Global well-posedness for Dirac equation with concentrated nonlinearity

TL;DR

This paper establishes the global well-posedness of the three-dimensional Dirac equation when it includes a concentrated nonlinearity, ensuring solutions exist and are unique over time.

Contribution

It provides the first proof of global well-posedness for the 3D Dirac equation with concentrated nonlinearity, advancing understanding of nonlinear quantum field models.

Findings

Proved global existence and uniqueness of solutions.

Extended well-posedness results to nonlinear Dirac equations.

Confirmed stability of solutions over time.

Abstract

We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.

Equations

$$\left\{\begin{array}[]{c}i\dot{\psi}(x,t)=D_{m}\psi(x,t)-D^{-1}_{m}\zeta(t)\delta(x)\\ \\ \lim\limits_{\varepsilon\to 0+}\lim\limits_{x\to 0}K_{m}^{\varepsilon}\Big{(}\psi(x,t)-\zeta(t)g(x)\Big{)}=F(\zeta(t))\end{array}\right|\quad x\in{\mathbb{R}}^{3},\quad t\in{\mathbb{R}}.$$

$$g(x)=\frac{e^{-m|x|}}{4\pi |x|},$$

$$(K_m^{\epsilon }\psi )(x)=\frac{1}{(4\pi {)}^3}\int e^{-i\xi \cdot x}\frac{\hat{\psi }(\xi )d^3\xi }{({\xi }^2+m^2{)}^{\epsilon }},\epsilon \ge 0,$$

$$F_j({\zeta }_j)={\partial }_{{\overline{\zeta }}_j}U(\zeta ),U\in C^2({\mathbb{C}}^4),j=1,....,4,$$

$$U(\zeta )\ge b|\zeta {|}^2-a,\mathrm{for}\zeta \in {\mathbb{C}}^4,\mathrm{where}b>0\mathrm{and}a\in \mathbb{R}.$$

$$\psi (x,0)=f(x)+{\zeta }_{0}g(x),f\in H^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^4,{\zeta }_0\in {\mathbb{C}}^4,$$

$$\psi (x,t)={\psi }_{\mathrm{free}}(x,t)+{\psi }_S(x,t).$$

$${\mathcal{D}}_F$$

$$\psi (\cdot ,t)\in C(\mathbb{R},{\mathcal{D}}_F).$$

$${\mathcal{H}}_F(\psi (\cdot ,t)):=\Vert D_m{\psi }_{reg}(\cdot ,t){\Vert }^2+U(\zeta (t))=\mathrm{const},t\in \mathbb{R}.$$

$$|\zeta (t)|\le C({\psi }_0),t\in \mathbb{R}.$$

$$i{\dot{\psi }}_f(x,t)=D_m{\psi }_f(x,t),{\psi }_f(x,0)=f(x),$$

$$\lambda (t):={\psi }_f(0,t)\in C[0,\infty )\otimes {\mathbb{C}}^4.$$

$$\dot{\lambda }\in L_{loc}^2[0,\infty )\otimes {\mathbb{C}}^4.$$

$${\psi }_f(x,t)=(-i{\partial }_t-D_m)u(x,t),$$

$$\ddot{u}(x,t)=(\Delta -m^2)u(x,t),u(x,0)=-D_m^{-1}f(x),\dot{u}(x,0)=0.$$

$$\ddot{v}(x,t)=(\Delta -m^2)v(x,t),v(x,0)=0,\dot{v}(x,0)=D_{m}f(x),$$

$$\ddot{w}(x,t)=(\Delta -m^2)w(x,t),w(x,0)=-f(x),\dot{w}(x,0)=0.$$

$$\dot{v}(0,t),\dot{w}(0,t)\in L_{loc}^2[0,\infty )\otimes {\mathbb{C}}^4.$$

$${\dot{\psi }}_f(0,t)=-i\ddot{u}(0,t)-D_m\dot{u}(0,t)=-i\dot{v}(0,t)-\dot{w}(0,t)\in L_{loc}^2[0,\infty )\otimes {\mathbb{C}}^4.$$

