Long Time Behavior of a Quasilinear Hyperbolic System Modelling Elastic Membranes

Abstract

The paper studies the long time behavior of a simplified model of an elastic membrane driven by surface tension and inner air pressure. The system is a degenerate quasilinear hyperbolic one that involves the mean curvature, and also includes a damping term that models the dissipative nature of genuine physical systems. With the presence of damping, a small perturbation of the sphere converges exponentially in time to the sphere, and without the damping the evolution that is $\varepsilon$-close to the sphere has a life span longer than $\varepsilon ^{-1/6}$. Both results are proved using a new Nash–Moser–Hörmander type theorem proved by Baldi and Haus.

Appendices

Appendix A. Weakly Hyperbolic Systems

In this appendix, we will sketch results on weakly hyperbolic systems and local well-posedness of (EQ0) and (EQWD).

Let M be a compact differential manifold, and let $E_1,E_2$ be smooth vector bundles on M. Let $\phi$ and $\psi$ be time-dependent sections of $E_1$ and $E_2$ respectively. A weakly hyperbolic linear system in $(\phi ,\psi )$ takes the following form:

$$\begin{aligned} \begin{aligned} \frac{\partial ^2}{\partial t^2}\phi (t)&=L(t)\phi (t)+M(t)\partial _t\phi (t)+P(t)\psi (t)+f_1(t),\\ \frac{\partial ^2}{\partial t^2}\psi (t)&=Q(t)\phi (t)+R(t)\psi (t)+f_2(t), \end{aligned} \end{aligned}$$

(A.1)

where given any time t, $L(t):\Gamma (E_1)\rightarrow \Gamma (E_1)$ is a second order elliptic operator, $P(t):\Gamma (E_2)\rightarrow \Gamma (E_2)$ and $Q(t):\Gamma (E_1)\rightarrow \Gamma (E_1)$ are first order pseudo-differential operators, $M(t):\Gamma (E_1)\rightarrow \Gamma (E_1)$ and $R(t):\Gamma (E_2)\rightarrow \Gamma (E_2)$ are zeroth order pseudo-differential operators, and $f_1,f_2$ are known time-dependent sections of $E_1$ and $E_2$. The notion of weakly hyperbolic linear system is just a rephrasing of Hamilton’s notion of weakly parabolic linear system in [13].

Proposition A.1

Fix Riemannian metrics on $E_1,E_2$ to measure function norms. Given smooth initial data $\phi [0],\psi [0]$ to (A.1), the equation is uniquely solvable on any time interval [0, T], and the solution satisfies the tame energy estimate

$$\begin{aligned} \begin{aligned} E_n(t)&\leqslant Ce^{Ct}E_n(0) +C\int _0^te^{C(t-s)}\left( \Vert f_1(s)\Vert _{H^{n}_x}^{(1)}+\Vert f_2(s)\Vert _{H^{n+1}_x}\right) \mathrm{d}s\\&\quad +C\int _0^te^{C(t-s)}\left( [L(s)]_n^{(1)}+[M(s)]_n^{(1)}+[P(s)]_n^{(1)}\right. \\&\left. \quad +[Q(s)]_{n+1}+[R(s)]_{n+1}\right) \left( E_{n_0}(0)+\Vert f_1(s)\Vert _{H^{n_0}_x}^{(1)}\right) \mathrm{d}s, \end{aligned} \end{aligned}$$

where $\Vert f(s)\Vert ^{(1)}:=\Vert f(s)\Vert +\Vert \partial _sf(s)\Vert$, the energy norm

$$\begin{aligned} E_n(t):=\Vert \partial _t^2\phi (t)\Vert _{H^n_x} +\Vert \partial _t\phi (t)\Vert _{H^{n+1}_x} +\Vert \phi (t)\Vert _{H^{n+1}_x}+\Vert \partial _t\psi (t)\Vert _{H^{n+1}_x}+\Vert \psi (t)\Vert _{H^{n+1}_x}, \end{aligned}$$

and $[\cdot ]_n$ denotes the Sobolev norm on jet bundles. The integer $n_0$ is the smallest integer to ensure Sobolev embedding $H^{n_0}\hookrightarrow C^2$. The constants C depend on n, the elliptic constants of $\{L(t)\}_{t\in [0,T]}$, and

$$\begin{aligned} \sup _{t\in [0,T]}\left( [L(t)]_{n_0}^{(1)}+[M(t)]_{n_0}^{(1)}+[P(t)]_{n_0}^{(1)}+[Q(t)]_{n_0}+[R(t)]_{n_0}\right) . \end{aligned}$$

