Existence Uniqueness Schrodinger Equation Solutions

Funkcialaj Ekvacioj, 23 (1980) 259-277

On Solutions of the Derivative NonlinearSchrodinger Equation.

Existence and Uniqueness Theorem

By

Masayoshi TSUTSUMI and Isamu FUKUDA

(Waseda University and Kokushikan University, Japan)

§1. Introduction

Consider the derivative nonlinear Schrodinger (DNLS) equation of the form

(1.1) $¥frac{¥partial u}{¥partial t}=i¥frac{¥partial^{2}u}{¥partial x^{2}}+¥frac{¥partial}{¥partial x}$$(|u|^{2}u)+g(x, t)$, $(¥chi, f)¥in R^{2}$

which has several applications in plasma physics ([8], [9]). In case $g¥equiv 0$ , Kaup andNewell [5] succeeded to apply the inverse scattering techniques for solving the DNLSequation and obtained the one-soliton solution as well as the infinite family of con-

servation laws. However, the existence, uniqueness and continuous-dependence-on-data of solutions of the initial value problem for the DNLS equation in appropriatefunction spaces are still to be established even when $g¥equiv 0$ .

In this paper we are concerned with two initial value problems for the DNLSequation: the pure Cauchy problem and the initial value problem with periodicboundary conditions; i.e. (1.1), (1.2) or (1.2)’, (1.3):

(1.2) $¥frac{¥partial^{k}u}{¥partial x^{k}}(x, t)¥rightarrow 0$ as $|x|¥rightarrow¥infty$ , $k=0,1,2$, $¥cdots$

(1.2)’ $¥frac{¥partial^{k}u}{¥partial x^{k}}(x+1, t)=¥frac{¥partial^{k}u}{¥partial x^{k}}(x, t)$ for $¥forall(¥chi, f)¥in R^{2}$

$k=0,1,2$,?,

(1.3) $u(x, 0)=u_{0}(x)$, $x$ $¥in R$ .

Our aim here is to establish the uniqueness and local existence of weak, strongand smooth solutions of these problems and to obtain weak global solutions of them.

Subsequent papers will deal with the global existence of strong and smooth solutionsand continuous-dependence-on-data. We employ a technique of parabolic regulari-zation, the $¥mathrm{e}¥mathrm{q}$ . (1.1) being approximated by

260 M. TSUTSUMI and I. FUKUDA

(1.4) $¥frac{¥partial u_{¥epsilon}}{¥partial t}=(i+¥epsilon)¥frac{¥partial^{2}u_{¥epsilon}}{¥partial x^{2}}+¥frac{¥partial}{¥partial x}(|u_{¥epsilon}|^{2}u_{¥epsilon})+g(x, t)$ $(¥epsilon>0)$ .

Note that conserved functionals of (1.1) with $g¥equiv 0$ derived from the conservationlaws are not positive definite except one. Thus we must require smallness of initialvalues and forcing terms in order to obtain global solutions of (1.1).

The content of this paper is as follows. Section 2 contains notations and pre-liminaries. In Section 3, the local existence theorem to the initial value problemsfor the regularized equation (1.4) are discussed. In Section 4, by letting $¥mathrm{e}$ tend tozero in (1.4) we shall establish the local existence theorems to the initial value prob-lem for (1.1), (1.2) or (1.2)’, (1.3). The final section is devoted to the global existencetheorem of weak solutions to (1.1), (1.2) or (1.2)’, (1.3).

§2. Notations and preliminaries

Let $s¥in R$ and $p¥in R$, $p¥geq 0$ . $L^{p}(R)(L^{p}(¥Pi))$ denotes the complex Lebesgue spaceon the real axis $R$ (the one$-¥dim$ . unit torus $¥Pi$) with the norm

(2.1) $|u|_{p}=(¥int_{-¥infty}^{¥infty}|u(x)|^{p}dx)^{1/p}$ or $|u|_{p}=(¥int_{0}^{1}|u(x)|^{p}dx)^{1/p}$

and $L^{¥infty}(R)(L^{¥infty}(¥Pi))$ the complex Lebesgue space with norm

(2.2) $|u|_{¥infty}=¥sup_{x¥in R}|u(x)|$ or $|u|_{¥infty}=¥sup_{x¥in[0,1]}|u(x)|$ .

Let $H^{s}(R)$ denote the complex Sobolev space of order $s$ defined by

$¥{u ¥in L^{2}(R): |¥xi|^{s/2}¥hat{u}(¥xi)¥in L^{2}(R)¥}$

where $¥wedge$ denotes the Fourier transform. $H^{s}(R)$ is a Hilbert space with the norm

(2.3) $||u||_{s}=¥{¥int_{-¥infty}^{¥infty}(1+|¥xi|^{2})^{s}|¥hat{u}(¥xi)|^{2}d¥xi¥}^{1/2}=|(1+|¥xi|^{2})^{s/2}¥hat{u}|_{2}$ .

Similarly, let $H^{s}(¥Pi)$ denote the complex Sobolev space of order $s$, which is the spaceof complex periodic functions with period one:

(2.4) $u(x)=¥sum_{k=-¥infty}^{¥infty}u_{k}¥exp(2¥pi ikx)$

such that

(2.5) $¥{¥sum_{k=-¥infty}^{¥infty}(1+|k|^{2})^{s}|u_{k}|^{2}¥}^{1/2}<+¥infty$ .

$H^{s}(¥Pi)$ is a Hilbert space with the norm defined by the left hand side of (2.5), whichis also denoted by $||u||_{s}$ .

Derivative Nonlinear Schrodinger Equation 261

Below, when there is no chance of confusion, and in order to treat simultane-ously the two initial value problem mentioned above, $L^{p}$ and $H^{s}$ will denote either$L^{p}(R)$ and $H^{s}(R)$ , or $L^{p}(¥Pi)$ and $H^{s}(¥Pi)$, respectively. Other function spaces arestandard and their notations and definitions are omitted, (see Lions [6]). By $c$ wedenote various positive constants which need not be the same throughout.

The following lemmas are essential for our arguments:Lemma 2.1. Let $m$ and $j$ be integers such that $0¥leq j<m$ , $ 0¥leq a<1,1¥leq p<¥infty$

and suppose that

(2.6) $¥frac{1}{p}=¥frac{1}{2}+j-am$ .

Then, for $alfu$ $¥in H^{m}$ , there exists a constant $C_{1}(=C_{1}(p, m, j))$ independent of $u$ ,

so that

(2.7) $|D^{j}u|_{p}¥leq C_{1}¥{|D^{m}u|_{2}+|u|_{2}¥}^{a}|u|^{1-a}$ $(D=¥frac{d}{dx})$ .

Proof. According to Peetre [11], $H^{s}¥subset L^{p}¥mathrm{i}¥mathrm{f}s=1/2-1/p$ $(¥geq 0)$ . Hence

(2.8) $|D^{j}u|_{p}¥leq c||D^{j}u||_{¥theta}¥leq c||u||_{am}$

where $c$ denotes various positive constants independent of $¥mathrm{v}$ . On the other hand,with notation of Lions-Magenes [7], chapter 1, we have

$[H^{m}, H^{0}]_{1-a}=H^{am}$ $(H^{0}=L^{2})$ .

Hence, we obtain

(2.9) $|D^{j}u|_{p}¥leq c||v||_{am}¥leq c||u||_{m}^{a}|u|_{2}^{1- a}$ .

Since $||u||_{m}$ and $|D^{m}u|_{2}+|u|_{2}$ are equivalent norms, (2.9) implies (2.7). Q.E.D.

Lemma 2.2. For all $¥in H^{1}$ , there exists a constant $C_{2}$ independent of $u$, so that

(2. 10) $|u|_{¥infty}¥leq C_{2}¥{|Du|_{2}+|u|_{2}¥}^{1/2}|u|_{2}^{1/2}$ .

Proof of Lemma 2.2 is found in Temam [16].

Lemma 2.3. Let $s$ and $r$ be real numbers such that $s>1$ and $r>1/2$ . For $alf$

$u¥in H^{¥max(s,r+1)}$ , $v¥in H^{¥max(s,r)}$ , there exists a constant $C(s, r)$ independent of $u$ and $v$ sothat

(2. 11) $||uu||_{s}¥leq C(s, r)(||u||_{s}||u||_{r}+||u||_{¥gamma}||u||_{s}+||u||_{1+r}||v||_{s-1})$ .


