The Vanishing Surface Tension Limit of the Muskat Problem

Abstract

The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where the latter is due to surface tension. To leading order, both the Muskat problems with and without surface tension effect are scaling invariant in the Sobolev space $H^{1+\frac{d}{2}}({\mathbb {R}}^d)$, where d is the dimension of the interface. We prove that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with surface tension ${\mathfrak {s}}$ converge to the unique solution of the Muskat problem without surface tension locally in time with the rate $\sqrt{\mathfrak {s}}$ when ${\mathfrak {s}}\rightarrow 0$. This allows for initial interfaces that have unbounded or even not locally square integrable curvature. If in addition the initial curvature is square integrable, we obtain the convergence with optimal rate ${\mathfrak {s}}$.

Appendix A. Paradifferential Calculus

In this appendix, we recall the symbolic calculus of Bony’s paradifferential calculus. See [11, 42].

Definition A.1

1. (Paradifferential symbols) Given $\rho \in [0, \infty )$ and $m\in {\mathbb {R}}$, $\Gamma _{\rho }^{m}({\mathbb {R}}^d)$ denotes the space of locally bounded functions $a(x,\xi )$ on ${\mathbb {R}}^d\times ({\mathbb {R}}^d{\setminus } 0)$, which are $C^\infty$ with respect to $\xi$ for $\xi \ne 0$ and such that, for all $\alpha \in {\mathbb {N}}^d$ and all $\xi \ne 0$, the function $x\mapsto \partial _\xi ^\alpha a(x,\xi )$ belongs to $W^{\rho ,\infty }({\mathbb {R}}^d)$ and there exists a constant $C_\alpha$ such that,

$$\begin{aligned} \forall |\xi |\ge \frac{1}{2},\quad \Vert \partial _\xi ^\alpha a(\cdot ,\xi )\Vert _{W^{\rho ,\infty }({\mathbb {R}}^d)}\le C_\alpha (1+|\xi |)^{m-|\alpha |}. \end{aligned}$$

Let $a\in \Gamma _{\rho }^{m}({\mathbb {R}}^d)$, we define the semi-norm

$$\begin{aligned} M_{\rho }^{m}(a)= \sup _{|\alpha |\le 2(d+2) +\rho ~}\sup _{|\xi | \ge \frac{1}{2}~} \Vert (1+|\xi |)^{|\alpha |-m}\partial _\xi ^\alpha a(\cdot ,\xi )\Vert _{W^{\rho ,\infty }({\mathbb {R}}^d)}. \end{aligned}$$

(A.1)

2. (Paradifferential operators) Given a symbol a, we define the paradifferential operator $T_a$ by

$$\begin{aligned} \widehat{T_a u}(\xi )=(2\pi )^{-d}\int \chi (\xi -\eta ,\eta ){\widehat{a}}(\xi -\eta ,\eta ) \Psi (\eta ){\widehat{u}}(\eta ) \, d\eta , \end{aligned}$$

(A.2)

where ${\widehat{a}}(\theta ,\xi )=\int e^{-ix\cdot \theta }a(x,\xi )\, dx$ is the Fourier transform of a with respect to the first variable; $\chi$ and $\Psi$ are two fixed $C^\infty$ functions such that:

$$\begin{aligned} \Psi (\eta )=0\quad \text {for } |\eta |\le \frac{1}{5},\qquad \Psi (\eta )=1\quad \text {for }|\eta |\ge \frac{1}{4}, \end{aligned}$$

(A.3)

and $\chi (\theta ,\eta )$ satisfies, for $0<\varepsilon _1<\varepsilon _2$ small enough,

$$\begin{aligned} \chi (\theta ,\eta )=1 \quad \text {if}\quad |\theta |\le \varepsilon _1| \eta |,\qquad \chi (\theta ,\eta )=0 \quad \text {if}\quad |\theta |\ge \varepsilon _2|\eta |, \end{aligned}$$

and such that

$$\begin{aligned} \forall (\theta ,\eta ), \qquad | \partial _\theta ^\alpha \partial _\eta ^\beta \chi (\theta ,\eta )|\le C_{\alpha ,\beta }(1+| \eta |)^{-|\alpha |-|\beta |}. \end{aligned}$$