This is enough for establishing the local well-posedness result for (EQ0) and (EQWD), defined on a general compact surface M. In fact, these two equations both fall into the class of “evolutionary problems with an integrability condition”, treated by Hamilton in [13]. The linearization of both gives a weakly hyperbolic system, whose unique solution satisfies a tame estimate. By the Nash–Moser theorem, the following is true:

Theorem A.1

Let M be a compact oriented surface, and let $u_0:M\hookrightarrow {\mathbb {R}}^3$ be a smooth embedding. Fix constants $\kappa >0$, $b\geqslant 0$. Let $u_1:M\rightarrow {\mathbb {R}}^3$ be any smooth mapping. Then there exists a $T>0$ depending on $u_0,u_1$ such that the Cauchy problem

$$\begin{aligned} \frac{\partial ^2u}{\partial t^2}+b\frac{\partial u}{\partial t} =\frac{\mathrm{d}\mu (u)}{\mathrm{d}\mu _0}\left( -H(u)+\frac{\kappa }{\mathrm {Vol}(u)}\right) N(u),\, \left( \begin{matrix} u(0,x)\\ \partial _tu(0,x) \end{matrix} \right) = \left( \begin{matrix} u_0(x)\\ u_1(x) \end{matrix} \right) \end{aligned}$$

has a unique smooth solution $u:[0,T]\times M\rightarrow {\mathbb {R}}^3$.

Appendix B. Discussion on “Elementary” Methods in Establishing Long-time Results

In this appendix, we shall sketch the method of estimating lifespan for (EQ0) in [31], and explain why it cannot be improved, thus why the Nash–Moser method in the proof of Theorem 1.2 is unavoidable.

In [31], the lifespan of equation (EQ0) with initial data $\varepsilon$-close to a static solution was estimated using a standard continuous induction argument, based on energy estimates for the equation satisfied by geometric quantities (the second fundamental form and components of the velocity etc.). The equation is a complicated quasilinear weakly hyperbolic system. The lifespan so obtained was $\sim \log 1/\varepsilon$.

Following [31], we consider the evolution of various geometric quantities. We shall write $w=i_0+u$, where u is a perturbation, and suppose w solves the Cauchy problem of (EQ0) on a time interval [0, T]:

$$\begin{aligned} \frac{\partial ^2u}{\partial t^2} =\frac{\mathrm{d}\mu (w)}{\mathrm{d}\mu _0}\left( -H(w)+\frac{\kappa }{\mathrm {Vol}(w)}\right) N(w),\, \left( \begin{matrix} w(0,x)\\ \partial _tu(0,x) \end{matrix} \right) = \left( \begin{matrix} i_0(x)+u_0(x)\\ u_1(x) \end{matrix} \right) . \end{aligned}$$

(EQ0')

Let

$$\begin{aligned} \sigma =\perp _w\partial _tw,\, S=\top _w\partial _tu,\, B_{ij}=({\bar{\nabla }}_{\partial _iw}\partial _tu,\partial _jw), \end{aligned}$$

where ${\bar{\nabla }}$ denotes the connection in ${\mathbb {R}}^3$. Then $\sigma$ is a scalar function, S is a vector field, and $B=B_{ij}$ is a symmetric second order tensor. Given any local coordinate on $S^2$, the difference between the Christoffel symbols of g(w) and $g_0$ is denoted by $\Gamma _{ij}^k(w)-\Gamma _{ij}^k(g_0)$, which is a tensor.

Now define the following tensors:

$$\begin{aligned} \zeta =\left( \begin{matrix} \sigma \\ h_{ij}(w)-h_{ij}(g_0) \end{matrix}\right) ,\, \xi =\left( \begin{matrix} u\\ du\\ S\\ B_i^j(w)\\ \Gamma _{ij}^k(w)-\Gamma _{ij}^k(g_0) \end{matrix}\right) . \end{aligned}$$

Here $B_i^j(w)=g^{ik}(w)B_k^j(w)$. We shall directly quote from [31] the evolution equations of $\zeta$ and $\xi$. For convenience we omit the dependence on w, and add an upper circle for geometric quantities induced by $g_0$.

