Global Unique Solutions for the Inhomogeneous Navier-Stokes equations with only Bounded Density, in Critical Regularity Spaces

Abstract

We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where the initial density $\rho _0$ is discontinuous and the initial velocity $u_0$ has critical regularity. Assuming that $\rho _0$ is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity $u_0$ belongs to the critical homogeneous Besov space $\dot{B}^{-1+2/p}_{p,1}({\mathbb {R}}^{2})$ $(1<p<2)$ and, in the three-dimensional case, if $u_0$ is small in $\dot{B}^{-1+3/p}_{p,1}({\mathbb {R}}^{3})\ (1<p<3)$. Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by Zhang (Adv Math 363:107007, 2020) are unique. Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system.

Appendix A

For the reader’s convenience, we here list some results involving Besov spaces and Lorentz spaces, prove maximal regularity estimates in Lorentz spaces for (2.4), and product estimates that were needed in the last section.

The following properties of Lorentz spaces may be found in [22] (and [25, Th2:1.18.6] for the first item):

Proposition A.1

(Properties of Lorentz spaces). There holds:

• (1) Interpolation: For all $1\le r,q\le \infty$ and $\theta \in (0,1)$, we have

  $$\begin{aligned} \left( L_{p_{1}}({\mathbb {R}}_{+};L_{q}({\mathbb {R}}^{d}));L_{p_{2}}({\mathbb {R}}_{+};L_{q}({\mathbb {R}}^{d})) \right) _{\theta ,r}=L_{p,r}({\mathbb {R}}_{+};L_{q}({\mathbb {R}}^{d}))), \end{aligned}$$

  where $1<p_{1}<p<p_{2}<\infty$ are such that $\frac{1}{p}=\frac{(1-\theta )}{p_{1}}+\frac{\theta }{p_{2}}\cdot$

• (2) Embedding: $L_{p,r_{1}}\hookrightarrow L_{p,r_{2}} \ \text {if}\ r_{1}\le r_{2},$ and $L_{p,p}=L_{p}.$

• (3) Hölder inequality: for $1<p,p_{1},p_{2}<\infty$ and $1 \le r,r_{1},r_{2}\le \infty ,$ we have

  $$\begin{aligned} \Vert {fg}\Vert _{L_{p,r}}\lesssim \Vert {f}\Vert _{L_{p_{1},r_{1}}}\Vert {g}\Vert _{L_{p_{2},r_{2}}} \quad \hbox {if}\quad \frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\quad \hbox {and}\quad \frac{1}{r} =\frac{1}{r_{1}}+\frac{1}{r_{2}}\cdot \end{aligned}$$

  This still holds for couples (1, 1) and $(\infty ,\infty )$ with the convention $L_{1,1}=L_{1}$ and $L_{\infty ,\infty }=L_{\infty }.$

• (4) For any $\alpha >0$ and nonnegative measurable function f,  we have $\Vert {f^{\alpha }}\Vert _{L_{p,r}} =\Vert {f}\Vert ^{\alpha }_{L_{p\alpha ,r\alpha }}$.

• (5) For any $k>0$, we have $\Vert {x^{-k}1_{{\mathbb {R}}_+}}\Vert _{L_{1/k,\infty }}=1$.

Next, let us state a few classical properties of Besov spaces.

Proposition A.2

(Besov embedding). There holds:

• (1) For any (p, q) in $[1,\infty ]^{2}$ such that $p\le q,$ we have

  $$\begin{aligned} \dot{B}^{d/p-d/q}_{p,1}({\mathbb {R}}^{d})\hookrightarrow L_{q}({\mathbb {R}}^{d}). \end{aligned}$$

• (2) Let $1\le p_{1}\le p_{2}\le \infty$ and $1\le r_{1}\le r_{2}\le \infty .$ Then, for any real number s,

  $$\begin{aligned} \dot{B}^{s}_{p_{1},r_{1}}({\mathbb {R}}^{d})\hookrightarrow \dot{B}^{s-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}({\mathbb {R}}^{d}). \end{aligned}$$

The interpolation theory in Besov spaces played an important role in our paper. Below are listed some results that we used (see details in [21, Prop. 2.22] or in [25, chapter 2.4.2]).