Corollary of Lemma 2.3. If $s>3/2$, then, for u, u $¥in H^{s}$

$||uu||_{s}¥leq c(s)||u||_{s}||u||_{s}$ .

Thus, $H^{s}$ if a Banach algebra.

Proof ofLemma 2.3. Since in case of periodic functions, the proof is a discreteversion of the one for functions on the entire real axis, we only prove (2.11) in caseof the Sobolev spaces on $R$ .

We have for $u$, $u¥in¥ovalbox{¥tt¥small REJECT}(R)$

$||uv||_{s}=¥{¥int_{-¥infty}^{¥infty}(1+|¥xi|^{2})^{s}|¥int_{-¥infty}^{¥infty}¥hat{u}(¥xi-¥eta)¥hat{v}(¥eta)d¥eta|^{2}d¥xi¥}^{1/2}$

$=¥{¥int_{-¥infty}^{¥infty}|¥int_{-¥infty}^{¥infty}(1+|¥xi|^{2})^{s/2}¥hat{u}(¥xi-¥eta)¥hat{v}(¥eta)d¥eta|d¥xi¥}^{1/2}$

On the other hand, we get

$|(1+|¥xi|^{2})^{s/2}-(1+|¥eta|^{2})^{s/2}|=|¥int_{0}^{1}¥frac{¥partial}{¥partial¥theta}(1+|¥eta+¥theta(¥xi-¥eta)|^{2})^{s/2}d¥theta|$

$¥leq c(s)[(1+|¥eta|^{2})^{(s-1)/2}(1+|¥xi-¥eta|^{2})^{1/2}+(1+|¥xi-¥eta|^{2})^{s/2}]$ .

Hence, by Hausdorff-Young’s inequality, we have

(2. 12) $||uu||_{s}¥leq c(s)[||u||_{s}¥int_{-¥infty}^{¥infty}|¥hat{v}(¥xi)|d¥xi+||v||_{s}¥int_{-¥infty}^{¥infty}|¥hat{u}(¥xi)|d¥xi$

$+||u||_{s-1}¥int_{-¥infty}^{¥infty}(1+|¥xi|^{2})^{1/2}|¥hat{u}(¥xi)|d¥xi]$.

Schwarz’s inequality yields

(2. 13) $¥int_{-¥infty}^{¥infty}|¥hat{u}(¥xi)|d¥xi=¥int_{-¥infty}^{¥infty}(1+|¥xi|^{2})^{r/2}|¥hat{u}(¥xi)|(1+|¥xi|^{2})^{-r/2}d¥xi$

$¥leq c(r)||u||_{r}$ , $r>¥frac{1}{2}$ .

From (2. 12) and (2. 13), we obtain (2. 11). Q.E.D.

Let $A=-¥partial^{2}/¥partial x^{2}$ with domain $D(¥mathrm{A})=H^{2}$ . $A$ is nonnegative selfadjoint operatorin $L^{2}$ . We define the fractional power $A^{s}$ of order $s$ $(s ¥in R, s¥geq 0)$ by

(2.14) $A^{s}u=¥swarrow^{¥varpi-1}(|¥xi|^{2S}¥swarrow^{¥varpi}u)$ if $u¥in H^{2s}(R)$ ,

(2.15) $A^{s}u=¥sum_{k=-¥infty}^{¥infty}|k|^{2s}u_{k}¥exp(2i¥pi kx)$ if $u¥in H^{2S}(¥Pi)$ ,

where $¥ovalbox{¥tt¥small REJECT}$ denotes the Fourier transform. The following lemma isadirect consequ-ence of definitions.

The following lemma is a direct consequ-


Lemma 2.4. 1) A $s_{l}+s_{¥mathit{2}}u=A^{S_{l}}A^{S_{¥mathit{2}}}u$ for $u¥in H^{s_{1}+S2}(s_{1}, s_{2}¥geq 0)$ . 2) $A^{s}Du=DA^{s}u$,

for $u¥in H^{s+1}$ . 3) $[u]_{s}=(|A^{s/2}u|_{2}^{2}+|u|_{2}^{2})^{1/2}$ and $||u||_{s}$ are equivalent norms.

The next lemma is due to Saut and Temam [12].

Lemma 2.5. Let $s$ and $r$ be real numbers such that $s>1/2$ and $r>1/2$. For $aff$

$u$, $v¥in D(A_{s})(=H^{2s})¥cap H^{r+1}$

(2. 16) $|$ A $S(uv)-uA^{s}u|_{¥mathit{2}}¥leq c(s, r)(||u||_{2s}||u||_{r}+||u||_{r+1}||v||_{2s1}¥_)$.

Proof of Lemma 2.5 is similar to that of Lemma 2.3.

Lemma 2.6. Let $¥sigma(t)$ and $¥phi(t)$ be two nonnegative measurable functions definedon $[0, T]$ $(T>0)$ such that $¥phi(t)¥sigma(t)$ is integrable on $[0, T]$ . Let $a$ be nonnegative con-

stant. If$¥sigma(t)^{2}¥leq a^{2}+2¥int_{0}^{t}¥phi(s)¥sigma(s)ds$ $(0¥leq t¥leq T)$ ,

then, we have

$¥sigma(t)¥leq a+¥int_{0}^{t}¥phi(s)ds$ $(0¥leq t¥leq T)$ .

Proof is easy and omitted.

§3. Parabolic regularization

Let

(3.1) $A=-¥frac{¥partial^{2}}{¥partial x^{2}}$

with domain

(3.2) $D(¥mathrm{A})=H^{2}(R)$ (or $H^{2}(¥Pi)$).

Then, $-iA$ is a skew selfadjoint operator in $L^{2}(R)$ (or $L^{2}(¥Pi)$) and is a generator ofone parameter family of strong continuous unitary group.

Define

(3.3) $J_{0}u=¥frac{¥partial}{¥partial x}$ $(|u|^{2}u)$ for $u¥in H^{1}$ .

Inclusion $H^{1}¥subset L^{¥infty}$ yields that $J_{0}$ is a nonlinear operator from $H^{1}$ into $L^{2}$ (and from$H^{s}$ into $H^{s-1}$ , $s>1$ ). We can rewirte (1.1), (1.2) or (1.2)’, (1.3) as

(3.4) $u_{t}=-iAu+J_{¥mathit{0}}u+g$ ,


(3.5) $u(0)=u_{0}$ .

Since the nonlinear term $J_{0}u$ is very singular, Segal’s theory of nonlinear semi-groups [13] does not seem applicable to (3.4), (3.5) (Note that $J_{0}$ is not continuousfrom $H^{s}$ into $H^{s}$ for any $s¥geq 0$).

We now consider the parabolic regularizations to (3.4), (3.5);

(3.6) $u_{t}=-A_{¥epsilon}u+Ju+g$,

(3.7) $u(0)=u_{0}$ ,

where

(3.8) $A_{¥epsilon}u=(i+¥epsilon)Au+u$, $¥epsilon>0$

with $D(A_{¥epsilon})=D(¥mathrm{A})$ and

(3.9) $Ju=J_{0}u-u$.

It is easily seen that $¥mathrm{A}_{¥epsilon}$ is an $¥mathrm{m}$-sectorial operator with a vertex 0 and a semi-angle $(¥pi/2-¥arctan¥epsilon)$ (for definition, see Kato [4]) and hence $A_{¥epsilon}$ generates holomor-phic semi-groups. We can define the fractional power $A_{¥mathrm{e}}^{a}$ of $A_{¥epsilon}$ for $¥alpha¥in R$, $¥alpha>0$ by

(3. 10) $¥mathrm{A}_{¥epsilon}^{a}u=¥ovalbox{¥tt¥small REJECT}^{-1}P[¥{(i+¥epsilon)¥xi^{2}+1¥}^{¥alpha}]¥ovalbox{¥tt¥small REJECT} u$

with $D(¥mathrm{A}_{¥mathrm{e}}^{¥alpha})=H^{2¥alpha}(R)$ $(=D(¥mathrm{A}^{¥alpha}))$ or

(3.11) $A_{¥mathrm{e}}^{a}u=¥sum_{k=-¥infty}^{¥infty}u_{k}P[¥{(i+¥epsilon)k^{2}+1¥}^{¥alpha}]¥exp(2i¥pi kx)$

with $D(A_{¥epsilon}^{¥alpha})=H^{2a}(¥Pi)$ $(=D(A^{¥alpha}))$ where $P[z^{a}]$ denotes the principal value of $¥mathrm{z}^{¥alpha}$ . Then,from the well-established theory of semi-linear parabolic equations (for example,Sobolevskii [14], Asano [1]$)$ , we can obtain a local existence theorem of strict solu-tions to the initial value problems (3.6), (3.7) provided that $J$ satisfies a condition:there exists real numbers $¥alpha$ , $¥beta$ with $0¥leq¥alpha<1$ , $¥alpha¥leq¥beta¥leq 1$ such that $Ju$ is defined for$¥forall u¥in D(A_{¥epsilon}^{¥beta})$ and

(3.12) $|Ju-Ju|_{2}¥leq h$( $|$ A $¥epsilon¥rho u|+|¥mathrm{A}_{¥mathrm{g}}^{¥beta}v|_{2}$) $|$ $¥mathrm{A}_{¥mathrm{g}}^{¥alpha}u-A_{¥epsilon}^{a}v|_{2}$ ,

where $h$ is a positive nondecreasing function defined on [0, $¥infty$ ).