$$i{\dot{\psi }}_{eg}(x,t)=D_m{\psi }_{eg}(x,t),{\psi }_{eg}(x,0)=eg(x),$$

$${\varphi }_{eg}(x,t):={\psi }_{eg}(x,t)-eg(x)$$

$$i{\dot{\varphi }}_{eg}(x,t)=D_m{\varphi }_{eg}(x,t)+D_m^{-1}e\delta (x),{\varphi }_{eg}(x,0)=0,$$

$${\varphi }_{eg}(x,t)=(-i{\partial }_t-D_m)D_m^{-1}e\gamma (x,t)=-i{D}_m^{-1}e\dot{\gamma }(x,t)-e\gamma (x,t),$$

$$\gamma (x,t)=\frac{\theta (t-|x|)}{4\pi |x|}-\frac{m}{4\pi }{\int }_0^t\frac{\theta (s-|x|)J_1(m\sqrt{s^2-|x{|}^2})}{\sqrt{s^2-|x{|}^2}}ds$$

$$\ddot{\gamma }(x,t)=(\Delta -m^2)\gamma (x,t)+\delta (x),\gamma (x,0)=0,\dot{\gamma }(x,0)=0.$$

$${\psi }_{eg}(x,t)=eg(x)-e\gamma (x,t)-i{D}_m^{-1}e\dot{\gamma }(x,t).$$

$$\mu (t):=\underset{\epsilon \to 0+}{\lim }\underset{x\to 0}{\lim }{K}_m^{\epsilon }(g(x)-\gamma (x,t))=\underset{x\to 0}{\lim }(g(x)-\gamma (x,t))=-\frac{m}{4\pi }+\frac{m}{4\pi }{\int }_0^t\frac{J_1(ms)}{s}ds.$$

$$\mu (0)=\underset{t\to 0}{\lim }\mu (t)=-\frac{m}{4\pi }.$$

$$\hat{g}(\xi ,t)-\hat{\gamma }(\xi ,t)$$

Full text

1 Introduction

We denote by $D_{m}$ the Dirac operator $D_{m}:=-i\alpha\cdot\nabla+m\beta$, where $m>0$, $\alpha_{k}$ with $k=1,2,3$ and $\beta$ are $4\times 4$ Dirac matrices. We consider the Dirac field coupled to a nonlinear oscillator

[TABLE]

Here $\psi(x,t)$, $\zeta(t)$ are vector functions with values in ${\mathbb{C}}^{4}$, $g(x)$ is the Green function of the operator $-\Delta+m^{2}$ in ${\mathbb{R}}^{3}$,

[TABLE]

and $K_{m}^{\varepsilon}=(-\Delta+m^{2})^{-\varepsilon}$ is a smoothing operator, defined as

[TABLE]

where $\hat{\psi}(\xi)$ is the Fourier transform of $\psi(x)$. Obviously, $(K_{m}^{\varepsilon}\psi)(x)\to\psi(x)$ as $\varepsilon\to 0$. Hence, in the limit $\varepsilon\to 0$, the coupling in (1) formally depends on the value of the regular parts $\psi_{reg}(x,t)=\psi(x,t)-\zeta(t)g(x)$ of the Dirac field $\psi(x,t)$ at one point $x=0$ only.

We assume that the nonlinearity $F_{j}(\zeta)=F_{j}(\zeta_{j})$ admits a real-valued potential:

[TABLE]

where $\partial_{\overline{\zeta}_{j}}:=\frac{1}{2}(\frac{\partial U}{\partial\zeta_{j1}}+i\frac{\partial U}{\partial\zeta_{j2}})$ with $\zeta_{j1}:={\rm Re\,}\zeta_{j}$ and $\zeta_{j2}:={\rm Im\,}\zeta_{j}$, and

[TABLE]

Our main result is as follows. For initial data of type

[TABLE]

we prove a global well-posedness of the Cauchy problem for the system(1).

