On the Free Surface Motion of Highly Subsonic Heat-Conducting Inviscid Flows

Abstract

For the free surface problem of highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables, are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations, for which a geometric approach was introduced by Christodoulou and Lindblad [11]. Due to the loss of the one more derivative and loss of the symmetry of equations which create significant difficulties in closing the estimates in Sobolev spaces of finite regularity, the geometric approach in [11] needs to be substantially developed and extended by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

Appendix: Sobolev Lemmas, Interpolation Inequalities and Elliptic Estiamtes

We list here Sobolev lemmas and interpolation inequalities with the dependence of the Sobolev constants on the lower bound of $1/\iota _1$; and the elliptic estimates for the boundary problem. Before that, recall the following notations for easier reference. The $L^p$-norms of a (0, r)-tensor $\alpha$ on $\Omega$ and $\partial \Omega$ are denoted, respectively, by $\Vert \alpha \Vert _{L^p}$ and $| \alpha | _{L^p}$:

$$\begin{aligned} \Vert \alpha \Vert _{L^p}= \left( \int _\Omega |\alpha |^p {\mathrm{d}}\mu _g\right) ^{1/p} \ \ \mathrm{for} \ \ 1\le p < \infty , \ \ \Vert \alpha \Vert _{L^\infty }={ \mathrm{ess} \ \mathrm{sup} }_\Omega |\alpha | \end{aligned}$$

and

$$\begin{aligned} | \alpha |_{L^p}= \left( \int _{\partial \Omega } |\alpha |^p {\mathrm{d}}\mu _\zeta \right) ^{1/p} \ \ \mathrm{for} \ \ 1\le p < \infty , \ \ | \alpha |_{L^\infty }={ \mathrm{ess} \ \mathrm{sup} }_{\partial \Omega } |\alpha | . \end{aligned}$$

Lemma A-1

(Lemmas A.1-A.4 in [11]) Let $\alpha$ be a (0, r) tensor and $\iota _1\ge 1/K_1$. Assume k, m are positive integers, and $p\ge 1$. We have

• (i) if $2\le p\le s\le q\le \infty$ and $m/s=k/p+(m-k)/q$,

  $$\begin{aligned}&| \overline{\nabla }^k \alpha |_{L^s}^{m} \le C(k,m,n,s) | \alpha |_{L^q}^{m-k} | \overline{\nabla }^m \alpha |_{L^p}^{k}, \end{aligned}$$

  (A-1a)
  $$\begin{aligned}&\left( \sum _{i=0}^k \Vert \nabla ^i \alpha \Vert _{L^s} \right) ^{m} \le C(k,m,n,s) \Vert \alpha \Vert _{L^q}^{m-k} \left( \sum _{i=0}^m K_1^{m-i} \Vert \nabla ^i \alpha \Vert _{L^p} \right) ^{k}; \end{aligned}$$

  (A-1b)

• (ii) for any $\delta >0$,

  $$\begin{aligned}&| \alpha |_{L^{{(n-1)p}/{(n-1-kp)}}} \le C(k,n,p) \sum _{i=0}^k K_1^{k-i} | \nabla ^i \alpha |_{L^{p}}, \ \ 1\le p < (n-1)/k, \end{aligned}$$

  (A-2a)
  $$\begin{aligned}&| \alpha |_{L^{\infty }} \le \delta | \nabla ^k \alpha |_{L^{p}} + C(\delta ^{-1}, K_1, k, n,p) \sum _{i=0}^{k-1} | \nabla ^i \alpha |_{L^{p}}, \ \ p>(n-1)/k, \end{aligned}$$

  (A-2b)
  $$\begin{aligned}&\Vert \alpha \Vert _{L^{{np}/{(n-kp)}}} \le C(k,n,p) \sum _{i=0}^k K_1^{k-i} \Vert \nabla ^i \alpha \Vert _{L^{p}}, \ \ 1\le p < n/k, \end{aligned}$$

  (A-2c)
  $$\begin{aligned}&\Vert \alpha \Vert _{L^{\infty }} \le C(k,n,p) \sum _{i=0}^k K_1^{k-i} \Vert \nabla ^i \alpha \Vert _{L^{p}}, \ \ p > n/k. \end{aligned}$$

  (A-2d)

Lemma A-2

(Lemmas 5.5-5.6 in [11]) Let w be a (0, 1) tensor and define a scalar $\mathrm{div} w=g^{ab}\nabla _a w_b$ and a (0, 2) tensor $\mathrm{curl} w_{ab}=\nabla _a w_b -\nabla _b w_a$. If $|\theta |+1/\iota _0\le K$, then for any nonnegative integer r,

$$\begin{aligned} |\nabla ^{r+1} w|^2&\le C\left( g^{ij} \zeta ^{kl} \zeta ^{IJ} (\nabla _k \nabla ^r_I w_i) \nabla _l \nabla ^r_J w_j + |\nabla ^ r \mathrm{div} w|^2 + |\nabla ^r \mathrm{curl} w|^2 \right) , \end{aligned}$$

(A-3a)

$$\begin{aligned} \Vert \nabla ^{r+1} w\Vert _{L^2}^2&\le C\int _{\Omega } \widetilde{N}^i\widetilde{N}^j g^{kl} \zeta ^{IJ} (\nabla _k\nabla ^{r }_I w_i)\nabla _l\nabla ^{r }_J w_j {\mathrm{d}}\mu _g \nonumber \\&\qquad \quad + C\left( \Vert \nabla ^{r}\mathrm{div} w\Vert _{L^2}^2 + \Vert \nabla ^{r}\mathrm{curl} w\Vert _{L^2}^2 + K^2 \Vert \nabla ^{r} w\Vert _{L^2}^2 \right) ,\end{aligned}$$

