数理解析研究所講究録
1495 巻 2006 年 103-111

103

Remarks on Quasi-linear Wave Equations
Related to Gas Dynamics

dedicated to Professor Tai-Ping Liu on his
60th birthday
Tetu Makino(Faculty of Engineering, Yamaguchi University)
This is a joint work with Cheng-Hsiung Hsu (National Central Univ.,Taiwan) and Song-Sun Lin (National Chiao-Rng Univ., Taiwan).

1

Introduction

We consider the equation(1)$y_{tt}-(G(y_{x}))_{x}=0$

on $0<x<1$ with the boundary conditions(2)$y(t, 0)=y(t, 1)=0$ .

Here we suppose that $G(v)$ is real analytic in. As example we keep in mind
and $G’(r))>0$ for$|v|<\delta,$ $G(\mathrm{O})=0,$$|?\mathit{1}|<\delta$$G(\uparrow))=1-(1+\tau))^{-\gamma}$,$(|v|<1)$.

The linearized problem is$y_{1,tt}-\gamma y_{1,rx},$$=0$ ,$y_{1}(t, 0)=y_{1}(t,, 1)=0$,$G’(0)=\gamma>0$104which has smooth time periodic solutions.

$y_{1}= \sum_{n=1}^{N}a_{n}\mathrm{s}’\ln(n\pi\sqrt{\gamma}(t+\theta_{n}))\sin n\pi x$(3)periodic solutions for (1)(2) nearThus we have the problcm: Are thcreproblem?
We have not yet obtained
linaerized
theto the periodic solution ofan answer to this question. In this report we give some related observations
to this problem. Detailed discussion can be found in [2].
$\mathrm{t}\mathrm{i}_{\mathrm{I}}\mathrm{n}\mathrm{e}$2Derivation of the problem

We consider one-dimensional movement of polytropic gas without external
force governed by the compressible Euler equation$\rho_{t}+(\rho u)_{x}=0$,$(\rho u)_{t}+(\rho u^{2}+P)_{x}=0$

on a fixed interval $0<x<L$ with the boundary conditions$\rho u|_{x=0}=\rho u|_{x=L}=0$.

, where and are positive constants such that
We assume that. $>0,$ $u=0$ .
$1<\gamma\leq 2$ , Equilibria are constant densities
Lct us introduce the Lagrangean coordinate$A$$P=A\rho^{\gamma}$$\gamma$$\rho=\overline{\rho}=Const$$m= \int_{0}^{x}\rho dx$.

Then the equation is reduced to$x_{tt}+(A(x_{m})^{-\gamma})_{m}=0$,where $x=x(t, m)$ is new unknown function, while$u= \frac{\partial x,}{\partial t}$

We consider the perturbation$y$,$\frac{1}{\rho}=\frac{\partial x}{\partial m}$.

near the equilibrium$x(t, m)= \overline{x}+y=\frac{L}{M}m+y$,

the independent variable and norwherc $M$ is the total xnass. Taking
malizing the variables, we have (1) and (2) with$\overline{x}\mathrm{a}_{\iota}\mathrm{s}$$G(v)=1-(1+v)^{-\gamma}$.

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3Existence of smooth solutions on long time
Let us fix a smooth time periodic solution $y_{1}(t, x)$ of the form (3) of thelinearized problem.

andTheorem 1. For any positive number there are positive constants$(Z)$$y(t,x)$(1)on
we have -solution
such that for anyof$0\leq t\leq T$ such that$T$

$C^{2}$$0<\epsilon\leq\Xi^{*}$$C$$\epsilon^{*}$$|y(t, x)-\epsilon y_{1}(t, x)|\leq C\epsilon^{2}$for$0\leq t\leq T$and $0\leq x\leq 1$ . Here$y(\mathrm{O}, x)=\epsilon y_{1}(0, x)$,$y_{t}(0, x)=\epsilon y_{1,t}(0, x)$.

Before proving this theorem we consider the problem by extending solutions as$y(t, x)=-y(t, -x)$ ,$y(t, x+2n)=y(t, x)$

for any $n\in$ Z. Putting$u_{1}=y_{1,t}$,$\uparrow)_{1}=y_{1,x}$,,$y_{t}=\epsilon u_{1}+U$,$y_{x}=\epsilon v_{1}+V$

we have$V_{t}-U_{x}=0$,.