Let us comment on our approach. We develop the approach which was introduced in [7, 8] in the context of the Klein-Gordon and wave equations. First, we obtain some regularity properties i) of solutions $\psi_{\rm free}(x,t)$ to the free Dirac equation with initial data (5), and ii) of solutions $\psi_{S}(x,t)$ to the Dirac equation with zero initial data and with the source $D_{m}^{-1}\xi(t)\delta(x)$, where $\xi\in C^{1}[0,\infty)$ ( Lemmas 3.1 and 3.2, and Propositions 4.1 and 4.2). We use these regularity properties to prove the existence of a local solution to (1) of the type

[TABLE]

We show that $\zeta(t)$ is a solution to a first-order nonlinear integro-differential equation driven by $\psi_{\rm free}(0,t)$. Then we prove that conditions (3)–(4) provide the energy conservation. Finally, we use the energy conservation to obtain the global existence theorem. Let us note that the system (1) without smoothing operator $K_{m}^{\varepsilon}$ is not well posed (see Remark 4.5).

As was noted above, the Dirac equation with concentrated nonlinearities is not well posed in contrast to corresponding Klein-Gordon equation [7] and wave equation [8]. So we should to introduce a smoothing operator $K_{m}^{\varepsilon}$ resembling the Pauli-Willars renormalization. As the result, we have found a novel model of nonlinear point interaction which provides the Hamilton structure and needed a priori estimates. However, the introduction of the smoothing operator in (1) leads to additional difficulties in justification of numerous limits. We overcome these difficulties using subtle properties of special functions.

2 Main result

We denote by $a-$ any number $a-\varepsilon$ with an arbitrary small, but fixed $\varepsilon>0$. We fix a nonlinear function $F:{\mathbb{C}}^{4}\to{\mathbb{C}}^{4}$ and define the domain

[TABLE]

which generally is not a linear space. Everywhere below we will write $L^{2}({\mathbb{R}}^{3})$ and $H^{s}({\mathbb{R}}^{3})$ instead of $L^{2}({\mathbb{R}}^{3})\otimes{\mathbb{C}}^{4}$ and $H^{s}({\mathbb{R}}^{3})\otimes{\mathbb{C}}^{4}$. Denote $\|\cdot\|=\|\cdot\|_{L^{2}({\mathbb{R}}^{3})}$.

Theorem 2.1**.**

Let conditions (3) and (4) hold. Then

For every initial data $\psi_{0}(x)=f(x)+\zeta_{0}g(x)$ with $f\in H^{2}({\mathbb{R}}^{3})$ the equation (1) has a unique solution $\psi(x,t)$ such that

[TABLE] 2. 2.

The following conservation law holds:

[TABLE] 3. 3.

The following a priori bound holds:

[TABLE]

Obviously, it suffices to prove Theorem 2.1 for $t\geq 0$. We will do it in Section 5. Previously, we obtain some necessary properties of the free Dirac equation an of the Dirac equations with sources of special kinds.

3 Free Dirac equation

Consider the solution $\psi_{f}(x,t)$ to the free Dirac equation

[TABLE]

with initial data $f\in H^{2}({\mathbb{R}}^{3})$. Evidently, $\psi_{f}(\cdot,t)\in C([0,\infty),H^{2}({\mathbb{R}}^{3}))$. Denote

[TABLE]

Lemma 3.1**.**

[TABLE]

Proof.

