Striated Regularity of 2-D Inhomogeneous Incompressible Navier–Stokes System with Variable Viscosity

Abstract

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier–Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier–Stokes equations with constant viscosity, the main difficulty here lies in the fact that the $L^1$ in time Lipschitz estimate of the velocity field can not be obtained by energy method (see Danchin and Mucha in The incompressible Navier–Stokes equations in vacuum; Liao and Zhang in Arch Ration Mech Anal 220:937–981, 2016; Commun Pure Appl Math 72:835–884, 2019 for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid (Chemin in Invent Math 103:599–629, 1991; Ann Sci École Norm Sup 26(4):517–542, 1993), namely, striated regularity can help to get the $L^\infty$ boundedness of the double Riesz transform, we derive the a priori$L^1$ in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space. As an application, we shall prove the propagation of $H^{\frac{5}{2}}$ regularity of the interface between fluids with different densities and viscosities.

Appendices

Appendix A. The Commutative Estimate

Let f be a locally integrable function and $\mathrm{M}f(x)$ be its maximal function given by Definition 5.2. We first recall the following lemma from [17].

Lemma A.1

Let $p, q\in ]1,\infty [$ or $p=q=\infty .$ Let $\{f_j\}_{j\in {{{\mathbb {Z}}}}}$ be a sequence of functions in $L^p({{\mathbb {R}}}^d)$ so that $\left( f_j(x)\right) _{\ell ^q({{{\mathbb {Z}}}})}\in L^p({{\mathbb {R}}}^d).$ Then there holds

$$\begin{aligned} \bigl \Vert \left( M(f_j)(x)\right) _{\ell ^q}\bigr \Vert _{L^p}\le C\bigl \Vert \left( f_j(x)\right) _{\ell ^q}\bigr \Vert _{L^p}. \end{aligned}$$

Proposition A.1

Let $p, r\in ]1,\infty [$ and $q\in ]1, \infty ]$ satisfying $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}.$ Let $X=(X^1,\cdots , X^d)\in \dot{W}^{1,p}({{\mathbb {R}}}^d)$ with ${\mathrm{div}}X=0,$$g\in L^q({{\mathbb {R}}}^d),$$R_i=\partial _i(-\Delta )^{-\frac{1}{2}}$ be the Riesz transform. Then one has

$$\begin{aligned} \bigl \Vert [\partial _X; R_iR_j]g\bigr \Vert _{L^r}\le C\Vert \nabla X\Vert _{L^p}\Vert g\Vert _{L^q}. \end{aligned}$$

(A.1)

Proof

We first get by, applying Bony’s decomposition (5.2) that

$$\begin{aligned} \begin{aligned}&[\partial _X; R_iR_j]g=[T_{X^k}; R_iR_j]\partial _kg+T_{\partial _kR_iR_jg}X^k-R_iR_j(T_{\partial _kg}X^k)\\&\quad +R(X^k,\partial _kR_iR_jg)-R_iR_j(R(X^k,\partial _kg)). \end{aligned} \end{aligned}$$

(A.2)

In view of (2.27) of [5], one has

$$\begin{aligned} |\Delta _\ell (\partial _kR_iR_jg)(x)|\le C2^\ell \mathrm{M}g(x)\Longrightarrow |S_\ell (\partial _kR_iR_jg)(x)|\le C2^\ell \mathrm{M}g(x), \end{aligned}$$

(A.3)

from which, we infer

$$\begin{aligned} \begin{aligned} \bigl |\Delta _\ell \bigl (T_{\partial _kR_iR_jg}X^k\bigr )(x)\bigr |&\le C\sum _{|\ell '-\ell |\le 4}\mathrm{M}\big (S_{\ell '-1}(\partial _kR_iR_jg)\Delta _{\ell '}X^k\bigr )(x)\\&\le C\sum _{|\ell '-\ell |\le 4}\mathrm{M}\big (\mathrm{M}g 2^{\ell '}\Delta _{\ell '}X^k\bigr )(x). \end{aligned} \end{aligned}$$