Theorem3.1. Let $1/2¥leq¥alpha<1$ and $T>0$ . Let $g(t)(0¥leq t¥leq T)$ be Hofder continu-ous in $L^{¥mathit{2}}$ -norm with Hofder exponent $(1-¥alpha)$ and $fetu_{0}¥in D(A_{¥epsilon}^{¥alpha})$ . Then for every $¥epsilon>0$

$(3.6)$ , (3.7) has a unique local strict sofution

(3.13) $u¥in C([0, T_{0}]:D(¥mathrm{A}_{¥epsilon}^{a}))¥cap C^{1}((0, T_{0}]:L^{2})¥cap C((0, T_{0}]:D(A_{¥epsilon}))$

where $T_{0}(0<T_{0}¥leq T)$ is some constant depending only on $|¥mathrm{A}_{6}^{a}u_{0}|_{2}$ .


Remark 3.1. $T_{0}=T_{0}(|¥mathrm{A}_{¥epsilon}^{a}u_{0}|_{¥mathrm{z}})$ is a monotone decreasing $¥mathrm{f}¥mathrm{u}¥mathrm{n}¥mathrm{c}¥mathrm{t}¥tilde{¥mathrm{i}}¥mathrm{o}¥mathrm{n}$ of $|A_{¥epsilon}^{¥alpha}u_{0}|_{2}$ .

Theorem 3.2. If $g¥in C^{¥infty}([0, T]:D(¥mathrm{A}^{¥infty}))$ and $u_{0}¥in D(A^{¥infty})$ , then

(3.14) $u¥in C^{¥infty}([0, T_{0}]:D(¥mathrm{A}^{¥infty}))$ ,

where we denote $D(A^{¥infty})=¥bigcap_{a}D(A^{a})$ .

Remark 3.2. $D(¥mathrm{A}^{¥infty})=H^{¥infty}=¥bigcap_{s}H^{s}$

Proof of Theorem 3.1. It suffices to show that $J$ satisfies (3. 12). Using Lemma2.2, we have

(3.15) $|Ju-Ju|_{2}¥leq|u^{2}(¥overline{u}_{x}-¥overline{u}_{x})|_{2}+|u_{x}(u+u)(u-u)|_{2}$

$+2||u|^{2}(u_{x}-u_{x})|_{2}+2|(|u|^{2}-|v|^{2})u_{x}|_{2}+|u-u|_{2}$

$¥leq h(||u||_{1}+||v||_{1})||u-u||_{1}$

which implies (3.12) since

(3.16) $||u||_{1}¥leq c|A_{¥epsilon}^{¥alpha}u|_{2}$ for $¥alpha¥geq¥frac{1}{2}$ . Q.E.D.

Prior to the proof of Theorem 3.2, we state the following lemma.

Lemma 3.1. Let $w(t)¥in D(¥mathrm{A}_{¥epsilon}^{a})$ and $A_{¥epsilon}^{¥alpha}w(t)$ be continuous in $[0, T]$ for some $¥alpha>0$.Put

(3.17) $ u(t)=¥int_{0}^{t}e^{-(t-¥tau)A_{¥epsilon}}w(¥tau)d¥tau$.

Then, for any $¥beta$ with $0<¥beta<1$ , $v(t)¥in D(A_{¥mathrm{e}}^{¥alpha+¥beta})$ for $t¥in[0, T]$ and

(3.18) $A_{¥epsilon}^{a+¥rho}u(t)=¥int_{0}^{t}A_{¥mathrm{e}}^{¥beta}e^{-(t-¥tau)A_{¥epsilon}}¥mathrm{A}_{¥epsilon}^{a}w(¥tau)d_{T}$ $(0¥leq t¥leq T)$ .

Moreover $A_{¥epsilon}^{a+¥beta}u(t)$ is Hofder continuous in $[0, T]$ with exponent $(1-¥alpha)$ .

The proof of Lemma 3.1 may be found in [1].

Proof of Theorem 3.2. From (3.6), (3.7), we have

(3.19) $¥mathcal{U}(f)=e^{-A_{¥epsilon}t}u_{0}+¥int_{0}^{t}e^{-(t-¥tau)A_{¥epsilon(Ju)(¥tau)d_{T}+¥int_{0}^{t}e^{-(t-¥tau)A_{6}}g(¥tau)d¥tau}}$ .

Assumptions on $u_{0}$ and $g$ imply that the first and third terms belong to $C^{¥infty}([0, T]$ :$D(A_{¥epsilon}^{¥infty}))$ . Note that $u¥in C([0, T_{0}]:D(A_{¥epsilon}^{a}))$ for $¥forall¥alpha$ with $1/2¥leq¥alpha<1$ . If we take $¥alpha$ sothat $3/4<¥alpha<1$ , then in virtue of Lemma 2.3 and its corollary, we see that $(Ju)(t)$

$¥in D(A_{¥mathrm{e}}^{¥alpha-1/2})$ and $(A_{¥epsilon}^{¥alpha-1/2}Ju)(t)$ is continuous on $[0, T_{0}]$ . Indeed, we have


$|A_{¥mathrm{e}}^{a-1/2}(Ju(t)-Ju(t^{¥prime}))|_{2}¥leq||$ $(|u|^{2}u)(t)-(|u|^{2}u)(t^{¥prime})||_{2a}+|A_{¥epsilon}^{a-1/2}(u(t)-u(t^{¥prime}))|_{2}$

$¥leq C(¥alpha)(||u(t)||_{2a}^{2}+||u(t^{¥prime})||_{2a}^{2})||u(t)-u(t^{¥prime})||_{2a}+|A_{¥epsilon}^{a}(u(t)-u(t^{¥prime}))|_{2}$

$¥leq C^{¥prime}(¥alpha)¥max_{0¥leq t¥leq T_{0}}|$$¥mathrm{A}_{8}^{a}u(t)|_{2}^{2}|¥mathrm{A}_{¥mathrm{g}}^{a}(u(t)-u(t^{¥prime}))|_{2}$

since $|¥mathrm{A}_{¥mathrm{e}}^{a}u|_{2}$ and $||u||_{2a}$ are equivalent norms in $H^{2¥alpha}$ . Then, in virtue of (3.19) fromLemma 3.1 we conclude that for any $¥beta¥in(0,1)$

(3.20) $u¥in C([0, T_{0}]:D(A_{¥epsilon}^{a+¥beta-1/2}))$ .

In much the same way as above, we see that $(Ju)(t)¥in D(A_{¥epsilon}^{a+¥beta-1})$ and $A_{¥epsilon}^{a+¥beta-1}Ju(t)$ iscontinuous on $[0, T]$ . Then, applying Lemma 3.1, for any $¥beta^{¥prime}¥in(0,1)$ we have

(3.21) $u¥in C([0, T_{0}]:D(A_{¥epsilon}^{a+¥beta+¥beta^{J}-1}))$ .

Repeating this argument yields

(3.22) $u¥in C([0, T_{0}]:D(A_{¥epsilon}^{¥infty}))$ .

Then, in virtue of the $¥mathrm{e}¥mathrm{q}$ . (3.6) we obtain

(3.23) $u_{t}¥in C([0, T_{0}]:D(¥mathrm{A}_{¥mathrm{e}}^{¥infty}))$.