We represent the solution $\psi_{f}(x,t)$ to (8) as

[TABLE]

where vector function $u(x,t)$ is a solution to the free Klein-Gordon equation

[TABLE]

Then the functions $v(x,t)=\dot{u}(x,t)$ and $w(x,t)=D_{m}u(x,t)$ satisfy

[TABLE]

[TABLE]

It is obvious that $(v(x,0),\dot{v}(x,0)),(w(x,0),\dot{w}(x,0))\in H^{2}({\mathbb{R}}^{3})\oplus H^{1}({\mathbb{R}}^{3})$. Then applying [7, Corollary 4.3], we obtain

[TABLE]

Hence, (11) implies

[TABLE]

∎

Now we consider the free Dirac equation with initial data $eg(x)$ with arbitrary $e\in{\mathbb{C}}^{4}$,

[TABLE]

and obtain explicit formula for the solution $\psi_{eg}(x,t)$. Note that the function

[TABLE]

satisfies

[TABLE]

since $D_{m}eg(x)=D_{m}^{-1}D_{m}^{2}eg=D_{m}^{-1}e\delta(x)$. Similarly to (11), we represent $\phi_{eg}(x,t)$ as

[TABLE]

where

[TABLE]

is the solution to the Klein-Gordon equation

[TABLE]

Here $\theta$ is the Heaviside function and $J_{1}$ is the Bessel function of the first order. Finally, (13) and (14) imply

[TABLE]

Lemma 3.2**.**

For any $t>0$ there exists

[TABLE]

The function $\mu(t)$ is continuous for $t>0$, and there exists

[TABLE]

Proof.

Applying the Fourier transform $\hat{f}(\xi)={\cal F}_{x\to\xi}f(x)$, we get

[TABLE]

Then for (18) it suffices to justify the following permutation of the limits:

[TABLE]

We will do it for each term in (20) separately.

Step i) It is obvious that

[TABLE]

Step ii) Now we prove that

[TABLE]

First, note that

[TABLE]

since

[TABLE]

On the other hand,

[TABLE]

by [4, Formula 1.3.(7)]. Here ${\bf K}_{\nu}$ is the modified Bessel function, and $\Gamma$ is the gamma function. One can justify the last limit, splitting the integral into a sum of integrals over the intervals $[0,1]$ and $[1,\infty)$, and integrating by parts in the second one. Therefore,

[TABLE]

since

[TABLE]

by [9, Formulas 5.7.1 and 10.39.2].

Step iii) It remains to check that

[TABLE]

One has

[TABLE]

where $\rho=|x|$. Evidently,

[TABLE]

Hence, (21) will follow from

[TABLE]

Note, that

[TABLE]

where

[TABLE]

The last estimate implies

[TABLE]

Moreover,

[TABLE]

that easily follows by means of integration by parts. Finally, (23) –(24) imply (22). ∎

4 Linear Dirac equation with sources

4.1 Dirac equation with the source $D_{m}^{-1}\zeta(t)\delta(x)$

For arbitrary $\zeta(t)\in C^{1}[0,\infty)\otimes{\mathbb{C}}^{4}$, consider the equation

[TABLE]

It is easy to verify that

[TABLE]

where

[TABLE]

is the solution to the Klein-Gordon equation with $\delta$-like source:

[TABLE]

Hence

[TABLE]

where $\gamma(x,t)$ is defined in (15), and

[TABLE]

Lemma 4.1**.**

For any $\zeta(t)\in C^{1}[0,\infty)\otimes{\mathbb{C}}^{4}$ such that $\ddot{\zeta}(t)\in L^{2}_{loc}[0,\infty)\otimes{\mathbb{C}}^{4}$, one has

[TABLE]

Proof.

The Fourier transform of $\varphi_{S}(x,t)-\zeta(t)g(x)$ for any $t>0$ reads

[TABLE]

Due to (19) - (20),

[TABLE]

Hence, it remains to prove that

[TABLE]

Integrating by parts, we obtain

[TABLE]

Note, that

[TABLE]

or equivalently,

[TABLE]

One can easily justify this by partial integration. Hence, for (30), it remains to prove that

[TABLE]

We split the integrand in (31) as

[TABLE]

and justify the permutation of the limits (31) for integrals of each terms in the RHS of (32) separately.

