数理解析研究所講究録 1208 巻 2001 年 135-139

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Nonrelativistic limit of scattering theory
for nonlinear Klein-Gordon equations
中西賢次 (Kenji Nakanishi)
Fuculty of Science, Kobe University

神戸大学理学部

We study the scattering theory in the nonrelativistic limit for the nonlinear KleinGordon equation:$\dot{v}/c^{2}-\Delta v+c^{2}v+f(v)=0$(1),is the unknown function, $c\gg 1$ denotes the propagawhere $v=v(t, x)$ :tion speed, namely the speed of light, and $f(u)=|u|^{p}u$ is agiven nonlinearity with
$p>0$ . Actually we can deal with the power $p\in(4/n, 4/(n-2))$ (without the upperbound when $n\leq 2$ ). We can easily anticipate from the simpler equation$\mathbb{R}^{1+n}arrow \mathbb{C}$$\dot{v}/c^{2}+c^{2}v=0$(2),. So wecauses time oscillation of the form
that the nonrelativistic limit, which obeys the following
first eliminate this time oscillation by putting
modulated equation:$e^{\pm ic^{2}t}$$carrow\infty$$u:=e^{-ic^{2}}{}^{t}v$$\text{\"{u}}/c^{\mathit{2}}+2\mathrm{i}\mathrm{v}-\triangle u+f(u)=0$.

(3)

Then we can take the singular limit as $carrow\infty$ to the nonlinear Schr\"odinger equation:$2\mathrm{i}\mathrm{v}-\triangle v+f(v)=0$.

(4)

Our main goal is to see if we can describe the asymptotic behavior of solutions
to (3) via nonrelativistic approximation by (4). It was proved in [4] that every
finite energy solution to the Cauchy problem for (3) converges to the corresponding
solution of (4) in the energy space, locally uniformly in time. The nonrelativisticlimit can not approximate solutions globally in time for the free equation, neither forthe nonlinear one in case every solution behaves asymptotically free. Nevertheless,
we can show that the wave operators, their inverses and the scattering operator for(3) converge to those for (4). This means that the time-asymptotic behavior can beapproximated through the nonrelativistic equation

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Now we breifly recall the most important conserved quantities, namely the energy
and the charge, for (3) and (4). The energy for (3) and (4) is given respectively by$E^{c}(u)= \int_{\mathrm{R}^{n}}|\dot{u}/c|^{2}+|\nabla u|^{2}+F(u)dx=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$,(5)$E(v)= \int_{\mathrm{R}^{n}}|\nabla v|^{2}+F(v)dx=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$,where $F(u):=2|u|^{p+2}/(p+2)$ . The charge for (3) and (4) is given by$Q^{c}(u)= \int_{\mathrm{R}^{n}}|u|^{2}+\Im\dot{u}\overline{u}/c^{2}dx=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$,(6)$Q(v)= \int_{\mathrm{R}^{n}}|v|^{2}dx=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$.

For any space-time function , we denote$u$$\tilde{u}$$:=(u,\dot{u}/c)$,(7)

and define $E:=H^{1}\oplus L^{2}$ . Then the above conservation laws ensure global bound of.
solution to (3) in
Next we define the wave operators for (3) and (4).
$\tilde{u}\in E$$u$

Definition 1. The wave operatorsfor (3) are maps from $E$ into itself which
map the initial dataof any finite energy solutionof the free modulated
Klein-Gordon:$W_{\pm}^{c}$$\tilde{u}_{0}(0)$$u_{0}$$\dot{u}_{0}/c^{2}+2\mathrm{i}\mathrm{i}\mathrm{i}0-\Delta u_{0}=0$

into the initial data$(u_{\pm}(0),\dot{u}_{\pm}(0))$,(8)

of the solution of (3) satisfying$\lim_{tarrow\pm\infty}||\overline{u}_{\pm}(t)-\tilde{u}_{0}(t)||_{E}=0$,(9)respectively. Similarly, the wave operatorsfor (4) are defined as maps from
into itself which map the initial data of any finite energy solution of the free
Schr\"odinger:$W_{\pm}$$H^{1}$$2\mathrm{i}\mathrm{v}-\Delta v=0$

into that of (4) asymptotic as t
and $S:=M_{+}W_{-}$ .