Further regularities with respect to $t$ follows immediately from (3.22) and (3.23).Thus, we have the theorem.

Remark 3.3. The assertions mentioned in Theorem 3.1 and Theorem 3.2 holdvalid for more general nonlinear terms $Ju$ satisfying (3.12) and

(3.24) $|A_{¥epsilon}^{a-1/2}(Ju-Ju)|_{2}¥leq h_{1}(|¥mathrm{A}_{¥epsilon}^{a}u|_{2}+|¥mathrm{A}_{¥mathrm{g}}^{a}u|_{2})|A_{¥epsilon}^{¥alpha}(u-u)|_{2}$

for any $¥alpha>3/4$, where $h_{1}$ is a positive nondecreasing function defined on [0, $¥infty$ ).For example, $Ju=¥partial/¥partial x$ $(|u|^{p}u)$ , $p¥geq 1$ , satisfies these conditions.

§4. Local existence theorem

$D(A^{s})(s¥geq 0)$ is a Hilbert space equipped with norm $[u]_{2s}$ .

Theorem 4.1. Let $¥alpha$ be any real number $>3/4$ and $T>0$ . Let $u_{0}¥in D(¥mathrm{A}^{a})$ and$g¥in L^{1}(0, T:D(¥mathrm{A}^{a}))$ . Then, there exists a positive constant $T_{1}$ with $0<T_{1}<T$ such thatthe initial value problem (3.4), (3.5) has a unique solution $u=u(t)$ in the interval $[0, T_{1}]$

such that

(4. 1) $u¥in L^{¥infty}(0, T_{1} : D(A^{a}))¥cap C([0, T_{1}]:D(¥mathrm{A}^{a-1}))$

with


(4.2) $u_{t}¥in L^{1}(0, T_{1} : D(¥mathrm{A}^{a-1}))$ .

$T_{1}$ depends only on $[u_{0}]_{¥gamma}$ and $¥int_{0}^{T}[g(t)]_{¥gamma}dt$ , where $¥gamma$ is an arbitraryfixed number such that$ 3/2<¥gamma<2¥alpha$ . Moreover, for each bounded set of the form

$¥{¥{u, g¥}:[u]_{¥gamma}¥leq a$, $¥int_{0}^{T}[g(t)]_{¥gamma}dt¥leq b$ , $¥frac{3}{2}<¥gamma<2¥alpha¥}$

$T_{1}$ can be chosen uniformly for $afl$ $¥{u_{0}, g¥}$ in the set.

Remark 4.1. Due to Strauss [15], $u(t)$ is weakly continuous on $[0, T_{1}]$ withvalues in $D(A^{a})$ .

The following corollary is immediate.

Corollary of Theorem 4.1. In addition to the assumptions in Theorem 4.1, if weassume that $¥alpha¥geq 1$ and $g¥in L^{1}(0, T:D(¥mathrm{A}^{a}))¥cap C([0, T]:D(¥mathrm{A}^{a-1}))$ , then (3.4), (3.5) has $a$

continuously differentiable {strong) solution for $t¥in[0, T_{1}]$ . If $u_{0}¥in D(A^{¥infty})$ and $g¥in C^{¥infty}$

$([0, T]:D(A^{¥infty}))$, then $u¥in C^{¥infty}([0, T_{1}]:D(A^{¥infty}))$ .

Proof of Theorem 4. 1. The proof of uniqueness is simple and omitted. Since$D(A^{¥infty})$ is dense in $D(A^{a})=H^{2a}$ , we find sequences $¥{u_{0¥mathrm{e}}¥}¥subset D(A^{¥infty})$ and $¥{g_{¥epsilon}¥}¥subset C^{¥infty}([0, T]$ :$D(A^{¥infty}))$ such as $¥epsilon¥searrow 0$

(4.3) $u_{0¥mathrm{e}}¥rightarrow u_{0}$ strongly in $D(¥mathrm{A}^{a})$

and

(4.4) $g_{¥epsilon}$? $g$ strongly in $L^{1}(0, T:D(A^{a}))$ .

Let $u_{¥epsilon}(t)$ be a solution of the initial value problem

(4.5) $(u_{¥epsilon})_{t}=-A_{¥epsilon}u_{¥mathrm{e}}+Ju_{¥epsilon}+g_{¥epsilon}$,

(4.6) $u_{¥epsilon}(0)=u_{0¥epsilon}$ .

Then, Theorem 3.2 yields $u_{¥epsilon}¥in C^{¥infty}([0, T_{0}]:D(A^{¥infty}))$.Taking the inner product of (4.5) with $u_{¥mathrm{e}}$ in $L^{2}$ and adding the complex conju-

gation of the produce, we find

(4.7) $¥frac{d}{dt}|u_{¥epsilon}|_{2}^{2}+2¥epsilon|A^{1/2}u_{¥epsilon}|_{2}^{2}=2¥mathrm{R}¥mathrm{e}(g_{¥epsilon}, u_{¥epsilon})¥leq 2|g_{¥mathrm{e}}|_{2}|u_{¥epsilon}|_{2}$.

Using Lemma 2.6, from (4.7) we have for $0<¥forall t<T_{0}$

(4.8) $|u_{¥epsilon}(t)|_{2}¥leq|u_{0¥epsilon}|_{2}+¥int_{0}^{T}|g_{¥mathrm{e}}(¥tau)|_{2}d¥tau$


and

(4.9) $ 2¥epsilon¥int_{0}^{t}|¥mathrm{A}^{1/2}u_{¥epsilon}(¥tau)|_{2}^{2}d_{T}¥leq|u_{0¥epsilon}|_{2}^{2}+2(|u_{0¥epsilon}|^{2}+¥int_{0}^{T}|g_{¥epsilon}(¥tau)|_{2}d¥tau)¥int_{0}^{T}|g_{¥epsilon}(¥tau)|_{2}d¥tau$.

Hence $¥{u_{¥epsilon}¥}$ and $¥{¥sqrt{¥epsilon}A^{1/2}u_{¥epsilon}¥}$ remain bounded in $L^{¥infty}(0, T_{0} : L^{2})$ and $L^{2}(0, T_{0} : L^{2})$ , re-spectively, that is,

(4. 10) $||u_{¥epsilon}||_{L^{¥infty}(0,T_{0}:L2)}¥leq c_{1}$ ,

(4. 11) $¥vee¥overline{¥epsilon}||$$¥mathrm{A}^{1/2}u_{¥mathrm{e}}||_{L2(0,T¥mathrm{o}:L2)}¥leq c_{2}$ ,

where and in the sequel of this section, $c_{k}(k=1,2,3, -)$ and $c$ denote variouspositive constants independent of $¥epsilon$ in any finite interval $[0, _{¥epsilon_{0}}]$ .

Applying $A^{s}$ the both sides of (4.5), taking the inner product of the resultantequation with $A^{s}u_{¥epsilon}$ in $L^{2}$ and adding the complex conjugation of the produce, we have

(4. 12) $¥frac{d}{dt}|$ A $su_{¥epsilon}|_{2}^{2}+2¥epsilon|A^{S+1/2}u_{¥epsilon}|_{2}^{2}=2¥mathrm{R}¥mathrm{e}$ ($¥mathrm{A}^{s}Ju_{¥epsilon}$ , A $Su_{¥epsilon}$ ) $+2¥mathrm{R}¥mathrm{e}(A^{s}g_{¥epsilon}, A^{s}u_{¥epsilon})$ .

We rewrite the first term of the right hand side of (4. 10) in the form

(4.13) $4¥mathrm{R}¥mathrm{e}(¥mathrm{A}^{s}(|u_{¥epsilon}|^{2}Du_{¥xi})-|u_{¥epsilon}|^{2}A^{s}Du_{¥epsilon}, A^{s}u_{¥xi})+4¥mathrm{R}¥mathrm{e}(|u_{6}|^{2}¥mathrm{A}^{s}Du_{¥epsilon}, A^{s}u_{¥epsilon})$

$+2¥mathrm{R}¥mathrm{e}$ ($¥mathrm{A}^{s}(u_{¥epsilon}^{2}Du_{¥epsilon})-u_{¥mathrm{g}}^{2}¥mathrm{A}^{s}Du_{¥epsilon}$, A $S¥overline{u}_{¥epsilon}$ ) $+2¥mathrm{R}¥mathrm{e}$ ($u_{¥epsilon}^{2}¥mathrm{A}^{s}$Du $¥epsilon’ A^{s}u_{¥epsilon}$).