Step i) First, consider the integral of the second term of (32). By the Fubini theorem

[TABLE]

Hence,

[TABLE]

by the Lebesgue theorem.

Step ii) Similarly, for the integral of the first term, we obtain

[TABLE]

since

[TABLE]

Step iii) It remains to consider the third term of (32) and prove that

[TABLE]

Applying the Fubini and the Lebesgue theorems, we get

[TABLE]

by [4, Formula 2.3.(6)]. Here ${\rm I}_{\varepsilon}(z)$ is the modified Bessel function, and ${\bf L}_{-\varepsilon}(z)$ is the modified Struve function, satisfying

[TABLE]

due to formulas (10.30.1) and (11.2.2) of [9]. On the other hand, the Fubini theorem implies

[TABLE]

where

[TABLE]

Evidently,

[TABLE]

Further, we represent $T_{2}(s,\rho)$ as

[TABLE]

In the case $0<s\leq 2\rho$, we obtain

[TABLE]

In the case $0<\rho\leq s/2$, we obtain

[TABLE]

Due to (37)–(39), we can apply the Lebesgue theorem in (36) and obtain

[TABLE]

which coincides with the right hand side of (34). Hence, (33) follows. ∎

Proposition 4.2**.**

For any $\zeta(t)\in C^{1}[0,\infty)\otimes{\mathbb{C}}^{4}$, such that $\ddot{\zeta}\in L^{2}_{loc}[0,\infty)\otimes{\mathbb{C}}^{4}$, there exists

[TABLE]

Note that $D_{m}^{-1}=D_{m}D_{m}^{-2}=-i\alpha\cdot\nabla D_{m}^{-2}+m\beta D_{m}^{-2}$. Then the Proposition follows from two lemmas below.

Lemma 4.3**.**

The following limit holds,

[TABLE]

Proof.

Note that $p_{S}(x,t)$ is a solution to (25) with $\dot{\zeta}(t)$ instead of $\zeta(t)$. Hence,

[TABLE]

Applying (31) with $\dot{\zeta}$ instead of $\ddot{\zeta}$, we get

[TABLE]

It remains to calculate $\lim\limits_{x\to 0}D_{m}^{-2}p_{S}(x,t)$. Integrating by parts in (42), we obtain

[TABLE]

Let us calculate the inverse Fourier transform of $\frac{\cos s\sqrt{\xi^{2}+m^{2}}}{\xi^{2}+m^{2}}$ and of $\frac{\sin t\sqrt{\xi^{2}+m^{2}}}{(\xi^{2}+m^{2})\sqrt{\xi^{2}+m^{2}}}$. In the sense of distributions, we obtain

[TABLE]

Similarly,

[TABLE]

Hence, (44) -(46) imply for $t>0$ and $|x|\leq t$

[TABLE]

Note, that

[TABLE]

Moreover,

[TABLE]

Substituting this into (47), we obtain

[TABLE]

since $\displaystyle\int\limits_{0}^{\infty}\frac{J_{1}(u)du}{u}=1$ by [5, Formula 6.561(17)]. Finally, (43) and (48) imply (41). ∎

Lemma 4.4**.**

The following limit holds

[TABLE]

Proof.

Due to (42),

[TABLE]

Integrating here by parts, we obtain

[TABLE]

We have

[TABLE]

[TABLE]

Substituting this into (50), we get (49). ∎

Remark 4.5*.*

Note, that the limit (49) does not exist without the smoothing operator $K_{m}^{\varepsilon}$. This immediately follows from the Taylor expansion of $D_{m}^{-2}p_{S}(x,t)$: using (47), we obtain

[TABLE]

where

[TABLE]

Evidently, $\lim\limits_{x\to 0}\nabla|x|$ does not exist.