Proposition A.3

(Interpolation). A constant C exists that satisfies the following properties. If $s_{1}$ and $s_{2}$ are real numbers such that $s_{1}<s_{2}$ and $\theta \in ]0,1[,$ then we have, for any $(p,r)\in [1,\infty ]^{2}$ and any tempered distribution u satisfying (2.3),

$$\begin{aligned} \Vert {u}\Vert _{\dot{B}^{\theta s_{1}+(1-\theta )s_{2}}_{p,r}({\mathbb {R}}^{d})} \le \Vert {u}\Vert ^{\theta }_{\dot{B}^{s_{1}}_{p,r}}\Vert {u}\Vert ^{1-\theta }_{\dot{B}^{s_{2}}_{p,r} ({\mathbb {R}}^{d})} \end{aligned}$$

and, for some constant C depending only on $\theta$ and $s_2-s_1,$

$$\begin{aligned} \Vert {u}\Vert _{\dot{B}^{\theta s_{1}+(1-\theta )s_{2}}_{p,1}({\mathbb {R}}^{d})} \le C\Vert {u}\Vert ^{\theta }_{\dot{B}^{s_{1}}_{p,\infty }({\mathbb {R}}^{d})} \Vert {u}\Vert ^{1-\theta }_{\dot{B}^{s_{2}}_{p,\infty }({\mathbb {R}}^{d})}. \end{aligned}$$

Furthermore, we have for all $s\in (0,1)$ and $(p,q)\in [1,\infty ]^2:$

$$\begin{aligned} \dot{B}^{s}_{p,q}({\mathbb {R}}^d)=\bigl (L_{p}({\mathbb {R}}^d);\dot{W}^{1}_{p}({\mathbb {R}}^d)\bigr )_{s,q}. \end{aligned}$$

The following proposition has been used several times.

Proposition A.4

Let $1\le q<\infty$, $1\le p<r\le \infty$ and $\theta \in (0,1)$ such that

$$\begin{aligned} \frac{1}{r}+\frac{1}{d}-\frac{2\theta }{dq}=\frac{1}{p}\cdot \end{aligned}$$

(6.16)

Then, there exists C so that the following inequality holds true

$$\begin{aligned} \Vert {\nabla u}\Vert _{L_r({\mathbb {R}}^d)}\le C\Vert {\nabla ^2 u}\Vert ^{\theta }_{L_p({\mathbb {R}}^d)} \Vert {u}\Vert ^{1-\theta }_{\dot{B}^{2-2/q}_{p,\infty }({\mathbb {R}}^d)}\cdot \end{aligned}$$

Proof

Proposition A.3 tells us in particular that

$$\begin{aligned} \Vert u\Vert _{\dot{B}^{2-\frac{2\theta }{q}}_{p,1}}\lesssim \Vert u\Vert _{\dot{B}^2_{p,\infty }}^{1-\theta } \Vert u\Vert _{\dot{B}^{2-\frac{2}{q}}_{p,\infty }}^{\theta }. \end{aligned}$$

It is obvious that

$$\begin{aligned} \Vert u\Vert _{\dot{B}^2_{p,\infty }}\lesssim \Vert \nabla ^2 u\Vert _{L_p} \end{aligned}$$

and, according to Proposition A.2 and to the definition of $\theta ,$ we have

$$\begin{aligned} \dot{B}^{2-\frac{2\theta }{q}}_{p,1}({\mathbb {R}}^d)\hookrightarrow \dot{B}^{2+\frac{d}{r}-\frac{d}{p}-\frac{2\theta }{q}}_{r,1}({\mathbb {R}}^d) =\dot{B}^1_{r,1}({\mathbb {R}}^d). \end{aligned}$$

As $\dot{B}^1_{r,1}({\mathbb {R}}^d)\hookrightarrow \dot{W}^1_r({\mathbb {R}}^d),$ we get the desired inequality. $\square$

The following result that is an easy adaptation of [20, Prop. 2.1] played a key role in Sects. 3 and 4. Note that estimates in the same spirit but for source terms valued in Besov spaces have been proved by Kozono and Shimizu in [26].