As a result, we deduce from Lemma A.1 that

$$\begin{aligned} \begin{aligned} \bigl \Vert T_{\partial _kR_iR_jg}X^k\bigr \Vert _{L^r}&\le C\Bigl \Vert \left\{ \sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [\mathrm{M}\big (\mathrm{M}g 2^{\ell }\Delta _{\ell }X^k\bigr )\bigr ]^2\right\} ^{\frac{1}{2}}\Bigr \Vert _{L^r}\\&\le C\left\| \mathrm{M}g\left\{ \sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [ 2^{\ell }\Delta _{\ell }X^k\bigr ]^2\right\} ^{\frac{1}{2}}\right\| _{L^r}\\&\le C\Vert \mathrm{M}g\Vert _{L^q}\left\| \left\{ \sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [ 2^{\ell }\Delta _{\ell }X^k\bigr ]^2\right\} ^{\frac{1}{2}}\right\| _{L^p}\le C\Vert g\Vert _{L^q}\Vert \nabla X\Vert _{L^p}. \end{aligned} \end{aligned}$$

The same estimate holds for $T_{\partial _kg}X^k,$ so that we obtain

$$\begin{aligned} \Vert R_iR_j(T_{\partial _kg}X^k)\Vert _{L^r}\le C\Vert T_{\partial _kg}X^k\Vert _{L^r}\le C\Vert g\Vert _{L^q}\Vert \nabla X\Vert _{L^p}. \end{aligned}$$

While due to ${\mathrm{div}}X=0$ and (A.3), we write

$$\begin{aligned} \begin{aligned} |\Delta _\ell R(X^k,\partial _kR_iR_jg)(x)|&=|\partial _k\Delta _\ell R(X^k, R_iR_jg)(x)|\\&\le C2^\ell \mathrm{M}\left( \sum _{\ell '\ge \ell -5}\Delta _{\ell '}X^k\widetilde{\Delta }_{\ell '}R_iR_jg\right) (x)\\&\le C\mathrm{M}\left( \sum _{\ell '\ge \ell -5}2^{\ell -\ell '}|2^{\ell '}\Delta _{\ell '}X^k| \mathrm{M}g\right) (x), \end{aligned} \end{aligned}$$

from which, we infer

$$\begin{aligned} \begin{aligned} \bigl \Vert R(X^k,\partial _kR_iR_jg)\bigr \Vert _{L^r}&\le \Bigl \Vert \Bigl \{\sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [\sum _{\ell '\ge \ell -5}2^{\ell -\ell '}|2^{\ell '}\Delta _{\ell '}X^k| \mathrm{M}g\bigr ]^2\Bigr \}^{\frac{1}{2}}\Bigr \Vert _{L^r}\\&\le C\Bigl \Vert \mathrm{M}g\Bigl \{\sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [ 2^{\ell }\Delta _{\ell }X^k\bigr ]^2\Bigr \}^{\frac{1}{2}}\Bigr \Vert _{L^r}\le C\Vert g\Vert _{L^q}\Vert \nabla X\Vert _{L^p}. \end{aligned} \end{aligned}$$

The same estimate holds for $R_iR_j(R(X^k,\partial _kg)).$

Finally let us turn to the first term on the right hand side of (A.2). We first get, by a similar derivation of (5.6), that

$$\begin{aligned}{}[T_{X^k}; R_iR_j]\partial _kg= \sum _{|\ell -\ell '|\le 4}2^{2\ell '}\int _{{{\mathbb {R}}}^2}{\check{\theta }}(2^{\ell '}z)\int _0^1\Delta _\ell \nabla X(x-\tau z)\cdot z\,d\tau S_\ell \partial _kg(x-z)\,dz, \end{aligned}$$

where $\theta (\xi ) \xi _i\xi _j|\xi |^{-2}\phi (\xi ).$ Then we deduce from (A.3) that