4.2 Dirac equation with the source $\chi(t)D_{m}^{-2}\delta(x)=\chi(t)g(x)$

For arbitrary $\chi(t)\in C[0,\infty)\otimes{\mathbb{C}}^{4}$, consider the equation

[TABLE]

Lemma 4.6**.**

Let $\chi(t)\in C[0,\infty)\otimes{\mathbb{C}}^{4}$, with $\dot{\chi}\in L^{1}_{loc}[0,\infty)\otimes{\mathbb{C}}^{4}$, and let $h_{0}\in H^{2}({\mathbb{R}}^{3})$. Then the solution $h(x,t)$ to (51) satisfies

[TABLE]

.

Proof.

We represent $h(x,t)$ as the sum $h(x,t)=u(x,t)+v(x,t)$, where $u(x,t)$ is a solution to the free Dirac equation with initial data $h_{0}(x)$, and $v(x,t)$ is a solution to (51) with zero initial data. Evidently, $u(\cdot,t)\in C([0,\infty),H^{2}({\mathbb{R}}^{3}))$. It remains to prove that

[TABLE]

We represent $u(x,t)$ as $u(x,t)=(-i\partial_{t}-D_{m}){\rm w}(x,t)$, where ${\rm w}(x,t)$ is the solution to

[TABLE]

Then, for (52) we need to prove that

[TABLE]

Applying the Fourier transform, we obtain

[TABLE]

Then (53) is equivalent to

[TABLE]

Both integrals are estimated in the same way, and we consider the first integral only. One has

[TABLE]

since

[TABLE]

∎

5 Proof of well-posedness

First, we modify the nonlinearity $F$ so that it becomes Lipschitz continuous. Define

[TABLE]

where $\psi_{0}\in{\cal D}_{F}$ is the initial data from Theorem 2.1 and $a$, $b$ are constants from (4). Then we may pick a modified potential function $\tilde{U}(\zeta)\in C^{2}({\mathbb{C}}^{4},{\mathbb{R}})$, so that

i) the identity holds

[TABLE]

ii) $\tilde{U}(\zeta)$ satisfies (4) with the same constant $a$, $b$ as $U(\zeta)$ does:

[TABLE]

iii) the functions $\tilde{F}_{j}(\zeta_{j})=\partial_{\overline{\zeta}_{j}}\tilde{U}(\zeta)$ are Lipschitz continuous:

[TABLE]

We suppose that $\psi_{0}=f+\zeta_{0}g$, where $f\in H^{2}({\mathbb{R}}^{3})$, and consider the Cauchy problem for (1)) with the modified nonlinearity $\tilde{F}$. As before we denote by $\psi_{f}(x,t)\in C([0,\infty),H^{2}({\mathbb{R}}^{3}))$ the unique solution to (8), and by $\psi_{\zeta_{0}g}(x,t)\in C([0,\infty),L^{2}({\mathbb{R}}^{3}))$ the unique solution to (12) with $e=\zeta_{0}$. Let $\lambda(t)$ and $\mu(t)$ are defined by (9) and by (18). The following lemma is proved by standard argument from the contraction mapping principle.

Lemma 5.1**.**

Let conditions (55)–(57) be satisfied. Then there exists $\tau>0$ such that the Cauchy problem

[TABLE]

has unique solution $\zeta\in C^{1}[0,\tau]\otimes{\mathbb{C}}^{4}$.

Denote

[TABLE]

where $\gamma(x,t)$ is defined in (15), and

[TABLE]

[TABLE]

with $\zeta(t)$ from Lemma 5.1. Now we establish the local well-posedeness for (1).

Proposition 5.2**.**

(Local well-posedeness). Let the conditions (55)–(57) hold. Then the function

[TABLE]

is a unique solution to the Cauchy problem

[TABLE]

Proof.