Let $s>1/2$ . Then, in virtue of Lemma 2.3-2.5, the first and second terms arebounded $¥mathrm{b}¥mathrm{y}^{-}$

(4. 14) $c[||u_{¥epsilon}^{2}||_{2s}|| Du_{¥epsilon}||_{¥gamma}+||u_{¥epsilon}^{2}||_{r+1}|| Du_{6}||_{2s-1}]$ $|A^{s}u_{¥epsilon}|_{2}$

$¥leq c[(2||u_{¥epsilon}||_{2s}||u_{¥epsilon}||_{r}+||u_{¥epsilon}||_{2s-1}||u_{¥epsilon}||_{r+1})||u_{¥epsilon}||_{r+1}+||u_{¥epsilon}||_{r+1}||u_{¥epsilon}||_{2s}]|$A $su_{¥epsilon}|_{2}$

$¥leq c||u_{¥epsilon}||_{r+1}^{2}[u_{¥epsilon}]_{2s}^{2}$

for any $r>1/2$ . Since A $s$ and $D$ commute, integration by parts gives

(4. 15) $4¥mathrm{R}¥mathrm{e}$ ( $|u_{¥epsilon}|^{2}¥mathrm{A}^{s}Du_{¥epsilon}$ , A $Su_{¥epsilon}$ ) $=-2(A^{s}u_{¥epsilon}D|u_{¥epsilon}|^{2}, A^{s}u_{¥epsilon})$

which is bounded by

(4.16) $c|u_{¥epsilon}|_{¥infty}|Du_{¥epsilon}|_{¥infty}|¥mathrm{A}^{S}u_{¥epsilon}|_{2}^{2}¥leq c||u_{¥epsilon}||_{r+1}^{2}[u_{¥epsilon}]_{2s}^{2}$, $r>¥frac{1}{2}$ .

Here we have used the fact that $H^{r}¥subset L^{¥infty}$ if $r>1/2$ . Analogously the fourth term of(4. 13) is majorized by $c||u_{¥epsilon}||_{r+1}^{2}[u_{¥epsilon}]_{2s}^{2}$ . Thus we have

(4. 17) $¥frac{d}{dt}|$ A $Su_{¥epsilon}|_{2}^{2}+2¥epsilon|$ A $s+1/2u_{¥epsilon}|_{2}^{2}¥leq c||u_{¥epsilon}||_{r+1}^{2}[u_{¥epsilon}]_{s}^{2}+2|A^{s}g_{¥epsilon}|_{2}|$ A $su_{¥epsilon}|_{2}$ .


Combining (4.7) and (4. 17), we get

(4. 18) $¥frac{d}{dt}[u_{¥epsilon}]_{2s}^{2}+2¥epsilon[u_{¥epsilon}]_{2s+1}^{2}¥leq c||u_{¥epsilon}||_{r+1}^{2}[u_{¥epsilon}]_{2s}^{2}+2[g_{¥mathrm{e}}]_{2s}[u_{¥mathrm{e}}]_{2s}$ .

Since $s$ and $r$ are arbitrary numbers such that $s>1/2$, $r>1/2$, we can choose $s$ and $r$

in (4. 18) so that $ 2s=r+1=¥gamma<2¥alpha$ . Then we obtain

(4. 19) $¥frac{d}{dt}[u_{¥epsilon}]_{¥gamma}^{2}+2¥epsilon[u_{¥epsilon}]_{¥gamma+1}^{2}¥leq c[u_{¥epsilon}]_{r}^{4}+2[g_{¥epsilon}]_{¥gamma}[u_{¥epsilon}]_{¥Upsilon}$

from which it follows that

(4.20) $¥frac{d}{dt}[u_{¥epsilon}]_{¥gamma}¥leq c[u_{¥epsilon}]_{¥gamma}^{3}+[g_{¥epsilon}]_{¥gamma}$.

Integrating both sides of (4.20) from 0 to $t$, we get

(4.21) $[u_{¥epsilon}(t)]_{¥gamma}¥leq[u_{0¥epsilon}]_{¥gamma}+¥int_{0}^{t}[g_{¥epsilon}(¥tau)]_{¥gamma}d_{T}+c¥int_{0}^{t}[u_{¥epsilon}(¥tau)]_{¥gamma}^{3}d¥tau$ .

In virtue of Opial’s inequality, the solution $[u_{¥epsilon}(t)]_{¥gamma}$ of the integral inequality (4.21)can be majorized by the solution of

(4.22) $ y(t)=¥eta+c¥int_{0}^{t}y^{3}(¥tau)d¥tau$

with

(4.23) $¥eta=[u_{0¥epsilon}]_{¥gamma}+¥int_{0}^{T}[g_{¥epsilon}(¥tau)]_{¥gamma}d¥tau$ .

The solution $y(t)$ of (4.22) exists only for $0¥leq t<T_{*}=1/2c¥eta^{2}$ . Hence, $¥{u_{¥epsilon}¥}$ remainsbounded in $L^{¥infty}(0, T_{1} : H^{¥gamma})$ as $¥epsilon¥searrow 0$ provided that we choose $T_{1}$ so that $T_{1}<¥min(T_{0}$ ,$T_{*})$ . Thus, we have

(4.24) $||u_{¥epsilon}||_{L^{¥infty}(0,T_{1}:H¥gamma)}¥leq c_{3}$ .

Integration of (4. 19) with respect to $t$ yields

(4.25) $¥vee¥overline{¥epsilon}||u_{¥epsilon}||_{L2(0,T_{1}:Hr+1)}¥leq c_{4}$.

We next choose $s$ and $r$ in (4. 18) so that $ s=¥alpha$, $ r+1=¥gamma$ . In virtue of (4.24) weobtain

(4.26) $¥frac{d}{dt}[u_{¥epsilon}]_{2a}^{2}+2¥epsilon[u_{¥mathrm{e}}]_{2a+1}^{2}¥leq c^{¥prime}[u_{¥epsilon}]_{2a}^{2}+[g_{¥epsilon}]_{2a}^{2}$ for $0¥leq t¥leq T_{1}$ ,


where $c^{¥prime}$ is a constant depending on $c_{3}$ , but independent of $¥epsilon$ . Gronwall’s inequalitygives

(4.27) $||u_{¥epsilon}||_{L^{¥infty}(0,T_{1}:D(A^{a}))}¥leq c_{5}$

from which it immediately follows that

(4.28) $||A_{¥epsilon}u_{¥epsilon}||_{L^{¥infty}(0,T_{1}:D(Aa-1))}¥leq c_{6}$

and

(4.29) $||Ju_{¥epsilon}||_{L^{¥infty}(0,T_{1}:D(Aa-1/2))}¥leq c_{7}$ .

In view of the $¥mathrm{e}¥mathrm{q}$ . (4.5), we get

(4.30) $||(u_{¥epsilon})_{t}||_{L^{1}(0,T:D(A^{a-1))}}¥leq c_{8}$.

We now show the convergence of the approximate solutions $¥{u_{¥epsilon}(t)¥}$ . Set $w(t)$

$=u_{¥epsilon}(t)-u_{¥delta}(t)$ . We have

(4.31) $w_{t}=A_{¥epsilon}w+Ju_{¥epsilon}-Ju_{¥delta}+g_{¥epsilon}-g_{¥delta}+(¥epsilon-¥delta)Au_{¥delta}$.

Taking the $¥mathrm{L}^{2}$-inner product of (4.31) with $w$ and adding the complex conjugation ofthe produce, we get

(4.32) $¥frac{d}{dt}|w|_{2}^{2}+2¥epsilon|A^{1/2}w|_{2}^{2}=2¥mathrm{R}¥mathrm{e}(Ju_{¥epsilon}-Ju_{¥delta}, w)+2¥mathrm{R}¥mathrm{e}(g_{¥epsilon}-g_{¥delta}, w)$

$+2(¥epsilon-¥delta)¥mathrm{R}¥mathrm{e}(A^{1/2}u, A^{1/2}w)$.