The evolution of $\zeta$ is given by

$$\begin{aligned} \begin{aligned} \partial _t^2\sigma&=\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left[ \Delta \sigma +|h|^2\sigma -\frac{\kappa }{\mathrm {Vol}(w)^2}\int _{S^2}\sigma \mathrm{d}\mu +\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) (\text {div}S+H)\sigma \right] \\&\quad +\sigma \left( |\nabla \sigma |^2+S^kS^lh^j_kh_{jl}-2h_k^i\partial _i\sigma S^k\right) -2\partial _t S^k\partial _k\sigma +2h_{ik}S^i\partial _tS^k, \\ \partial _t^2h_{ij}&=\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left\{ \Delta h_{ij} +|h|^2h_{ij}-Hh_{il}h^{l}_j\right. \\&\left. \quad +\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) \left[ \nabla _i(\Gamma _{jl}^l-\mathring{\Gamma }_{jl}^l )+(\Gamma _{ik}^k-\mathring{\Gamma }_{ik}^k )(\Gamma _{jl}^l-\mathring{\Gamma }_{jl}^l)\right] \right\} \\&\quad +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\nabla _jH(\Gamma _{ik}^k-\mathring{\Gamma }_{ik}^k) +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\nabla _iH(\Gamma _{jl}^l-\mathring{\Gamma }_{jl}^l) +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) h_{ik}h^{k}_j\\&\quad +h_{ij}\left( |\nabla \sigma |^2+h^j_kh_{jl}S^kS^l-2h^i_k\partial _i\sigma S^k\right) +2\partial _t\Gamma ^k_{ij}(\partial _k\sigma -h_{kl}S^l). \end{aligned} \end{aligned}$$

Here the operator $\Delta$ acting on $h_{ij}$ is the trace Laplacian $g_{ij}\nabla ^i\nabla ^j$.

The evolution of $\xi$ is given by

$$\begin{aligned} \begin{aligned} \partial _t^2u&=\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) N(w), \\ \partial _t^2\partial _ku&=\partial _k\left[ \frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) \right] N(w) +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) h_k^l\partial _lu, \\ \partial _t^2S^m&=\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) \left[ -\nabla ^m\sigma +\sigma \left( \Gamma _{ij}^k-\mathring{\Gamma }_{ij}^k\right) g^{km}\right] \\&\quad +2\partial _t\sigma \left( \nabla ^m\sigma -h^m_kS^k\right) -\sigma \frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\nabla ^mH-2\sigma B_i^m(\nabla ^i\sigma -h^i_jS^j) -2\partial _tS^kB_k^m, \\ \partial _t^2B_i^k&=-\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}(\nabla ^k\sigma -h^k_lS^l)\partial _iH +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) (\Gamma _{il}^l-\mathring{\Gamma }_{il}^l)(\nabla ^k\sigma -h^k_mS^m)\\&\quad +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left[ -\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\partial _tH-\frac{\kappa }{\mathrm {Vol}(w)^2}\int _{S^2}\sigma \mathrm{d}\mu +\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) g^{ml}B_{ml}\right] h_i^k\\&\quad +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) (\partial _th^k_i+h_i^lB_l^k-B_i^lh_l^k)\\&\quad -2\partial _tB_i^lB_l^k+2\left[ -B_i^l(\partial _l\sigma -h_{lm}S^m)\right] (\nabla ^k\sigma -h_l^kS^l)\\&\quad -(\partial _i\sigma -h_{il}S^l)\left[ \frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\nabla ^kH+2(\nabla ^m\sigma -h_j^mS^j)B_m^k\right] \\&\quad +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) g^{km}(\Gamma _{ml}^l-\mathring{\Gamma }_{ml}^l)(\partial _i\sigma -h_{ij}S^j), \\ \partial _t^2\Gamma _{ij}^k&=\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}(-\nabla _iHh_j^k-\nabla _jHh_i^k+\nabla ^kHh_{ij}) +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) \nabla _ih_j^k\\&\quad +\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}\left( -H+\frac{\kappa }{\mathrm {Vol}(w)}\right) \left[ h_j^k(\Gamma _{il}^l-\mathring{\Gamma }_{il}^l)+h_i^k(\Gamma _{jl}^l-\mathring{\Gamma }_{jl}^l)-h_{ij}g^{km}(\Gamma _{ml}^l-\mathring{\Gamma }_{ml}^l)\right] \\&\quad -2\partial _t\Gamma _{ij}^lB_l^k-2\partial _th_{ij}(\nabla ^k\sigma -h_m^kS^m)+2h_{ij}(\nabla ^l\sigma -h_m^lS^m)B_l^k. \end{aligned} \end{aligned}$$