$arrow\pm\infty$,. We denote(10)$M_{*}^{*}:=(W_{*}^{*})^{-1}$,$S^{c}:=M_{+}^{c}W_{-}^{c}$

We review the known results about these wave operators. If $4/n<p<4/(n-2)$ ,are well defined as bijections, which was proved in $[2, 3]$ for $n\geq 3$
andthenandand in [6] for $n\leq 2$ . In the lower critical case $p=4/n$ , it is known thatare well defined as injections (see [3]). In the upper critical case $p=4/(n-2)$ ,$W_{\pm}^{c}$$W_{\pm}$$W_{\pm}^{c}$$W_{\pm}$

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is well defined as bijections [5], while W. is known to exist only for radiallysymmetric data [1] (the nonsymmetric case is an open problem).
Now we can state our main result.
$W\ovalbox{\tt\small REJECT}$

Theorem 2. AssumeSuppose$n\in \mathrm{N}$and $4/n<p<4/(n-2)$ . Let$\Phi^{c}arrow(\varphi, 0)$

as$carrow\infty$$\Phi^{c}\in E$

and$\varphi\in H^{1}$

in $E$ ,.

(11). Then we have
$W_{\pm}^{c}\Phi^{c}arrow(W_{\pm}\varphi, 0)$

in $E$ ,$M_{\pm}^{c}\Phi^{c}arrow(M_{\pm}\varphi, 0)$

in $E$ ,$S^{c}\Phi^{c}arrow(S\varphi, 0)$(12)

in $E$ .

Key ingredients in our proof are auniform decay estimate in the sense of spacetime norms for (8), compactness argument combined with the conservation laws andthe uniform Strichartz estimate derived in [4]. We use the space-time norms of thefollowing form:$||u||_{S|W\cap K}:=||\chi^{c}*u||_{S}+||\chi_{c}*u||_{W\cap K}$(13),, which is carried bysmoothly cuts off the higher frequency part
wherethe latter term $\chi_{c}*u=u-\chi^{c}*u$ . , $W$ and $K$ denote the space-time normsof Strichartz type for Schrodinger, wave and Klein-Gordon equations, respectively.
The following linear estimate of Strichartz type plays acrucial role.
$|\xi|_{\sim}>c$$\chi^{c}$$S$

Lemma 3([4]). Let$U^{c}(t):=e^{\pm ic\langle\nabla)_{c}t}$. For any c $>0$ , we have$||U^{c}(t)\varphi||_{S_{0}|(W_{0}\cap K_{0})}\leq C||\varphi||_{L^{2}}$

,

(14)

$|| \int_{0}^{t}U^{c}(t-s)f(s)ds||_{S_{0}|(W_{0}\cap K_{0})}\leq C||f||_{S_{1}’|(W_{1}’+K_{\acute{1}})}$,(15), and,
is a positive constant independent of ,denotewhereand .
satisfying the following conditions. Herearbitrary spaces of the form
$ho$ omogeneous Besov space.
Let $b:=1/p$ and $\alpha:=1/2-1/q$ . All, denotes theandmust obey
the spaces ,
$C$

$c$

$\varphi$$f$$S_{i}$$K_{i}$$W_{i}$$c^{-\mu}L^{p}(\mathbb{R};\dot{B}_{q,2}^{\sigma})$$\dot{B}_{**}^{*}$$S_{i}$$W_{i}$$K_{i}$$-2b+n\alpha+\sigma+\mu=0$ ,$0\leq 2b<1$ ,$0\leq 2\alpha\leq 1$,

(11)

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and each space should satisfy:$S_{i}$$\mu=0$,$2b\leq n\alpha$

,

(17)

$W_{i}$:$\mu=b$ ,$2b\leq(n-1)\alpha$$K_{i}$:$\mu=(1+2/n)b$ ,$2b\leq re\alpha$

,

(18)

,

(19)

respectively. $X’$ denotes the dual space of $X$ .
(16) shows that we have the same scaling both for the lower and the higher
frequency parts. Apart of regularity is transferred to the weight ofin the higher
frequency, compared with the lower part. We can recover this lost regularity because
we have akind of regularization property for the higher frequency in the associatedintegral equation:$c^{-1}$$u=u_{0}- \int_{\infty}^{t}e^{:c^{2}(t-s)}\sin\{c\langle\nabla\rangle_{c}(t-s)\}\frac{c}{\langle\nabla\rangle_{c}}f(u(s))ds$