Proposition A.5

Let $1<p,q< \infty$ and $1\le r\le \infty .$ Then, for any $u_{0}\in \dot{B}^{2-2/q}_{p,r}({\mathbb {R}}^{d})$ with $\textrm{div}\,u_0=0,$ and any $f\in L_{q,r}(0,T;L_{p}({\mathbb {R}}^{d})),$ the Stokes system (2.4) has a unique solution $(u,\nabla P)$ with $\nabla P\in L_{q,r}(0,T;L_p({\mathbb {R}}^d))$ and²u in the space $\dot{W}^{2,1}_{p,(q,r)}((0,T)\times {\mathbb {R}}^{d})$ defined by

$$\begin{aligned} \bigl \{u\in {\mathcal {C}}([0,T];\dot{B}^{2-2/q}_{p,r}( {\mathbb {R}}^{d})):u_{t}, \nabla ^{2}u\in L_{q,r}(0,T;L_{p}( {\mathbb {R}}^{d})) \bigr \}\cdot \end{aligned}$$

Furthermore, there exists a constant C independent of T such that

$$\begin{aligned}&\mu ^{1-1/q}\Vert {u}\Vert _{L_{\infty }(0,T;\dot{B}^{2-2/q}_{p,r}({\mathbb {R}}^{d}))} +\Vert {u_{t}, \mu \nabla ^{2}u,\nabla P}\Vert _{L_{q,r}(0,T;L_{p}({\mathbb {R}}^{d}))}\nonumber \\&\quad \le C\bigl (\mu ^{1-1/q}\Vert {u_{0}}\Vert _{\dot{B}^{2-2/q}_{p,r}({\mathbb {R}}^{d})} +\Vert {f}\Vert _{L_{q,r}(0,T;L_{p}({\mathbb {R}}^{d}))}\bigr )\cdot \end{aligned}$$

(A.1)

Let $\widetilde{s}>q$ be such that

$$\begin{aligned} \frac{1}{q}-\frac{1}{\widetilde{s}}\le \frac{1}{2}\quad \hbox {and}\quad \frac{d}{2p}+\frac{1}{q} -\frac{1}{\widetilde{s}}>\frac{1}{2}, \end{aligned}$$

and define $\widetilde{m}\ge p$ by the relation

$$\begin{aligned} \frac{d}{2\widetilde{m}}+\frac{1}{\widetilde{s}} =\frac{d}{2p}+\frac{1}{q}-\frac{1}{2}\cdot \end{aligned}$$

Then, the following inequality holds true:

$$\begin{aligned}&\mu ^{1+\frac{1}{\widetilde{s}}-\frac{1}{q}}\Vert {\nabla u}\Vert _{L_{\widetilde{s},r} (0,T;L_{\widetilde{m}}({\mathbb {R}}^{d}))}\nonumber \\&\quad \le C(\mu ^{1-1/q}\Vert {u}\Vert _{L_{\infty }(0,T;\dot{B}^{2-2/q}_{p,r}({\mathbb {R}}^{d}))} +\Vert {u_{t}, \mu \nabla ^{2}u}\Vert _{L_{q,r}(0,T;L_{p}({\mathbb {R}}^{d}))}). \end{aligned}$$

(A.2)

Finally, if $2/q+d/p>2,$ then for all $s\in (q,\infty )$ and $m\in (p,\infty )$ such that

$$\begin{aligned} \frac{d}{2m}+\frac{1}{s}=\frac{d}{2p}+\frac{1}{q}-1, \end{aligned}$$

it holds that

$$\begin{aligned}&\mu ^{1+\frac{1}{s}-\frac{1}{q}}\Vert {u}\Vert _{L_{s,r}(0,T;L_{m}({\mathbb {R}}^{d}))}\nonumber \\&\quad \le C\bigl (\mu ^{1-1/q}\Vert {u}\Vert _{L_{\infty }(0,T;\dot{B}^{2-2/q}_{p,r}({\mathbb {R}}^{d}))} +\Vert {u_{t}, \mu \nabla ^{2}u}\Vert _{L_{q,r}(0,T;L_{p}({\mathbb {R}}^{d}))}\bigr )\cdot \end{aligned}$$