Observe that all geometric quantities are smooth in $(\zeta ,\xi )$; for example, the induced metric g(w) and the Radon-Nikodym derivative $\mathrm{d}\mu /\mathrm{d}\mu _0$ are smooth functions of du, and the (scalar) mean curvature H is the contraction of g(w) with h(w), and

$$\begin{aligned} \partial _k\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0} =(\Gamma _{ik}^i-\mathring{\Gamma }_{ik}^i)\frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}. \end{aligned}$$

We thus reduce these evolution equations to much terser forms.

Proposition B.1

There is a $\delta _6>0$ such that if $|u|_{C_t^2C_x^2}<\delta _6$, then the system of $(\zeta ,\xi )$ can be re-arranged to a terser form as

$$\begin{aligned} \begin{aligned} \partial _t^2\zeta&=A\zeta +I_0^1(\zeta ,\xi ,\mathring{\nabla }\zeta ,\partial _t\xi ,\mathring{\nabla }\xi )\cdot \zeta +I_1^1(\zeta ,\xi ,\mathring{\nabla }\zeta ,\partial _t\xi ,\mathring{\nabla }\xi )\cdot \mathring{\nabla }\zeta \\&\quad +I_2^1(\zeta ,\xi ,\mathring{\nabla }\xi )\cdot \mathring{\nabla }^2\zeta ,\\ \partial _t^2\xi&=J^2(\zeta ,\partial _t\zeta ,\mathring{\nabla }\zeta ) +Q^2(\zeta ,\xi ,\partial _t\zeta ,\mathring{\nabla }\zeta ,\partial _t\xi ), \end{aligned} \end{aligned}$$

(B.1)

where A is the elliptic operator given by

$$\begin{aligned} A\left( \begin{matrix} \sigma \\ h_{ij} \end{matrix} \right) =\left( \begin{matrix} \displaystyle {\mathring{\Delta }\sigma +2\sigma -\frac{6}{4\pi }\int _{S^2}\sigma \mathrm{d}\mu _0}\\ \mathring{\Delta }h_{ij} \end{matrix} \right) , \end{aligned}$$

and the I, J’s are tensors that depend smoothly on their arguments, and vanish linearly when the arguments tend to zero; the $Q^2$ tensor depends smoothly on its arguments, and vanishes quadratically when the arguments tend to zero.

Proof

This is a direct calculation using Taylor’s formula. For the $\zeta$ component, the only difficulty is the equation satisfied by $h_{ij}$. However, note that for example for any symmetric section W of $T^*(S^2)\otimes T^*(S^2)$, we have

$$\begin{aligned} |\mathring{h}|^2W_{ij}-\mathring{H}\mathring{h}^l_jW_{il}\equiv 0, \end{aligned}$$

so

$$\begin{aligned} \frac{\mathrm{d}\mu }{\mathrm{d}\mu _0}(\Delta h_{ij} +|h|^2h_{ij}-Hh_{il}h^{l}_j) \end{aligned}$$

is reduced to the form

$$\begin{aligned} \mathring{\Delta }(h_{ij}-\mathring{h}_{ij}) +(\text {function of }du,\,\mathring{\nabla }du)\cdot \mathring{\nabla }^2h_{ij} +(\text {function of }h)\cdot (h-\mathring{h})\cdot h_{ij}. \end{aligned}$$