$$\begin{aligned} |[T_{X^k}; R_iR_j]\partial _kg|\le \sum _{|\ell -\ell '|\le 4}2^{\ell '}\int _{{{\mathbb {R}}}^2}\Psi (2^{\ell '}z)\int _0^1|\Delta _\ell \nabla X(x-\tau z)|\,d\tau 2^\ell \mathrm{M}g(x-z)\,dz, \end{aligned}$$

for $\Psi (z) |z||{\check{\theta }}(z)|.$ Now since $r\in ]1,\infty [$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}<1,$ we can choose $\alpha ,\beta \in ]0.1[$ satisfying

$$\begin{aligned} \alpha +\beta =1,\quad \alpha>\frac{1}{2},\quad p\alpha>1\quad \hbox {and}\quad q\beta >1. \end{aligned}$$

(A.4)

We get, by applying Hölder’s inequality, that

$$\begin{aligned} \begin{aligned} |[T_{X^k}; R_iR_j]\partial _kg(x)|&\le \sum _{|\ell -\ell '|\le 4} \left( \int _0^12^{2\ell '}\int _{{{\mathbb {R}}}^2}\Psi (2^{\ell '} z)|\Delta _\ell \nabla X(x-\tau z)|^{\frac{1}{\alpha }}\,dz\,d\tau \right) ^\alpha \\&\quad \times \left( 2^{2\ell '}\int _{{{\mathbb {R}}}^2}\Psi (2^{\ell '} z)\left[ \mathrm{M}g(x-z)\right] ^{\frac{1}{\beta }}\,dz\right) ^\beta \\&\le \sum _{\ell \in {{{\mathbb {Z}}}}}\left[ \mathrm{M}(|\Delta _\ell \nabla X|^{\frac{1}{\alpha }})\right] ^\alpha \left[ \mathrm{M}((\mathrm{M}g)^{\frac{1}{\beta }})\right] ^\beta , \end{aligned} \end{aligned}$$

from which, (A.4) and Lemma A.1, we deduce that

$$\begin{aligned} \begin{aligned} \Vert [T_{X^k}; R_iR_j]\partial _kg\Vert _{L^r}&\le \Bigl \Vert \Bigl \{\sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [\mathrm{M}(|\Delta _\ell \nabla X|^{\frac{1}{\alpha }})\bigr ]^{2\alpha }\Bigr \}^{\frac{1}{2\alpha }}\Bigr \Vert _{L^{p\alpha }}^\alpha \bigl \Vert \mathrm{M}(\mathrm{M}g)^{\frac{1}{\beta }})\bigr \Vert _{L^{q\beta }}^\beta \\&\le C\Bigl \Vert \Bigl \{\sum _{\ell \in {{{\mathbb {Z}}}}}\bigl [ 2^{\ell }\Delta _{\ell }X^k\bigr ]^2\Bigr \}^{\frac{1}{2}}\Bigr \Vert _{L^p}\bigl \Vert \mathrm{M}g^{\frac{1}{\beta }}\bigr \Vert _{L^{q\beta }}^\beta \\&\le C\Vert \nabla X\Vert _{L^p}\Vert g\Vert _{L^q}. \end{aligned} \end{aligned}$$

By summing up the above estimate, we conclude the proof of (A.1). $\quad \square$

Appendix B. Lipschitz Estimate of Elliptic Equation of Divergence Form

The goal of this appendix is to generalize Proposition 2.4 to elliptic equation of divergence form with bounded coefficients which may have a small gap across a surface. The main result reads