(A.3)

Proof

Let ${\mathbb {P}}$ and ${\mathbb {Q}}$ be the Helmholtz projectors defined in (3.27). As $u={\mathbb {P}}u,$ we have

$$\begin{aligned} u_t-\mu \Delta u={\mathbb {P}}f,\qquad u|_{t=0}= u_0. \end{aligned}$$

Applying [20, Prop 2.1] and using that ${\mathbb {P}}$ is continuous on $L_{q,r}(0,T;L_{p}( {\mathbb {R}}^{d}))$ gives (A.2) and (A.4) for u. Since $\nabla P={\mathbb {Q}}f,$ and ${\mathbb {Q}}$ is also continuous on $L_{q,r}(0,T;L_{p}( {\mathbb {R}}^{d}))$, $\nabla P$ satisfies (A.2) too.

In order to prove (A.3), take $q_0$ and $q_1$ such that $1<q_{0}<q<q_{1}<\infty$ and $2/q=1/q_{0}+1/q_{1}.$ From the mixed derivative theorem we have for all $\gamma \in (0,1)$ and $i=0,1,$

$$\begin{aligned} \dot{W}^{2,1}_{p,q_{i}}((0,T)\times {\mathbb {R}}^{d}) := \dot{W}^{2,1}_{p,(q_{i},q_{i})}((0,T)\times {\mathbb {R}}^{d}) \hookrightarrow \dot{W}^{\gamma }_{q_{i}}(0,T;\dot{W}^{2-2\gamma }_{p}({\mathbb {R}}^{d})). \end{aligned}$$

Let $\gamma :=1/q-1/\widetilde{s}$ (so that $d/\widetilde{m}=d/p+2\gamma -1$). As $\gamma \in (0,\frac{1}{2}]$ and $1-2\gamma <d/p,$ one can use the Sobolev embedding

$$\begin{aligned} \dot{W}^{\gamma }_{q_{i}}(0,T;\dot{W}^{2-2\gamma }_{p}({\mathbb {R}}^{d})) \hookrightarrow L_{\widetilde{s}_{i}}(0,T;\dot{W}^{1}_{\widetilde{m}}({\mathbb {R}}^{d})) \quad \hbox {with}\quad \frac{1}{\widetilde{s}_{i}}=\frac{1}{q_{i}}-\gamma . \end{aligned}$$

(A.4)

In the proof of [20, Prop. 2.1], it is pointed out that

$$\begin{aligned} \dot{W}^{2,1}_{p,(q,r)}((0,T)\times {\mathbb {R}}^{d})=\bigl (\dot{W}^{2,1}_{p,q_0} ((0,T)\times {\mathbb {R}}^{d});\dot{W}^{2,1}_{p,q_1}((0,T)\times {\mathbb {R}}^{d})\bigr )_{\frac{1}{2},r}. \end{aligned}$$

Consequently, the embeddings (A.5) with $i=0$ and $i=1$ imply that

$$\begin{aligned} \dot{W}^{2,1}_{p,(q,r)}((0,T)\times {\mathbb {R}}^{d})\hookrightarrow \bigl ( L_{\widetilde{s}_{0}}(0,T;\dot{W}^{1}_{\widetilde{m}} ({\mathbb {R}}^{d})); L_{\widetilde{s}_{1}}(0,T;\dot{W}^{1}_{\widetilde{m}} ({\mathbb {R}}^{d}))\bigr )_{\frac{1}{2},r}. \end{aligned}$$

(A.5)

Note that our definition of $\gamma$, $\widetilde{s}_0$, $\widetilde{s}_1$, $q_0$ and $q_1$ ensures that

$$\begin{aligned} \frac{1}{2}\left( \frac{1}{\widetilde{s}_{0}}+\frac{1}{\widetilde{s}_{1}}\right) =\frac{1}{2}\left( \frac{1}{q_{0}}+\frac{1}{q_{1}}\right) -\gamma =\frac{1}{\widetilde{s}}\cdot \end{aligned}$$