Similarly, the only difficulty for the $\xi$ component are terms involving $\mathrm{d}H$. However, we compute

$$\begin{aligned} \begin{aligned} \mathrm{d}H&=d\left[ (g^{ij}-\mathring{g}^{ij})h_{ij}+\mathring{g}^{ij}(h_{ij}-\mathring{h}_{ij})-2\right] \\&=-h_{ij}\left( \Gamma _{lk}^ig^{lj}-\mathring{\Gamma }_{lk}^i\mathring{g}^{lj} +\Gamma _{lk}^jg^{li}-\mathring{\Gamma }_{lk}^j\mathring{g}^{li}\right) \mathrm{d}x^k+(g^{ij}-\mathring{g}^{ij})\partial _kh_{ij}\mathrm{d}x^k\\&\quad +\partial _k\mathring{g}^{ij}(h_{ij}-\mathring{h}_{ij})\mathrm{d}x^k +\mathring{g}^{ij}\partial _k(h_{ij}-\mathring{h}_{ij})\mathrm{d}x^k, \end{aligned} \end{aligned}$$

where we used the compatibility condition

$$\begin{aligned} \mathrm{d}g_{ij}=g_{il}\Gamma _{jk}^l\mathrm{d}x^k+g_{jl}\Gamma _{ik}^l\mathrm{d}x^k. \end{aligned}$$

Therefore $\mathrm{d}H$ can be expressed linearly in first order derivatives of $\zeta$ and no derivative of $\xi$. $\quad \square$

Once the original equation (EQ0’) is reduced to the weakly hyperbolic quasilinear system (B.1), a standard fixed-point type argument with the aid of Proposition A.1 will give the local well-posedness result of that system. Nevertheless, we should note that (B.1) is derived from (EQ0’). Going back from the solution of (B.1) to the original unknown u in (EQ0’) shall encounter obstructions resulting from geometric constraints, namely the Gauss–Codazzi equations for evolving submanifolds. We do not yet know if the verification of these geometric-dynamical constraints is possible; should it be possible, it is certainly as lengthy as the Nash–Moser iteration scheme. From the analysis above, it is better illustrated why the Nash–Moser scheme is unavoidable in solving (EQ0).

However, if (EQ0’) is already proved to be locally well-posed, system (B.1) shall provide an estimate for the lifespan if some sufficiently high Sobolev norm of u[0] is $\varepsilon$-small. This is the method employed in [31] to estimate the lifespan, and it is applicable to generic closed CMC hypersurfaces in a generic ambient manifold. Nevertheless, the result produced by this approach for perturbation of $S^2$ in ${\mathbb {R}}^3$ is not optimal compared to our proof of Theorem 1.2. Let us briefly explain the reason below. The idea is to estimate the energy norm

$$\begin{aligned} E_n(t):=\Vert \partial _t^2\zeta (t)\Vert _{H^n_x} +\Vert \partial _t\zeta (t)\Vert _{H^{n+1}_x} +\Vert \zeta (t)\Vert _{H^{n+1}_x}+\Vert \partial _t\xi (t)\Vert _{H^{n+1}_x}+\Vert \xi (t)\Vert _{H^{n+1}_x} \end{aligned}$$

for some large n by proving inequalities of the form

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}E_n(t)\leqslant C(\varepsilon )E_n(t), \end{aligned}$$

and use a continuous induction argument to make the quantity $\varepsilon E_n(t)$ bounded. A problem then arises: in the evolution equations of u and du, we find

$$\begin{aligned} -H+\frac{\kappa }{\text {Vol}(w)} =(g^{ij}-\mathring{g}^{ij})h_{ij}+\mathring{g}^{ij}(h_{ij}-\mathring{h}_{ij})+\frac{\kappa }{\text {Vol}(w)}-2. \end{aligned}$$

Thus, as $\zeta ,\xi \rightarrow 0$, the difference $-H+\kappa /\text {Vol}(w)$ only vanishes linearly. Consequently, in establishing energy estimate for (B.1), we find that the information of (perturbed) non-growing modes is lost compared to the linearized problem (4.1), and the best to expect for energy estimate is

$$\begin{aligned} E_n(t)\lesssim e^{Ct}\varepsilon , \end{aligned}$$

where no smallness for C can be guaranteed even if u[0] is small. Thus the standard continuous induction argument, employed in for example [23] or section 6.4 of [16], will only give the lifespan estimate

$$\begin{aligned} T\simeq \log \frac{1}{\varepsilon }, \end{aligned}$$

as obtained in [31]. This loss of information on slow growth of modes near zero is because that the method does not employ stability of $S^2$, and thus the estimate $T\simeq \log 1/\varepsilon$ for lifespan is not optimal in this case.