Proposition B.1

Let $p\in ]2,\infty [,$$X=\bigl (X_\lambda \bigr )_{\lambda \in \Lambda }$ be a non-degenerate family of vector fields in the sense of Definition 1.1 with $X_\lambda \in C^1_b({{\mathbb {R}}}^2)$ and $\nabla X_\lambda \in L^p({{\mathbb {R}}}^2)$ for each $\lambda \in \Lambda .$ Let $a_{ij}\in L^\infty ({{\mathbb {R}}}^2)$ with $\sup _{\lambda \in \Lambda }\bigl \Vert \partial _{X_\lambda } a_{i,j}\bigr \Vert _{L^\infty }\le C$ and $\Vert (a_{ij})_{2\times 2}-Id\Vert _{L^\infty }\le \varepsilon _0$ for some $\varepsilon _0$ sufficiently small. We assume moreover that $f\in L^p\cap L^{\frac{2p}{2+p}}({{\mathbb {R}}}^2)$ and $\partial _{X_\lambda }f\in L^{\frac{2p}{2+p}}({{\mathbb {R}}}^2)$ for $\lambda \in \Lambda .$ Then the following equation

$$\begin{aligned} \sum _{i,j=1}^2\partial _i(a_{ij}\partial _j u)=f \quad \text{ for }\quad x\in {{\mathbb {R}}}^2, \end{aligned}$$

(B.1)

has a unique solution $u\in \dot{W}^{1,p}({{\mathbb {R}}}^2)\cap \dot{W}^{1,\infty }({{\mathbb {R}}}^2)$ so that for any $s\in ]2/p,1[,$

$$\begin{aligned} \begin{aligned} \Vert \nabla u\Vert _{L^\infty }&\le C_s\Bigl (\varepsilon _0\Bigl [1+C(s,p,X)\sup _{\lambda \in \Lambda }\Vert X_\lambda \Vert _{L^\infty }\bigl (\Vert \nabla X_\lambda \Vert _{L^p}^{\frac{2}{p-2}}+\Vert \nabla X_\lambda \Vert _{L^p}^{\frac{2}{ps-2}}\bigr ) \Bigr ]\Vert f\Vert _{L^\frac{2p}{2+p}}\\&\quad +\Vert f\Vert _{ L^{\frac{2p}{2+p}}}^{1-\frac{2}{p}}\Vert f\Vert _{L^p}^{\frac{2}{p}}\Bigr )+\frac{C}{I(X)}\sup _{\lambda \in \Lambda }\Bigl (\varepsilon _0\Vert X_\lambda \Vert _{L^\infty }\Vert f\Vert _{L^\frac{2p}{2+p}}\Bigr )^{1-\frac{2}{p}}\\&\quad \times \Bigl (\bigl [1+\varepsilon _0\Vert \nabla X_\lambda \Vert _{L^\infty }\bigr ]\Vert f\Vert _{L^\frac{2p}{2+p}}+\varepsilon _0\bigl \Vert \partial _{X_\lambda }f\bigr \Vert _{ L^{\frac{2p}{2+p}}}\Bigr )^{\frac{2}{p}}, \end{aligned} \end{aligned}$$

(B.2)

for C(s, p, X) given by (2.20).

Proof

The proof of this proposition consists in the estimate of the striated regularity of the solution (B.1) and then applying Proposition 2.4. Let us denote

$$\begin{aligned} \mathfrak {G} \left( (a_{ij})_{2\times 2}-\mathrm{Id}\right) \nabla u, \end{aligned}$$

For simplicity, we just present the a priori estimate for smooth enough solutions of (B.1). We first write

$$\begin{aligned} \nabla u=\sum _{i,j=1}^2\nabla (-\Delta )^{-1}\partial _i((a_{ij}-\delta _{ij})\partial _j u)+\nabla \Delta ^{-1}f, \end{aligned}$$

(B.3)

from which, we infer

$$\begin{aligned} \begin{aligned} \Vert \nabla u\Vert _{L^p}&\le C\bigl (\bigl \Vert \mathfrak {G}\Vert _{L^p}+\Vert f\Vert _{L^\frac{2p}{2+p}}\bigr ) \le C\bigl (\varepsilon _0\Vert \nabla u\Vert _{L^p}+\Vert f\Vert _{L^\frac{2p}{2+p}}\bigr ). \end{aligned} \end{aligned}$$