Integration by parts yields

$2¥mathrm{R}¥mathrm{e}(Ju_{¥epsilon}-Ju_{¥delta}, w)=(wD(|u_{¥epsilon}|^{2}+|u_{¥delta}|^{2}), w)+¥mathrm{R}¥mathrm{e}(¥overline{w}D(u_{¥epsilon}u_{¥delta}), w)$


$|2¥mathrm{R}¥mathrm{e}(Ju_{¥epsilon}-Ju_{¥delta}, w)|¥leq(2|u_{¥epsilon}|_{¥infty}|Du_{¥xi}|_{¥infty}+2|u_{¥delta}|_{¥infty}| Du_{¥delta}|_{¥infty}+|Du_{¥epsilon}|_{¥infty}|u_{¥delta}|_{¥infty}+|u_{¥epsilon}|_{¥infty}|Du_{¥delta}|_{¥infty})|w’|_{2}^{2}$

$¥leq c(||u_{¥epsilon}||_{2a}^{2}+||u_{¥delta}||_{2a}^{2})|w|_{2}^{2}$

since $H^{s}¥subset L^{¥infty}$ if $s>1/2$ . Hence we have for $0¥leq t¥leq T_{1}$

$|2¥mathrm{R}¥mathrm{e}(Ju_{¥mathrm{a}}-Ju_{¥delta}, w)|¥leq c^{¥prime¥prime}|w|_{2}^{2}$,

where $c^{¥prime¥prime}$ depends only on $c_{5}$ . Therefore, from (4.32) we deduce

(4.33) $¥frac{d}{dt}|w|_{2}^{2}¥leq c^{¥prime¥prime}|w|_{2}^{2}+2|g_{¥epsilon}-g_{¥delta}|_{2}|w|_{2}+2c^{¥prime¥prime¥prime}(¥epsilon+¥delta)$,

where $c^{¥prime¥prime¥prime}$ depends only on $c_{5}$ . Multiplying both sides of (4.33) by $c^{-c^{¥prime¥prime}t}$ and integra-


ting on (0,$ $i) $(0<t<T_{1})$, we get

$|w(t)|_{2}^{2}¥leq e^{c^{¥prime¥prime}T}(|w(0)|_{2}^{2}+2c^{¥prime¥prime¥prime}(¥epsilon+¥delta)T_{1}/c^{¥prime¥prime})+2e^{c^{¥prime¥prime}T}¥int_{0}^{t}|g_{¥epsilon}(¥tau)-g_{¥delta}(¥tau)|_{2}|w(¥tau)|_{2}d¥tau$.

Then, lemma 2.6 yields

(4.34) $|w(t)|_{2}¥leq c(|u_{0¥epsilon}-l¥mathit{1}_{0¥delta}|+¥epsilon+¥delta+¥int_{0}^{T}|g_{¥epsilon}(¥tau)-g_{¥delta}(¥tau)|_{2}d¥tau)$ for $0¥leq t¥leq T_{1}$ ,

which implies that $¥{u_{¥epsilon}¥}$ is a Cauchy sequence in $C([0, T_{1}]:L^{2})$ . Thus, there exists afunction $u(x, t)$ such that as $¥epsilon¥searrow 0$

(4.35) $u_{¥epsilon}$? $u$ strongly in $C([0, T_{1}]:L^{2})$ .

Furthermore, there exists a subsequence of $¥{u_{¥epsilon}¥}$ (also denoted by $¥{u_{¥epsilon}¥}$) such that

(4.36) $u_{¥epsilon}$? $u$ $¥mathrm{a}.¥mathrm{e}$ . in $[0, T_{1}]¥times R$ .

From the estimate (4.27)?(4.29), we may assume that

(4.37) $u_{¥epsilon}$? $u$ weakly star in $L^{¥infty}(0, T_{1} : D(A^{a}))$ ,

(4.38) $A_{¥epsilon}u_{¥epsilon}$? $iAu$ weakly star in $L^{¥infty}(0, T_{1} : D(A^{a-1}))$

and

(4.39) $Ju_{¥epsilon}¥rightarrow Ju$ wealy star in $L^{¥infty}(0, T_{1} : D(¥mathrm{A}^{a-1/2}))$ .

Taking the inner product of (4.5) with $u¥in D(¥mathrm{A}_{¥epsilon}^{*})=D(A)=H^{2}$ and integratingon $(0, t)$ , we find

$(u_{¥epsilon}(t), u)=(u_{0¥epsilon}, v)-¥int_{0}^{t}(u_{¥epsilon}(¥tau), A_{¥xi}^{*}u)d_{T}+¥int_{0}^{t}((Ju_{¥epsilon})(¥tau), u)d_{T}+¥int_{0}^{t}(g_{¥epsilon}(¥tau), u)d¥tau$.

Letting $¥epsilon$ tend to zero, we get

$(u(t), u)=(u_{0}, v)-¥int_{0}^{t}(u(¥tau), (-i¥mathrm{A}^{*})u)d_{T}+¥int_{0}^{t}(Ju(¥tau), u)d_{T}+¥int_{0}^{t}(g(¥tau), u)d¥tau$,


$¥langle u(t), u¥rangle=¥langle u_{0}+¥int_{0}^{t}(-i¥overline{A}u(¥tau)+Ju(¥tau)+g(¥tau))d¥tau, v¥rangle$

for every $u¥in D(A)$ , where $¥langle, ¥rangle$ denotes the duality between $D(¥mathrm{A})^{¥prime}(=H^{-2})$ and $D(A)$

$(=H^{2})$ and $¥overline{A}$ is an unique continuous linear extension of $A$ from $L^{2}$ into $D(A)^{¥prime}$ .One can consider that $¥overline{A}$ is a continuous linear operator from $D(¥mathrm{A}^{a})$ into $D(A^{a-1})$ ifwe take $D(¥mathrm{A}^{a-1/2})$ as the pivot space: $D(¥mathrm{A}^{a})¥subset D(¥mathrm{A}^{a-1/2})¥subset D(¥mathrm{A}^{a-1})$ . Hence we obtain


$ u(t)=u_{0}+¥int_{0}^{t}((-i¥overline{A}u)(¥tau)+(Ju)(_{¥overline{¥prime}})+g(¥tau))d¥tau$ $t¥in(0, T_{1})$ ,

which implies that $u(t)$ is strongly absolutely continuous in $D(A^{a-1})$ , strongly differ-entiable in $D(A^{a-1})$ at almost all $t¥in(0, T_{1})$, and

$¥frac{du}{dt}=-i¥overline{A}u+Ju+g¥in L^{1}(0, T_{1} : D(A^{a-1}))$.

Thus, we have the theorem.

Remark 4.2. The results in Theorem 4.1 and its Corollary extend to the case$Ju=(¥partial/¥partial x)(|u|^{p}u)$, $p>1$ , in general, $Ju=F(u,¥overline{u})(¥partial u/¥partial x)$ where $F(x, y)$ is a smoothfunction.

Remark 4.3. In order to prolong local solutions to global ones, it is essentialto construct the local existence theorem in such a manner that the existence timeinterval does not depend on regularity of solutions. Similar results to Theorem 4.1holds for the (generalized) $¥mathrm{K}¥mathrm{d}¥mathrm{V}$ equations, which may fill the gaps of the proofs in[3], [10] and [17] concerning the prolongation of local solutions.

Remark 4.4. Let $g¥in L^{2}(0, T:D(¥mathrm{A}^{a}))$ . Then the assertion of Theorem 4. 1 holdsvalid provided $u_{t}¥in L^{1}(0, T_{1} : D(A^{a-1}))$ is replaced by $u_{t}¥in L^{2}(0, T_{1} : D(A^{a-1}))$ . Theproof is easier than that of Theorem 4.1. In view of the $¥mathrm{e}¥mathrm{q}$ . (4.5), we have from(4.27)?(4.29),

$||(u_{¥epsilon})_{t}||_{L^{2}(0,T_{1}:D(Aa-1))}¥leq c$ .

Thus, from the compact argument (see Lions [6], Chapter 1) we see that

$u_{¥epsilon}$? $u$ strongly in $L^{2}(0, T_{1} : D(A^{a-¥eta}))$

for any $¥eta$ such that $0<¥eta<1$ and $¥eta¥leq¥alpha$ , from which it follows

$u_{¥epsilon}¥rightarrow u$ strongly in $L^{2}(Q)$

for any compact subset $Q$ of $R¥times[0, T_{1}]$ and (4.36). Moreover,

$(u_{¥epsilon})_{t}¥rightarrow u_{t}$ weakly in $L^{2}(0, T_{1} : D(¥mathrm{A}^{a-1}))$ .