Theorem A.2

For all $m\in {\mathbb {R}}$, if $a\in \Gamma ^m_0$ then

$$\begin{aligned} T_a=O_{Op^m}\big (M^m_0(a)\big ). \end{aligned}$$

(A.4)

Theorem A.3

(Symbolic calculus). Let $a \in \Gamma _r^{m}, a'\in \Gamma _r^{m'}$ and set $\delta = \min \{1,r\}$. Then,

(i)

$$\begin{aligned} T_{a}T_{a'} = T_{aa'} + O_{Op^{m+m'-\delta }}\Big ( M_r^{m}(a)M_0^{m'}(a')+M_0^{m_1}(a)M_r^{m'}(a')\Big ); \end{aligned}$$

(A.5)

(ii)

$$\begin{aligned} T_{a}^* = T_{{\overline{a}}}+ O_{Op^{m-\delta }}\big (M_r^m(a)\big ). \end{aligned}$$

(A.6)

Remark A.4

In the definition (A.2) of paradifferential operators, the cut-off $\Psi$ removes the low frequency part of u. In particular, if $a\in \Gamma ^m_0$ then

$$\begin{aligned} \Vert T_a u\Vert _{H^\sigma }\le CM_0^m(a)\Vert \nabla u\Vert _{H^{\sigma +m-1}}=CM_0^m(a)\Vert u\Vert _{H^{\sigma +m, 1}}, \end{aligned}$$

and similarly for other estimates involving paradifferential operators.

Proposition A.5

(Gåarding’s inequality). Assume $a\in \Gamma ^m_r$ with $m\in {\mathbb {R}}$ and $r\in (0, 1]$ such that for some $c>0$

$$\begin{aligned} \inf _{(x,\xi )\in {\mathbb {R}}^d\times ({\mathbb {R}}^d{\setminus }\{0\})} \mathrm {Re}(a(x,\xi )) \ge c|\xi |^m. \end{aligned}$$

(A.7)

Then, for all $\sigma \in {\mathbb {R}}$, there exists ${\mathcal {F}}:{\mathbb {R}}^+\times {\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ nondecreasing such that

$$\begin{aligned} \Vert \Psi (D)u\Vert _{H^\frac{m}{2}}^2\le {\mathcal {F}}(M_r^m(a),c^{-1})\Big (\mathrm {Re}(T_a u,u)_{L^2} + \Vert u\Vert _{H^{1, \frac{m-r}{2}}}^2\Big ) \end{aligned}$$

(A.8)

and

$$\begin{aligned} \Vert \Psi (D)u\Vert _{H^\frac{m}{2}}^2\le {\mathcal {F}}(M_r^m(a),c^{-1})\Big (\mathrm {Re}(T_a u,u)_{L^2} + \Vert u\Vert _{H^{\frac{m}{2}}}\Vert u\Vert _{H^{1, \frac{m}{2}-r}}\Big ) \end{aligned}$$

(A.9)

provided that both sides are finite. Here, $\Psi (D)$ is the Fourier multiplier with symbol $\Psi$ given by (A.3).

Proof

We have

$$\begin{aligned} \mathrm {Re}(T_a u,u)_{L^2}&=\frac{1}{2}\big ((T_a u,u)_{L^2}+(T^*_a u,u)_{L^2}\Big )\\&=(T_{\mathrm {Re}(a)} u,u)_{L^2}+\frac{1}{2}((T_a^*-T_{{\overline{a}}})u,u)_{L^2}). \end{aligned}$$

According to Theorem A.3 (ii), $T_a^*-T_{{\overline{a}}}$ is of order $m-r$ and

$$\begin{aligned} \Vert ((T_a^*-T_{{\overline{a}}})u,u)_{L^2})\Vert \le \Vert (T_a^*-T_{{\overline{a}}})u\Vert _{H^{\frac{m-r}{2}}}\Vert u\Vert _{H^{\frac{m-r}{2}}}\le CM^m_r(a) \Vert u\Vert ^2_{H^{\frac{m-r}{2}}}. \end{aligned}$$