(A-3b)

$$\begin{aligned} | \nabla ^r w |_{L^2}^2&\le C \left( \Vert \nabla ^{r+1}w\Vert _{L^2} + K \Vert \nabla ^r w\Vert _{L^2} \right) \Vert \nabla ^r w\Vert _{L^2},\end{aligned}$$

(A-3c)

$$\begin{aligned} | \nabla ^r w |_{L^2}^2&\le C| \Pi \nabla ^r w |_{L^2}^2 + C\left( \Vert \nabla ^{r}\mathrm{div} w\Vert _{L^2} + \Vert \nabla ^{r}\mathrm{curl} w\Vert _{L^2} + K \Vert \nabla ^r w\Vert _{L^2} \right) \Vert \nabla ^r w\Vert _{L^2},\end{aligned}$$

(A-3d)

$$\begin{aligned} \Vert \nabla ^{r+1} w\Vert _{L^2}^2&\le C| \nabla ^{r+1} w |_{L^2} | \nabla ^{r} w |_{L^2} + C\left( \Vert \nabla ^{r}\mathrm{div} w\Vert _{L^2}^2 + \Vert \nabla ^{r}\mathrm{curl} w\Vert _{L^2}^2 \right) ,\end{aligned}$$

(A-3e)

$$\begin{aligned} \Vert \nabla ^{r+1} w\Vert _{L^2}^2&\le C| \Pi \nabla ^{r+1} w |_{L^2} | \Pi (N^i \nabla ^{r} w_i) |_{L^2} \nonumber \\&\qquad \quad + C\left( \Vert \nabla ^{r}\mathrm{div} w\Vert _{L^2}^2 + \Vert \nabla ^{r}\mathrm{curl} w\Vert _{L^2}^2 + K^2 \Vert \nabla ^{r} w\Vert _{L^2}^2 \right) ,\end{aligned}$$

(A-3f)

$$\begin{aligned} \Vert \nabla ^{r+1} w\Vert _{L^2}^2&\le C| \Pi (N^i \nabla ^{r+1} w_i) |_{L^2}| \Pi \nabla ^{r} w |_{L^2} \nonumber \\&\qquad \quad + C\left( \Vert \nabla ^{r}\mathrm{div} w\Vert _{L^2}^2 + \Vert \nabla ^{r}\mathrm{curl} w\Vert _{L^2}^2 + K^2 \Vert \nabla ^{r} w\Vert _{L^2}^2 \right) . \end{aligned}$$

(A-3g)

Indeed, the proof of (A-3a)–(A-3b) can also be found in [36]. The proof of (A-3c)–(A-3g) are based on the divergence theorem, and (A-3f)–(A-3g) are based additionally on (A-3b).

Lemma A-3

(Lemma A.5 in [11]) Suppose that $q=0$ on $\partial \Omega$. Then

$$\begin{aligned} \Vert q\Vert _{L^2} \le C (\mathrm{Vol}\Omega )^{1/n} \Vert \nabla q \Vert _{L^2} \ \ \mathrm{and} \ \ \Vert \nabla q\Vert _{L^2} \le C (\mathrm{Vol}\Omega )^{1/n} \Vert \Delta q \Vert _{L^2} . \end{aligned}$$

(A-4)

As a consequence of Lemmas A-2 and A-3, we have

Corollary A-4

Let $q=q_b$ on $\partial \Omega$ with $q_b$ being a constant. If $|\theta |+1/\iota _0\le K$, we have for any $r\ge 2$ and $\delta >0$,

$$\begin{aligned}&\Vert q-q_b\Vert _{L^2} \le C(\mathrm{Vol}\Omega )^{1/n} \Vert \nabla q\Vert _{L^2} , \ \ \Vert \nabla q\Vert _{L^2} + \Vert \nabla ^2 q\Vert _{L^2} \le C(K, \mathrm{Vol}\Omega ) \Vert \Delta q\Vert _{L^2} , \end{aligned}$$

(A-5a)

$$\begin{aligned}&\Vert \nabla ^r q\Vert _{L^2} + | \nabla ^r q |_{L^2} \le C| \Pi \nabla ^r q |_{L^2} + C(K, \mathrm{Vol}\Omega ) \sum _{s=0}^{r-1} \Vert \nabla ^s \Delta q \Vert _{L^2} , \end{aligned}$$

(A-5b)

$$\begin{aligned}&\Vert \nabla ^r q\Vert _{L^2} + | \nabla ^{r-1} q |_{L^2} \le \delta | \Pi \nabla ^r q |_{L^2} + C(\delta ^{-1}, K, \mathrm{Vol}\Omega ) \sum _{s=0}^{r-2} \Vert \nabla ^s \Delta q \Vert _{L^2} . \end{aligned}$$

(A-5c)

Clearly, (A-5a) is a consequence of (A-4) and (A-3g). The proof of (A-5b) and (A-5c) can be found in Proposition 5.8, [11]. (Indeed, (A-5b) follows from (A-3c)–(A-3e) and (A-4), and (A-5c) follows from (A-3c), (A-3e), (A-3f) and (A-4).)