,

(20)

where
is the free solution asymptotic to as $tarrow\infty$ and the regularization iscaused by the operator, whereand $\varphi(\nabla):=F^{-1}\varphi(i\xi)F$
denotes the Fourier multiplier.
We demonstrate the outline of the main estimate in asimple case where $n=3$
and $p=2$ (for the general case, see [7]). Then we can choose the norm for thelower frequency and $K$ for the higher frequency as$u$$u_{0}$$\langle\xi\rangle_{c}:=\sqrt{|\xi|^{2}+c^{2}}$$c/\langle\nabla\rangle_{c}$$S$$S:=L_{t}^{4}(W^{1,3})$,$K:=c^{-5/12}L_{t}^{4}(B_{3,2}^{7/12})$

,

(21)

,
whereand , denotes the inhomogeneous Sobolev and Besov spaces, respectively. We do not need the space of wave type, since the cubic nonlinearity is quite
regular forsolution when $n=3$ . Using the Sobolev embeddingand, we can estimate the nonlinearity as$W^{**}$$B_{**}^{*}$$H^{1}$$W^{1,3}\subset L^{6}$$B_{3,2}^{7/12}\subset L^{6}$$|||u|^{2}u||_{L^{4/3}}(W^{1,3/2}+c^{-5/12}B_{3.2}^{7/12})\sim|<|u||_{L^{4}(L^{6})}^{2}||u||_{S|K}\sim|<|u||_{S|\kappa}^{3}$Then we can use the regularizing property of$c/\langle\nabla\rangle_{c}$.

(22)

as.

$||c\langle\nabla\rangle_{c}^{-1}f(u)||_{L^{4/3}}(W^{1,3/-\epsilon/\sim}2|cB_{3,2}^{7/12}12)|<|f(u)||_{L^{4/3}}(W^{1.3/2}+c^{-b/12}B_{3,2}^{7/12})$(23)

Finally, we obtain by the Strichartz estimate$||u-u_{0}||_{S|K(T,\infty)}\leq||u||_{S|K(T,\infty)}^{3}$.

(24)Thus, we deduce uniform estimates for and large from the estimate for the freesolution , which can be derived from linear decay estimates$u$$u_{0}$$T$

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can be proved as follows. Let be the solution ofThen the convergence of,(3) withbe the free solution of (8) asymptotic to as $tarrow\infty$ , bethe solution of (4) with $v(0)=\varphi$ and be the free solution of (10) asymptotic toin $E$ as $carrow\infty$ . Prom theas $tarrow\infty$ . What we want to prove isin the energy
above argument, we can uniformly approximate byand by
spaces for $t>T$ and $c>c_{0}$ with some and Co. If we take $c>c_{0}$ sufficiently large,
then, by the time-local convergence result in [4],is very close to $(v(T), 0)$ , soisto $(v_{0}(T), 0)$ . Taking large again if necessary, we can approximate
by $(v_{0}(0), 0)$ , as desired.
, we use the compactness argument for the sequence,
For the proof ofwhere $K^{c}(t)$ denotes the matrix valued free propagator for .
Our result reflects that the nonrelativistic limit converges globally in space-timenorms and that high-frequency modification does not effect the nonlinearity so much.
It is also possible to retrace the argument in [6] to derive auniform global estimate
for space-time norms in terms of the energy only. It would be interesting if we could
get the same result in the Sobolev critical case $p=4/(n-2)$ , where we know only
the estimate dependent on and global wellposedness for (4) with general data is
still open (see [1] for radial data).
$u$$M_{+}^{c}$$\vec{u}(0)=\Phi^{c}$$u$$u_{0}$$v$$v$$v_{0}$$\tilde{u}_{0}(0)arrow(v_{0}(0), 0)$

$u$

$v$

$u_{0}$$v_{0}$$T$$\tilde{u}(T)$$\vec{u}_{0}(T)$$\tilde{u}_{0}(0)$$c$$W^{c}$$K^{c}(-t)\tilde{u}(t)$$\vec{u}_{0}$$c$REFERENCES
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Schr\"odinger equations, J. Math. Pures Appl. 64, (1985), 363-401.
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Klein-Gordon equations, preprint.
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${\rm Res}$