$U_{t}-G’(\epsilon v_{1}+V)V_{x}=(G’(\epsilon v_{1}+V)-\gamma)\epsilon?\prime_{1,x}$

The variables$W=U+\hat{G}(\epsilon v_{1}+V)-\hat{G}(\epsilon v_{1})$,$Z=U-\hat{G}(\epsilon v_{1}+V)+\hat{G}(\epsilon v_{1})$where$\hat{G}(v)=\int_{0}^{v}\sqrt{G’(s)}ds$,

reduce the equation to the diagonalized equation$W_{t}-\Lambda(\epsilon v_{1}+V)W_{x}=L_{-}$,$Z_{t}+\Lambda(\epsilon v_{1}+V)Z_{x}=L_{+}$

where$\Lambda(v)=\sqrt{G^{J}(\tau))}$

,,,

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and.

$L_{\pm}=(-\gamma+\Lambda(\epsilon v_{1}+V)\Lambda(\epsilon v_{1}))\epsilon v_{1,x}\pm(\Lambda(\epsilon v_{1})-\Lambda(\epsilon v_{1}+V))\epsilon u_{1,x}$

We look for solutions $W,$

$Z$

such that$W(t, x+2n)=W(t, x)$ .

$W(t, x)=-Z(t, -x)$ ,For simplicity we consider solutions which satisfy the initial conditions$W(0, x)=Z(0, x)=0$ .

We construct solutions by iterations. Given $V(t, x)$ , we solve,$\tilde{W}_{t}-\Lambda(\epsilon v_{1}+V)\tilde{W}_{x}=L_{-}(t, x, V(t, x))$$\tilde{Z}_{t}+\Lambda(\epsilon v_{1}+V)\tilde{Z}_{x}=L_{+}(t, x, V(t, x))$and find$\tilde{U},\tilde{V}$,by$\tilde{W}=\tilde{U}+\hat{G}(\epsilon v_{1}+\tilde{V})-\hat{G}(\epsilon v_{1})$,$\tilde{Z}=\tilde{U}-\hat{G}(\epsilon v_{1}+\tilde{V})+\hat{G}(\epsilon v_{1})$.

The solution of the above problem is given by the integral along the characteristic curves: if $\xi(r)=\xi(\tau;t, x)$ is the solution of$\frac{d\xi}{d\tau}=\Lambda(\epsilon v_{1}+V)(\tau, \xi(\tau))$,$\xi(t)=x$,then$\tilde{Z}(t, x)=\int_{0}^{t}L_{+}(\tau, \xi(\tau),$ $V(\tau, \xi(\tau)))d\tau$,and so on. Through tedious computations we can prove the following lemmas.
Lemma 1. Thare exist $M_{0}>0,$, then$\epsilon_{0}>0$$||\tilde{U}||\leq\epsilon^{2}M_{0},$ $||\tilde{V}||\leq\epsilon^{2}M_{0}$$\epsilon^{2}M_{0}$such that if

and$0<\epsilon\leq\epsilon_{0}$$\epsilon||v_{1}||+\epsilon^{2}M_{0}\leq\delta/2$

Here$||f||= \sup\{|f(\theta, x)| |0\leq t\leq T, x\in \mathbb{R}\}$

Lemma 2. There exist, then$0<\epsilon_{1}(\leq\epsilon_{0}),$ $M_{1}$$\epsilon^{2}M_{0},$$||V_{x}||\leq\epsilon^{2}M_{1}$$||\tilde{U}_{x}||,$$||\overline{U}_{t}||,$

such that if$||\tilde{V}_{x,}||,$.

and $||V||\leq$.

$0\leq\epsilon\leq\epsilon_{1},$$||\tilde{V}_{t}||\leq\epsilon^{2}M_{1}$and$||V||\leq$$\epsilon^{2}M_{1}<1$.

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Lemma 3. There exist$0<\epsilon_{2}(\leq\epsilon_{1}),$$M_{2}$

such that if$||V||\leq\epsilon^{2}M_{0},$ $||V_{x}||\leq\epsilon^{2}M_{1},$then$||\tilde{U}_{xx}||,$$||\tilde{U}_{tx}||,$

Moreover, if
prove that$||\tilde{V}_{xx}||,$$V^{1},$ $V^{0}$$||V_{xx}||\leq\epsilon^{2}\Lambda l_{2}$$||\tilde{V}_{tt}||\leq\epsilon^{2}M_{2}$$||\tilde{V}_{xt}||,$

and,

,

$\epsilon^{2}M_{2}<1$.

satisfy the conditions of the lemmas, then we can$|| \tilde{V}^{1}-\tilde{V}^{0}||\leq\frac{1}{2}||V^{1}-V^{0}||$if$0<\epsilon\leq\epsilon_{2}$,, which is sufficiently small.
Thus we consider the iteration$\epsilon<\epsilon^{*}(\leq\epsilon_{2})$$V^{(0)}=0$

Then$V^{(n)}$,converges to the limit$V^{(n+1)}=\overline{V^{(n)}}$.

and$V$$y(t, x)= \epsilon y_{1}(t, x)+\int_{0}^{x}V(t, s)ds$

gives the required solution.