§5. Global existence theorem

If the data are small enough in some sense, the initial value problem (3.4), (3.5)(or (1.1), (1.2) or (1.2/, (1.3)) has global solutions.

Theorem 5.1. Let $u_{0}¥in D(A^{1/2})$ and $g¥in L^{¥infty}(0, T:L^{2})¥cap L^{2}(0, T:D(A^{1/2}))$ . Thereexists a $¥kappa>0$ such that if $[u_{0}]_{1’ 2}¥leq¥kappa$ and $||g||_{L^{1}(0,T:L^{2})}¥leq¥kappa$ , then the initial value problem


(3.4), (3.5) has a unique global weak solution $u$ which is weakly continuous on $[0, T]$

with values in $D(A^{1/2})$ ,

(5. 1) $u¥in L^{¥infty}(0, T:D(A^{1/2}))$

with

(5.2) $u_{t}¥in L^{¥infty}(0, T:D(A^{1/2})^{¥prime})$ .

Remark 5.1. The constant $¥kappa$ in Theorem 5.1 determines explicitly by $[u_{0}]_{1/2}$ and$||g||_{L^{1}(0,T:L^{2})}$ in the proof.

Remark 5.2. A weak solution $u$ with (5.1) and (5.2) satisfies the equation (3.4)in the following sense:

(5.3) $u_{t}=-i¥overline{A}u+Ju+g$ , $0<t<T$,

where $¥overline{A}$ is a unique continuous linear extension of $A$ from $D(¥mathrm{A}^{1/2})=H^{1}$ into $D(¥mathrm{A}^{1/2})^{¥prime}$

$=H^{-1}$ .Prior to the proof of Theorem 5.1, we state a lemma concerning conserved

functionals associated with a solution of (1.1) with $g¥equiv 0$ .

Lemma 5.1. Let $u(x, t)$ be a smooth solution of (1.1) with $g¥equiv 0$ . Then, follow-ing quantities are independent of $t$ .

(5.4) $I_{1}(u(t))=|u(t)|_{2}^{2}$ ,

(3.5) $I_{2}(u(t))=|$ Du(t) $|_{2}^{2}+¥frac{1}{2}|u(t)|_{6}^{6}+¥frac{3}{2}$ Im( $|u(t)|^{2}$ Du(t), $u(t)$).

Proof. A straightforward calculation yields

$¥frac{d}{dt}I_{1}(u(t))=¥frac{d}{dt}I_{2}(u(t))=0$ . Q.E.D.

Proof of Theorem 5.2. We find sequences $¥{u_{0¥epsilon}¥}¥subset D(A^{¥infty})$ and $¥{g_{¥epsilon}¥}¥subset C^{¥infty}([0, T]$ :$D(¥mathrm{A}^{¥infty}))$ such that as $¥epsilon¥searrow 0$

(5.6) $u_{0¥epsilon}¥rightarrow u_{0}$ strongly in $D(A^{1/2})$ ,

(5.7) $g_{¥epsilon}$? $g$ strongly in $L^{¥infty}(0, T:L^{2})¥cap L^{2}(0, T:D(¥mathrm{A}^{1/2}))$ .

Let $u_{¥epsilon}(t)$ be a solution of (4.5), (4.6). Then $u_{¥epsilon}¥in C^{¥infty}([0, T_{0}]:D(A^{¥infty}))$ .

First we have (4.8), (4.9). Since $L^{2}(0, T:L^{2})¥subset L^{1}(0, T:L^{2})$ , we get

(5.8) $||u_{¥epsilon}||_{L^{¥infty}(0,T¥mathrm{o}:L2)}¥leq c_{1}$ ,


(5.9) $¥vee¥overline{¥epsilon}||$ $¥mathrm{A}^{1/2}u_{¥epsilon}||_{L^{2}(0,T_{0}:L^{2})}¥leq c_{2}$ .

Here and in the sequel $c_{k}(k=1,2,3, -)$ and $c$ denote various constants independ-ent of $T_{0}$ and $¥epsilon$ in any fixed interval $[0, ¥epsilon_{0}]$ .

Differentiating $I_{2}(u_{¥epsilon}(t))$ with respect to $t$, using the $¥mathrm{e}¥mathrm{q}$ . (4.5) and integrating byparts with respect to $x$, we have

(5. 10) $¥frac{d}{dt}I_{2}(u_{¥epsilon}(t))+2¥epsilon|Au_{¥epsilon}|_{2}^{2}=3¥epsilon ¥mathrm{R}¥mathrm{e}(|u_{¥mathrm{e}}|^{4}u_{¥epsilon}, Au_{¥epsilon})-6¥epsilon$ Im $(|u_{¥epsilon}|^{2}Du_{¥epsilon}, Au_{¥epsilon})$

$+¥mathit{2}Re(Du_{e}, Dg_{¥epsilon})+3¥mathrm{R}¥mathrm{e}(|u_{¥epsilon}|^{4}u_{¥epsilon}, g_{¥epsilon})-6¥mathrm{I}¥mathrm{m}(|u_{¥epsilon}|^{2}Du_{¥epsilon}, g_{¥mathrm{g}})$.

In virtue of Lemma 2.1 and the estimate (5.8), the first term of the right hand sideof (5. 10) is bounded by

(5. 11) $3¥epsilon|u_{¥epsilon}|_{10}^{5}|Au_{¥epsilon}|_{2}¥leq 3¥epsilon C_{1}^{5}(10,2,0)(|Au_{6}|_{2}+|u_{¥epsilon}|_{2})|u_{¥epsilon}|_{2}^{4}|$ Au $¥epsilon|_{2}$

$¥leq 3¥epsilon c_{1}^{4}C_{1}^{5}(10,2,0)|Au_{¥epsilon}|_{2}^{2}+¥epsilon¥mu|Au_{¥epsilon}|_{¥mathrm{z}}^{2}+¥frac{c}{¥mu}$

for $0¥leq t¥leq T_{0}$ , where $¥mu$ is an arbitrary positive constant. Using Holder’s inequality,Lemma 2.1 and (5.8), we majorized the second term of the right hand side of (5.10)by

(5. 12) $6¥epsilon|u_{¥epsilon}|_{8}^{2}|$ $Du_{6}|_{4}|$ $Au_{¥xi}|_{2}¥leq 6¥epsilon C_{1}^{2}(8,2,0)C_{1}(4,2,1)(|Au_{¥epsilon}|_{2}+|u_{¥xi}|_{2})|u_{¥epsilon}|_{2}^{2}|Au_{¥mathrm{e}}|_{2}$

$¥leq 6¥epsilon c_{1}^{2}C_{1}^{2}(8,2,0)C_{1}(4,2,1)|$$Au_{¥epsilon}|_{2}^{2}+¥epsilon¥mu|Au_{¥mathrm{g}}|_{2}^{2}+¥frac{c}{¥mu}$

for $0¥leq t¥leq T_{0}$ . Analogously, we see that the fourth and fifth terms of the right handside of (5. 10) are bounded by

(5. 13) $3|u_{¥epsilon}|_{6}^{5}|g_{¥epsilon}|_{6}¥leq 3C_{2}^{6}(6,1,0)(|Du_{¥xi}|_{2}+|u_{¥epsilon}|_{2})^{5/3}|u_{¥epsilon}|^{10/3}(|Dg_{¥epsilon}|_{2}+|g_{¥epsilon}|_{2})^{1/3}|g_{¥epsilon}|_{2}^{2/3}$

$¥leq c|Du_{¥epsilon}|_{2}^{2}+c[g_{¥mathrm{g}}]_{1}^{2}+c$

and

(5. 14) $6|g_{¥epsilon}|_{¥infty}|u_{¥epsilon}|_{4}^{2}|Du_{¥mathrm{S}}|_{2}¥leq 6C_{1}^{2}(4,1,0)||g_{¥mathrm{e}}||_{1}^{1/2}|g_{¥epsilon}|_{2}^{1/¥mathrm{z}}(|Du_{6}|_{2}+|u_{¥epsilon}|_{2})^{1/2}|u_{¥epsilon}|_{2}^{3/2}|Du_{¥mathrm{g}}|_{2}$

$¥leq c|Du_{¥mathrm{g}}|_{2}^{2}+c[g_{¥epsilon}]_{1}^{2}+c$

for $0¥leq t¥leq T_{0}$ , respectively. Here we have used the fact that $||g_{¥mathrm{e}}||_{L^{¥infty}(0,T:L^{2})}¥leq c$.Using all these majorations of the right hand side of (5. 10), we find

