nabla v (\tau )\|_{L^{\infty }}}-1 \leq \frac{1}{C_{F} \mu _{visc}}$,

• 3. $T_{2}-T_{1} +\int _{T_{1}}^{T_{2}} \|\nabla v\|_{L^{\infty }} d\tau \leq C_{s, \nu _{0}}$

Then, there exists a constant $C_{\nu _{0},F}>0$ such that for all $r\in [1,\infty ]$ with $s+\frac{2}{r}\in ]-1,1[$ and $t\in [T_{1}, T _{2}]$,

$$\begin{aligned} &(\nu _{0}r)^{\frac{1}{r}}\|u\|_{\widetilde{L}^{r}([T_{1},t], B_{p,\infty }^{s+\frac{2}{r}})} \\ &\quad \leq C_{\nu _{0}, F} \biggl( \bigl\| u(T_{1})\bigr\| _{B_{p,\infty }^{s}} + \|F^{e}\|_{\widetilde{L}^{1}([T_{1},t],B_{p, \infty }^{s})} +\frac{1}{\nu _{0}}\|G^{e} \|_{\widetilde{L}^{\infty }([T _{1},t],B_{p,\infty }^{s-2})} \biggr) . \end{aligned}$$

(A.112)