So that by taking $\varepsilon _0$ sufficiently small, we obtain

$$\begin{aligned} \Vert \nabla u\Vert _{L^p}\le C\Vert f\Vert _{L^{\frac{2p}{2+p}}}. \end{aligned}$$

(B.4)

Whereas for any $C^1$ vector field X,  we get, by applying $\partial _X$ to (B.1), that

$$\begin{aligned} \sum _{i,j=1}^2\partial _i(a_{ij}\partial _j \partial _X u)=\partial _X f-\sum _{i,j=1}^2\partial _i(\partial _Xa_{ij}\partial _j u)-\sum _{i,j=1}^2\partial _i(a_{ij}\partial _j X\partial _j u). \end{aligned}$$

Then along the same line to proof of (B.4), we deduce

$$\begin{aligned} \begin{aligned} \Vert \nabla \partial _X u\Vert _{L^p}&\le C\Bigl (\Vert \partial _Xf\Vert _{L^{\frac{2p}{2+p}}}+\bigl (\Vert \partial _Xa_{ij}\Vert _{L^\infty }+\Vert \nabla X\Vert _{L^\infty }\bigr )\Vert \nabla u\Vert _{L^p}\Bigr )\\&\le C\Bigl (\Vert \partial _Xf\Vert _{L^{\frac{2p}{2+p}}}+\bigl (\Vert \partial _Xa_{ij}\Vert _{L^\infty }+\Vert \nabla X\Vert _{L^\infty }\bigr )\Vert f\Vert _{L^{\frac{2p}{2+p}}}\Bigr ). \end{aligned} \end{aligned}$$

(B.5)

On the other hand, for any $s\in ]2/p,1[,$ we deduce from Proposition 2.4 and (B.3) that

$$\begin{aligned} \begin{aligned}&\Vert \nabla u\Vert _{L^\infty }\le C\bigl (\bigl \Vert \mathfrak {G}\bigr \Vert _{L^p}+\Vert \mathfrak {G}\Vert _{L^\infty }\bigr )+\Vert \nabla \Delta ^{-1}f\Vert _{L^\infty }+\frac{C_s}{I(X)}\sup _{\lambda \in \Lambda }\Bigl \{\bigl (\Vert X_\lambda \Vert _{L^\infty }\Vert \mathfrak {G}\Vert _{L^p}\bigr )^{1-\frac{2}{p}}\\&\quad \times \bigl (\Vert \nabla X_\lambda \Vert _{L^p}\Vert \mathfrak {G}\Vert _{L^\infty }+\Vert \partial _{X_\lambda }\mathfrak {G}\Vert _{L^p}\bigr )^{\frac{2}{p}}+\bigl (\Vert X_\lambda \Vert _{L^\infty }\Vert \mathfrak {G}\Vert _{L^p}\bigr )^{1-\frac{2}{ps}}\bigl (\Vert \nabla X_\lambda \Vert _{L^p}\Vert \mathfrak {G}\Vert _{L^\infty }\bigr )^{\frac{2}{ps}}\Bigr \}. \end{aligned} \end{aligned}$$

Due to $p\in ]2,\infty [,$ we have

Applying Young’s inequality yields

As a result, it comes out

By taking $\varepsilon _0$ sufficiently small and inserting the Estimates (B.4) and (B.5) to the above inequality, we achieve (B.2). $\quad \square$

Remark B.1

By repeating the proof of Proposition B.1, we can prove the same Lipschitz estimate (B.2) for the solutions of the following Stokes type system:

$$\begin{aligned} \sum _{i,j=1}^2\partial _i\left( a_{ij}\partial _j u\right) =f-\nabla p\quad \hbox {and}\quad {\mathrm{div}}u=0 \quad \text{ for }\quad x\in {{\mathbb {R}}}^2. \end{aligned}$$

We can even work for the above problems in the multi-dimensional case.