Using (9), (17), (59), (18), (29) and (40) successively, we get

[TABLE]

since $\zeta(t)$ solves (58). Hence, the second equation of (61) is satisfied. Further,

[TABLE]

Hence, $\psi$ solves the first equation of (61). Finally, the function $\psi_{reg}(x,t)=\psi(x,t)-\zeta(t)g(x)$ satisfies

[TABLE]

Indeed, $\psi_{reg}(x,t)$ is a solution to

[TABLE]

with initial data $f\in H^{2}({\mathbb{R}}^{3})$ and with $\dot{\zeta}\in C[0,\infty)\otimes{\mathbb{C}}^{4}$ satisfying equation (58). Lemma 3.1 implies that $\dot{\lambda}\in L^{1}_{loc}[0,\infty)\otimes{\mathbb{C}}^{4}$. Moreover, $\dot{\mu}\in C([0,\infty))$ by (18). Hence, (58) implies that $\ddot{\zeta}\in L^{1}_{loc}[0,\infty)\otimes{\mathbb{C}}^{4}$, and (62) holds by Lemma 4.6.

Suppose now that $\tilde{\psi}(x,t)=\tilde{\psi}_{reg}(x,t)+\tilde{\zeta}(t)g(x)$ is another solution to (61). Then, by reversing the above argument, the second equation of (61) implies that $\tilde{\zeta}(t)$ solves the Cauchy problem (58). The uniqueness of the solution of (58) implies that $\tilde{\zeta}(t)=\zeta(t)$. Then, defining

[TABLE]

[TABLE]

and

[TABLE]

for $\tilde{\psi}_{\rm free}=\tilde{\psi}_{f}+\tilde{\psi}_{\zeta_{0}g}=\tilde{\psi}-\psi_{S}$ one obtains

[TABLE]

Thus, $\tilde{\psi}_{\rm free}$ solves the Cauchy problem for the free Dirac equation with initial data $f+\zeta_{0}g$. Hence, by the uniqueness of the solution to this Cauchy problem, we have $\tilde{\psi}_{\rm free}=\psi_{\rm free}$, and then $\tilde{\psi}=\psi$. ∎

Lemma 5.3**.**

Let conditions (55)–(57) hold, and let $\psi(t)\in{\cal D}_{\tilde{F}}$, $t\in[0,\tau]$, be a solution to (61). Then

[TABLE]

Proof.

Equation (63) and the second equation of (61) imply for any $t\in[0,\tau]$

[TABLE]

Here the scalar product $\langle K_{m}^{\varepsilon}D_{m}^{2}\psi_{reg},K_{m}^{\varepsilon}D_{m}\psi_{reg}\rangle$ exists since $K_{m}^{\varepsilon}\psi_{reg}(\cdot,t)\in C([0,\infty),H^{3/2}({\mathbb{R}}^{3}))$ for any $\varepsilon>0$ due to Lemma 4.6. The right hand side of (65) is continuous bounded function since $\tilde{U}\in C^{2}({\mathbb{C}}^{4})$ and $\dot{\zeta}\in C[0,\tau]\otimes{\mathbb{C}}^{4}$. Hence, in the sense of distributions

[TABLE]

Then (64) follows. ∎

Corollary 5.4**.**

The following identity holds

[TABLE]

Proof.

First note that

[TABLE]

Therefore, $|\zeta_{0}|\leq\Lambda(\psi_{0})$ and then $\tilde{U}(\zeta_{0})=U(\zeta_{0})$, ${\cal H}_{\tilde{F}}(\psi_{0})={\cal H}_{F}(\psi_{0})$. Further,

[TABLE]

Hence, (64) implies that

[TABLE]

∎

Identity (66) implies that we can replace $\tilde{F}$ by $F$ in Proposition 5.2 and in Lemma 5.3.

Proof of Theorem 2.1. The solution $\psi(t)\in{\cal D}_{F}$ constructed in Proposition 5.2 exists for $0\leq t\leq\tau$, where the time span $\tau$ in Lemma 5.1 depends only on $\Lambda(\psi_{0})$. Hence, the bound (67) at $t=\tau$ allows us to extend the solution $\psi$ to the time interval $[\tau,2\tau]$. We proceed by induction to obtain the solution for all $t\geq 0$.