4Non-existence of time global smooth solutions

Let us apply the arguemnet of [4].
Theorem 2. Suppose that $G”(v)<0for|v|<\delta$ or $G”\backslash (v)>0for|v|<\delta.$ If$y(t, x)\in C^{2}([0, +\infty)\cross[0,1])$ is a solutionof (1)(2) such that, then $y=0$ identically.
for $t\geq 0,$ $x\in[0,1]$ , where$|y_{x}(t, x)|\leq\delta_{1}$$0<\delta_{1}<\delta$

Proof. We can suppose that$y\in C^{2}([0, \infty)\mathrm{x}\mathbb{R})$$-y(t, -x),$ $y(t, x+2n)=y(t, x)$ for$n\in \mathbb{Z}$.

solves (1) and $y(t, x)=$Putting$w=y_{t}+\hat{G}(y_{x})$,$z=y_{t}-\hat{G}(y_{x})$,we reduce the equation to$w_{t}-\Lambda(y_{x})w_{x}=0$,$z_{t}+\Lambda(y_{x})z_{x}=0$.

108

By the assumption, we have$\frac{1}{C}\leq\Lambda(y_{x})\leq C$.

Now let $x=x(t)=x(t;a)$ solve$x(0)=a$ .

$\frac{dx}{dt}=\Lambda(y_{x}(t, x(t)),$ConsiderX$(t)= \frac{\partial}{\partial a}x(t;a)$.

Then$X(t)= \exp(\int_{0}^{t}\frac{\partial}{\partial x}\Lambda(y_{x}(\tau, x))d\tau)>0$.

On the other hand we can prove that
$X=( \frac{\Lambda(y_{x}(t_{\text{ノ}},x(i))}{\Lambda(y_{x}(0,a))})^{1/2}(1+z_{a}(0, a)\int_{0}^{t}Q(\tau)d\tau)$,where.

$Q( \tau)=-(\frac{\Lambda(y_{x}(0,a))}{\Lambda(y_{x}(\tau,x(\tau))})^{1/2}\frac{1}{4}\frac{G’’(y_{x}(\tau,x(\tau))}{G’(y_{x}(\tau,x(\tau))}$If $G”<0$ , then $Q\geq 1/c>0$ andhave $z_{a}(0, a)\geq 0$ for any . Since
By a similar discussion we have$a$. Thus weas.
is periodic, we have
. This implies the result.

$\int_{0}^{t}Q(\tau)d\tauarrow+\infty$$z(\mathrm{O}, .)$$tarrow\infty$$z(\mathrm{O}, .)=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$$w(\mathrm{O}, .)=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$5Exact solutions

In [3] F. John gave exact solutions to the equation$y_{tt}-(1+y_{x}^{2})y_{xx}=0$.

Along the idea of F. John we construct exact solutions of our general equation$y_{tt}-(G(y_{x}))_{x}=0$

Let be arbitrary function ofthe cquation$f$$C_{0}^{\infty}(\mathbb{R})$.

. Suppose that $v=v(t, x)$ solves$v=f^{j}(x-\Lambda(v)t)$

.

(4)

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Put
$y(t, x)=t\Phi(v(t, x))+f(x-\Lambda(v(t, x))t)$ ,where$\Phi(v)=\int_{0}^{v}\zeta\Lambda’(\zeta)d\zeta=v\Lambda(\tau’)-\hat{G}(v)$.

Then we have$y_{t}=-\hat{G}(v)$,$y_{tt}=-\Lambda v_{t}$,$y_{x}=v$,$y_{xx}=v_{x}$.

On the other hand$v_{t}=- \frac{\Lambda f’’}{1+f’\Lambda’t},$$v_{x}= \frac{f’’}{1+f’’\Lambda’t}$

as long as$1+f^{\prime l}(x-\Lambda(v)t)\Lambda’(v)t\neq 0$,.

Thus $y(t, x)$ satisfies the equation with the initial conditions$y(0, x)=f(x)$ ,Suppose that
and$f\in C_{0}^{\infty}(\mathbb{R})$.