Hence the real interpolation space in the right of (A.6) is nothing but $L_{q,r}(0,T;\dot{W}^1_{\widetilde{m}}({\mathbb {R}}^{d})),$ which completes the proof. $\square$

The usual product is continuous in many Besov spaces (see e.g. [9, 10, 27]). We here present a result that played a key role in the proof of uniqueness in dimension two. In order to prove it, we need to introduce the following so-called Bony decomposition (see [28]):

$$\begin{aligned} uv= T_u v+ T_v u+ R(u,v) \end{aligned}$$

with

$$\begin{aligned} T_u v\triangleq \sum _{j\ge 1} S_{j-1}u \Delta _j v \quad \hbox {and}\quad R(u,v)\triangleq \sum _{j\ge -1}\sum _{\left| {k-j}\right| \le 1} \Delta _j u \Delta _k v. \end{aligned}$$

Above, we used the notation $\Delta _j:=\dot{\Delta }_j$ for $j\ge 0$, $\Delta _{-1}:=\dot{S}_0$, $\Delta _j=0$ if $j\le -2$ and $S_j:=\sum _{j'\le j-1} \Delta _j.$

Operators T and R are called paraproduct and remainder, respectively. Their general properties of continuity may be found in [21, 25, 29]. The last inequality is new to the best of our knowledge.

Proposition A.6

Let $2\le p\le \infty$ and $1\le r_1,r_2\le \infty$ satisfy $\frac{1}{r_1}+\frac{1}{r_2}=1.$ Then, the following inequality holds true:

$$\begin{aligned} \Vert R(u,v)\Vert _{B^{-\frac{d}{p}}_{p,\infty }({\mathbb {R}}^d)} \lesssim \Vert u\Vert _{B^{\frac{d}{p}}_{p,r_1}({\mathbb {R}}^d)} \Vert v\Vert _{B^{-\frac{d}{p}}_{p,r_2}({\mathbb {R}}^d)}. \end{aligned}$$

(A.6)

In ${\mathbb {R}}^2$, it holds that

$$\begin{aligned} \Vert uv\Vert _{H^{-1}({\mathbb {R}}^2)}\lesssim \textrm{log}\,^{\frac{1}{2}} \Bigl (1+\frac{\Vert v\Vert _{L_2({\mathbb {R}}^2)}}{\Vert v\Vert _{H^{-1}({\mathbb {R}}^2)}}\Bigr ) \Vert u\Vert _{H^1({\mathbb {R}}^2)\cap L_\infty ({\mathbb {R}}^2)}\Vert v\Vert _{H^{-1}({\mathbb {R}}^2)}. \end{aligned}$$

(A.7)

Proof

To prove the first statement, we use that, by definition of the homogeneous remainder operator

$$\begin{aligned} R(u,v)=\sum _{j\ge -1}\widetilde{\Delta }_{j} u\Delta _j v\quad \hbox {with}\quad \widetilde{\Delta }_{j}\triangleq \Delta _{j-1}+\Delta _{j} +\Delta _{j+1}. \end{aligned}$$

Hence, owing to the support properties of the dyadic partition of unity, there exists an integer $N_0$ such that

$$\begin{aligned} \Delta _k R(u,v)=\sum _{j\ge k-N_0} \Delta _k (\widetilde{\Delta }_j u \Delta _j v) =\sum _{\nu \le N_0} \dot{\Delta }_k(\widetilde{\Delta }_{k-\nu } u\Delta _{k-\nu } v) \cdot \end{aligned}$$

(A.8)

As $2\le p\le \infty ,$ thanks to Bernstein’s inequality, we have

$$\begin{aligned} \Vert \Delta _k R(u,v)\Vert _{L_p({\mathbb {R}}^d)}\le 2^{k\frac{d}{p}}\Vert \Delta _k R(u,v) \Vert _{L_{p/2}({\mathbb {R}}^d)}\cdot \end{aligned}$$