(5. 15) $¥frac{d}{dt}I_{2}(u_{¥epsilon}(t))+¥epsilon¥{2-3c_{1}^{4}C_{1}^{5}(10,2,0)-6c_{1}^{2}C_{1}^{2}(8,2,0)C_{1}(4,2,1)¥}-2¥mu|Au_{6}|_{2}^{2}$

$¥leq c|Du_{¥mathrm{g}}|_{2}^{2}+c[g_{¥epsilon}]_{1}^{2}+c+¥frac{c}{¥mu}$


for $0¥leq t¥leq T_{0}$ . The constant $c_{1}$ depends only on $|u_{0}|_{2}$ and $||g||_{L^{1}(0,T:L^{2})}$ . It is easilyshown that there exists a $¥kappa>0$ so that if $¥{u_{0}, g¥}$ lies in the set

(5. 16) $¥{¥{u_{0}, g¥}:|u_{0}|_{2}<¥kappa, ||g||_{L^{1}(0,T:L^{2})}<¥kappa¥}$,

then $c_{1}$ can be taken so small that the inequality

(5. 17) $2-3c_{1}^{4}C_{1}^{5}(10,2,0)-6c_{1}^{2}C_{1}^{2}(8,2,0)C_{1}(4,2,1)>¥delta>0$

holds, where $¥delta$ is a sufficiently small constant. Since $¥mu$ is arbitrary, we choose $¥mu=$

$¥delta/4$ . Hence, from (5. 15) we have

(5. 18) $¥frac{d}{dt}L(u_{¥epsilon}(t))+¥frac{¥delta¥epsilon}{2}|Au_{6}|_{2}^{2}¥leq c|$ $Du_{¥epsilon}|_{2}^{2}+c[g_{¥epsilon}]_{2}^{2}+c$ .

Integration of (5. 18) on $[0, t]$ gives

(5. 19) $|Du_{¥epsilon}(t)|_{¥mathrm{z}^{2}}+¥frac{1}{2}|u_{8}(t)|_{6}^{6}+¥frac{3}{2}$ Im( $|u_{¥epsilon}(t)|^{2}$Du$8(t)$ , $u_{¥epsilon}(t)$) $+¥frac{¥delta¥epsilon}{2}¥int_{0}^{t}|Au_{¥epsilon}(¥tau)|_{2}^{2}d¥tau$

$¥leq c¥int_{0}^{t}|Du_{¥epsilon}(¥tau)|^{2}d_{T}+c¥int_{0}^{t}[g_{¥epsilon}(¥tau)]_{1}^{2}d_{T}+cT$

for $0¥leq t¥leq T_{0}¥leq T$. The third term of the left hand side of (5. 19) is majorized by

$¥frac{3}{2}|u_{¥epsilon}(t)|_{6}^{3}|Du_{¥xi}(t)|_{2}¥leq¥frac{3}{4}|u_{¥epsilon}(t)|_{6}^{6}+¥frac{3}{4}|Du_{8}(t)|_{2}^{2}$.

Hence, we obtain

(5.20) $|Du_{¥epsilon}(t)|_{2}^{2}+¥frac{¥delta¥epsilon}{2}¥int_{0}^{t}|Au_{¥epsilon}(¥tau)|_{2}^{2}d_{T}¥leq|u_{¥epsilon}(t)|_{6}^{6}+c¥int_{0}^{t}|Du_{¥epsilon}(¥tau)|_{2}^{2}d_{T}+c¥int_{0}^{t}[g_{¥epsilon}(¥tau)]_{1}^{2}d_{T}+cT$

for $0¥leq t¥leq T_{0}<T$.We again apply Lemma 2.1 to the first term of the right hand side of (5.20):

$|u_{¥epsilon}|_{6}^{6}¥leq C_{1}^{6}(6,1,0)(| Du_{¥epsilon}|_{2}+|u_{¥epsilon}|_{2})^{2}|u_{6}|_{2}^{4}$ . Hence,

(5.21) $(1-2C_{1}^{6}(6,1,0)c_{1}^{4})|Du_{¥epsilon}(t)|_{2}^{2}¥leq 2C_{1}^{6}(6,1,0)c_{1}^{6}+c¥int_{0}^{t}|Du_{¥epsilon}(¥tau)|^{2}d_{T}+¥int_{0}^{t}[g_{¥epsilon}(¥tau)]_{1}^{2}d_{T}+cT$

$¥leq c¥int_{0}^{t}|Du_{¥epsilon}(¥tau)|^{2}d_{T}+c$

for $0¥leq t¥leq T_{0}<T$.We choose $¥kappa$ so that

(5.22) $1-2¥mathrm{C}_{1}^{6}(6,1,0)c_{1}^{4}>¥delta>0$

provided that $|u_{0}|_{2}<¥kappa$ . Then, from (5.21) we obtain


(5.23) $¥delta|Du_{¥xi}(t)|_{2}^{2}+¥frac{¥delta¥epsilon}{2}¥int_{0}^{t}|Au_{¥epsilon}(¥tau)|_{2}^{2}d_{T}¥leq c¥int_{0}^{t}|Du_{¥epsilon}(¥tau)|_{2}^{2}d_{T}+c$

for $0¥leq t¥leq T_{0}<T$, from which it follows that

(5.24) $||$ $Du_{¥epsilon}||_{L^{¥infty}(0,T_{0}:L^{2})}¥leq c_{3}$

and

(5.25) $¥vee¥overline{¥epsilon}||$ $Au_{¥epsilon}||_{L^{2}(0,T_{0}:L^{2})}¥leq c_{4}$.

Since $c_{j}(j=1,2,3,4)$ are independent of $T_{0}$ , from the estimates (5.8), (5.9),(5.24) and (5.25), we conclude that the solution $u_{¥epsilon}(t)$ of the initial value problem (4. 5),(4.6) continues globally up to $T$. Therefore the estimates (5.8), (5.9), (5.24) and(5.25) with replacing $T_{0}$ by $T$ hold valid. That is, $¥{u_{¥epsilon}¥}$ and $¥{¥vee¥overline{¥epsilon}u_{¥epsilon}¥}$ remain boundedas $¥epsilon¥searrow 0$ in $L^{¥infty}(0, T:D(A^{1/2}))$ and $L^{2}(0, T:D(¥mathrm{A}))$ , respectively. Hence, there exist $u$

and a subsequence of $¥{u_{¥epsilon}¥}$ (also denoted by $¥{u_{¥epsilon}¥}$ ) such that as $¥epsilon¥searrow 0$

(5.26) $u_{¥epsilon}¥rightarrow u$ weakly star in $L^{¥infty}(0, T:D(¥mathrm{A}^{1/2}))$,

(5.27) $(u_{¥epsilon})_{t}¥rightarrow u_{t}$ weakly in $L^{2}(0, T:L^{2})$ .

According to the compactness lemma (see Lions [6], Chapter 1), we may assume that

(5.28) $u_{¥epsilon}$? $u$ strongly in $L^{2}(Q)$

for any compact subset $Q$ of $R¥times[0, T]$ and

(5.29) $u_{¥epsilon}¥rightarrow u$$¥mathrm{a}.¥mathrm{e}$ . in $R¥times[0, T]$ .

From (5.8) and (5.24), we see that $¥{Ju_{¥epsilon}¥}$ remains bounded as $¥epsilon¥searrow 0$ in $L^{¥infty}(0, T:L^{2})$,

and hence in virtue of (5.29)

(5.30) $Ju_{e}$? $Ju$ weakly star in $L^{¥infty}(0, T:L^{2})$ .

Thus, $u$ is a desired weak solution of (3.4), (3.5). This completes the proof.

Remark 5.3. The $¥mathrm{e}¥mathrm{q}$ . (1.1) with $g¥equiv 0$ has a infinite number of conserved func-tionals as noted in Introduction. Using other conserved functionals, we may obtainmore regular solutions of (1.1). This will be done in subsequent papers.


References

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nuna adreso:Tsutsumi, M.Division of Mathematical PhysicsDepartment of Applied PhysicsWaseda UniversityTokyo, JapanFukuda, I.Division of MathematicsFaculty of EngineeringKokushikan UniversityTokyo, Japan

(Ricevita la 16-an de aprilo, 1979)

Existence Uniqueness Schrodinger Equation Solutions

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