$y_{\mathrm{t}}(0, x)=-\hat{G}(f’(x))$

satisfies that$|f’(\xi)|\leq\delta_{0}(<\delta)$

for any$\xi\in \mathrm{R}$.

$-m= \min_{\xi}f’’(\xi)\Lambda’(f’(\xi))<0\leq\max f’’(\xi)\Lambda’(f’(\xi))\epsilon\leq m$

Put $T=1/m$ . The equation (4), which is equivalent to the equation$\xi=x-\Lambda(f’(\xi))t$,, we see thatadmits a unique solution as long as $0\leq t<T$ . As t–$y_{t}=-\hat{G}(v)$is a typicalThis.
remain to be bounded but$y_{x}=v,$
example of singularity which happens after a finite time for vmooth solutions.
$T-\mathrm{O}$$y_{xx}arrow\infty$

6

Estimate of life span of smooth solutions

Let us apply the theory of Lax [5] to find estimate of life span of smooth
solutions.
Consider the initial value problem:$y_{tt}-(G(y_{x}))_{x}=0$,$y(t, 0)=y(t, 1)=0$ ,110$y(0, x)=\phi(x)$ ,where$\phi,$$\psi$$y_{t}(0, x)=\psi(x)$,are smooth and$\phi(0)=\phi(1)=\psi(0)=\psi(1)=0$

Theorem 3. There exist constants

if$|\phi_{xx}(x)|$$\epsilon,$,$C$

such that if$|\psi_{x}(x)|\leq M$then there exists a solution $y(t, x)$ of class$C^{2}$.
$|\phi_{x}(x)|,$ $|\psi(x)|\leq\epsilon$,

as long as $0\leq t\leq 1/CM$ .

Observe$w=\tau/t+\hat{G}(y_{x})$,$z=y_{t}-\hat{G}(_{\mathrm{t}/x})$

which satisfy$w_{t}-\Lambda w_{x}=0$

Thus a priori estimates of $|w|,$
Consider the quantities

$|z|$

,

and

$z_{l}+\Lambda z_{x}=0$,.

are obvious.

$A=\sqrt{\Lambda}w_{x}$,$B=\sqrt{\Lambda}z_{x}$$A_{t}-\Lambda A_{x}+\mu A^{2}=0$,$B_{f,}+\Lambda B_{x}+/AB^{2}=0$,

which satisfy

,

where
$\mu=-\frac{1}{4}G’’(y_{x})G^{j}(y_{x})^{-5/4}$.

Note that $|\mu|\leq C$ a priori. As P. D. Lax said in [5]: “solution to initialvalue problems exists as long as one can place an a priori limitation on themagnitude of their first derivatives.” Thus this completes the proof.

7

Problem with vacuum

Originally we are interested in the equation$y_{tt}- \frac{1}{\rho}(PG(y_{x}))_{x}=0$

,

where
$G(v)=1-(1+v)^{-\gamma}$ ,$\rho=(1-x)^{\frac{1}{\gamma-1}}$,$P=(1-x)\overline{\gamma}-\mathrm{L}1$

.

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This equation is derived from the motion of gas under constant gavity governed by the equation$\rho_{t}+(\rho u)_{x}=0$

where$P=A\rho^{\gamma}$

and

$g$

,

$(\rho u)_{t}+(\rho u^{2}+P)_{x}=-\rho g$,

is positive constant. Equilibria are$A_{1}(L-x)^{\frac{1}{\gamma-1}}$$(0\leq x<L)$$\overline{\rho}=\{$

$0$

where

$L$

$(L<x)$ ,is positive constant determined by the total mass and$A_{1}=( \frac{g(\gamma-1)}{A\gamma})^{\frac{1}{\gamma-1}}$

The linearized problem has time periodic smooth solutions explicitely writtenby the Bessel function of order $1/(\gamma-1)$ . Detailed discussion can be foundin [1].
But we have not yet proven parallel results for this problem because of
the singularity at $x=1$ .

References[1] C.-H. Hsu, S.-S. Lin and T. Makino, Pcriodic solutions to the 1dimensional compressible Euler equation with gravity, to appear in Proc.

Hyp2004.

[2] C.-H. Hsu, S.-S.Lin and T. Makino, Smooth solutions to a class of quasilincar wave equations, to appear in J. Diff. Eqs.
[3] F. John, Delayed singularity formation in solutions of nonlinear wave
equatios in higher dimensions, Comm. Pure Appl. Math., XXIX(1976),

649-681.
[4] B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math., XIX(1966),371-420.
[5] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic differential equations, J. Math. Phys., 5(1964), 611-613.