Therefore, using convolution inequalities and (A.9), we discover that

$$\begin{aligned} 2^{-k\frac{d}{p}}\Vert \Delta _k R(u,v)\Vert _{L_p({\mathbb {R}}^d)}&\lesssim \sum _{\nu \le N_0}\Vert \widetilde{\Delta }_{k-\nu } u\Delta _{k-\nu } v\Vert _{L_{p/2}({\mathbb {R}}^d)}\\&\lesssim \sum _{\nu \le N_0} 2^{\frac{(k-\nu )d}{p}}\Vert \widetilde{\Delta }_{k-\nu } u\Vert _{L_p({\mathbb {R}}^d)} 2^{-\frac{(k-\nu )d}{p}} \Vert \Delta _{k-\nu } v \Vert _{L_p({\mathbb {R}}^d)} \end{aligned}$$

which gives (A.7).

In order to prove (A.8), we start from the following properties of continuity of the paraproduct operator (see e.g. [21, Chapter 2]):

$$\begin{aligned} \Vert T_{u}v\Vert _{H^{-1}({\mathbb {R}}^2)}&\lesssim \Vert u\Vert _{L_\infty ({\mathbb {R}}^2)}\Vert v\Vert _{H^{-1}({\mathbb {R}}^2)}, \end{aligned}$$

(A.9)

$$\begin{aligned} \Vert T_{v}u\Vert _{H^{-1}({\mathbb {R}}^2)}&\lesssim \Vert v\Vert _{H^{-1}({\mathbb {R}}^2)}\Vert u\Vert _{H^1({\mathbb {R}}^2)}\cdot \end{aligned}$$

(A.10)

Next, we decompose R(u, v) into low and high frequencies, using (A.7), to get for all $n\in {\mathbb {N}},$

$$\begin{aligned} \Vert R(u,v)\Vert ^2_{H^{-1}({\mathbb {R}}^2)}&=\sum _{j\ge -1} 2^{-2j} \Vert \Delta _j R(u,v)\Vert ^2_{L_2({\mathbb {R}}^2)}\\&=\sum _{-1\le j\le N} 2^{-2j}\Vert \Delta _j R(u,v) \Vert ^2_{L_2({\mathbb {R}}^2)}+\sum _{j>N} 2^{-2j}\Vert \Delta _j R(u,v)\Vert ^2_{L_2({\mathbb {R}}^2)}\\&\lesssim N \Vert R(u,v)\Vert ^2_{B^{-1}_{2,\infty }({\mathbb {R}}^2)}+2^{-2N}\Vert R(u,v) \Vert ^2_{B^0_{2,\infty }({\mathbb {R}}^2)}\\&\lesssim N \Vert u\Vert ^2_{H^1({\mathbb {R}}^2)}\Vert v\Vert ^2_{H^{-1}({\mathbb {R}}^2)}+2^{-2N} \Vert u\Vert ^2_{H^1({\mathbb {R}}^2)}\Vert v\Vert ^2_{B^{-1}_{\infty ,2}({\mathbb {R}}^2)}\cdot \end{aligned}$$

Then, choosing N to be the closest integer larger than $\textrm{log}\,_2 \Bigl (1\!+\!\frac{\Vert v\Vert _{B^{-1}_{\infty ,2}}}{\Vert v\Vert _{H^{-1}}}\Bigr )$ leads to

$$\begin{aligned} \Vert R(u,v)\Vert _{H^{-1}({\mathbb {R}}^2)}\lesssim \textrm{log}\,^{\frac{1}{2}} \biggl (1+\frac{\Vert v\Vert _{B^{-1}_{\infty ,2}({\mathbb {R}}^2)}}{\Vert v\Vert _{H^{-1}({\mathbb {R}}^2)}}\biggr ) \Vert u\Vert _{H^1({\mathbb {R}}^2)}\Vert v\Vert _{H^{-1}({\mathbb {R}}^2)}\cdot \end{aligned}$$

Combining with the embedding $B^0_{2,2}({\mathbb {R}}^2)\hookrightarrow B^{-1}_{\infty ,2}({\mathbb {R}}^2)$, and Inequalities (A.10), (A.11), this completes the proof of (A.8). $\square$