Set $b=(\mathrm {Re}(a))^\frac{1}{2}$. By virtue of (A.7) we have $b\in \Gamma ^{\frac{m}{2}}_r$ and $M^{\frac{m}{2}}_r(b)\le {\mathcal {F}}(M^m_r(a))$. We write

$$\begin{aligned} (T_{\mathrm {Re}(a)} u,u)_{L^2}&=(T_bT_bu, u)_{L^2}+((T_{b^2}-T_bT_b)u, u)_{L^2}\\&=(T_bu, T_b^*u)_{L^2}+((T_{b^2}-T_bT_b)u, u)_{L^2}\\&=\Vert T_bu\Vert _{L^2}^2+(T_bu, (T_b^*-T_b)u)_{L^2}+((T_{b^2}-T_bT_b)u, u)_{L^2}. \end{aligned}$$

Applying Theorem A.3 (ii) once again we deduce that $T_b^*-T_b$ is of order $\frac{m}{2}-r$ and

$$\begin{aligned} \big |(T_bu, (T_b^*-T_b)u)_{L^2}\big |\le \Vert T_bu\Vert _{H^{-\frac{r}{2}}} \Vert (T_b^*-T_b)u)_{L^2}\Vert _{H^\frac{r}{2}}\le {\mathcal {F}}(M^m_r(a), c^{-1}) \Vert u\Vert ^2_{H^{1, \frac{m-r}{2}}}, \end{aligned}$$

where we used Remark A.4 in the last inequality. On the other hand, an application of Theorem A.3 (i) yields

$$\begin{aligned} \big |((T_{b^2}-T_bT_b)u, u)_{L^2}\big |\le {\mathcal {F}}(M^m_r(a), c^{-1}) \Vert u\Vert ^2_{H^{1, \frac{m-r}{2}}}. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \Vert T_bu\Vert _{L^2}^2\le (T_{\mathrm {Re}(a)} u,u)_{L^2}+{\mathcal {F}}(M^m_r(a), c^{-1})\Vert u\Vert ^2_{H^{1, \frac{m-r}{2}}}. \end{aligned}$$

(A.10)

By shifting derivative differently in the above inner products, we have the variant

$$\begin{aligned} \Vert T_bu\Vert _{L^2}^2\le (T_{\mathrm {Re}(a)} u,u)_{L^2}+{\mathcal {F}}(M^m_r(a), c^{-1}) \Vert u\Vert _{H^{1, \frac{m}{2}}}\Vert u\Vert _{H^{1, \frac{m}{2}-r}}. \end{aligned}$$

(A.11)

Next we note that $T_{b^{-1}}T_b-\Psi (D)=T_{b^{-1}}T_b-T_1$ is of order $-r$ and

$$\begin{aligned} \Vert \Psi (D)u\Vert _{H^\frac{m}{2}}= & {} \Vert T_{b^{-1}} T_bu\Vert _{H^\frac{m}{2}}+\Vert (T_{b^{-1}}T_b-\Psi (D))u\Vert _{H^\frac{m}{2}}\nonumber \\&\le {\mathcal {F}}(M^m_r(a), c^{-1})\Vert T_bu\Vert _{L^2}+ {\mathcal {F}}(M^m_r(a), c^{-1})\Vert u\Vert _{H^{1, \frac{m}{2}-r}}. \end{aligned}$$

(A.12)

Finally, a combination of (A.10) and (A.12) leads to (A.8), and a combination of (A.11) and (A.12) leads to (A.9). $\quad \square$

The proof of (A.12) also proves the following lemma.

Lemma A.6

Let $a\in \Gamma ^m_r$, $r\in (0, 1]$, be a real symbol satisfying $a(x, \xi )\ge c|\xi |^m$ for all $(x, \xi )\in {\mathbb {R}}^d\times {\mathbb {R}}^d$. Then for all $s\in {\mathbb {R}}$ we have

$$\begin{aligned} \Vert \Psi (D)u\Vert _{H^{s}}\le {\mathcal {F}}(M^m_r(a), c^{-1})\Vert T_au\Vert _{H^{s-m}}+ {\mathcal {F}}(M^m_r(a), c^{-1})\Vert u\Vert _{H^{1, s-r}}. \end{aligned}$$

(A.13)