Global attractor for 3D Dirac equation with nonlinear point interaction

We prove global attraction to stationary orbits for 3D Dirac equation with concentrated nonlinearity. We show that each finite energy solution converges as t→±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \pm \infty $$\end{document} to the set of four-frequency “nonlinear eigenfunctions”. The global attraction is caused by nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation.


Introduction
In the last decades equations with point interactions became an intensively developing field of research, and this interest is driven by the possibility of investigating nonlinear problems in the context of solvable models. These equations are useful mathematical tool for modeling many phenomena in theoretical physics, (see introduction in [11]).
Remark 1.1. The nonlinearity in (1.1) is more singular than the nonlinearities considered in [26,27,30]. That's why we introduced the smoothing operator K ε m . In Sect. C.3.1, we show that without the operator K ε m , the limit as x → 0 in the second equation of (1.1) generally does not exist.
We note also that the 3D Schrödinger equation with concentrated nonlinearity was justified in [10] as a scaling limit of a regularized nonlinear Schrödinger dynamics. We suppose that for the Dirac equation a justification can be done by suitable modification of methods [10], but it still remains an open question.
Let us comment on our approach. For the proof of global well-posedness we develop the approach which was introduced in [26,32] in the context of the Klein-Gordon and wave equations. First, we obtain some regularity properties i) of solutions ϕ g (x, t) to the free Dirac equation with initial function ζ 0 g(x), and ii) of solutions ψ S (x, t) to the Dirac equation with zero initial function and with source D −1 m ζ(t)δ(x) (Lemma 2.2, and Propositions 2.4 and 2.5 ). We use these regularity properties to prove the existence of a local solution to (1.1) of the type is a solution to the free Dirac equation with initial function f (x). We show that ζ(t) is a solution to a first-order nonlinear integro-differential equation driven by ψ free (0, t). Then we prove that conditions (1.3)-(1.4) provide the conservation law (2.2). Finally, we use the conservation law to obtain the global existence theorem. Note that our system (1.1) gives a novel model of nonlinear point interaction which provides a conservation law and a priori estimates. The introduced smoothing operator K ε m leads to justification of numerous limit permutation.  [1], equation (7) in [2]).
Everywhere below we will write L 2 and H s instead of L 2 (R 3 ) ⊗ C 4 and H s (R 3 ) ⊗ C 4 . Denote · = · L 2 . In this section we will prove the following result.
The following conservation law holds: The following a priori bound holds: Obviously, it suffices to prove Theorem 2.1 for t ≥ 0. We split solutions to (1.1) as where ψ f (x, t) and ϕ g are the unique solutions to the free Dirac equation with initial functions f and ζ 0 g: Evidently, Hence, Moreover, the linear map f (·) → λ(·) is continuous H Now the existence and uniqueness of the solution ψ(·, t) ∈ C([0, ∞), D F of the system (1.1) is equivalent to the existence and uniqueness of the solution (ψ S (·, t), ζ(t)) to (2.5) Let us obtain an explicit formula for ϕ g (x, t). Note that the function Here J 1 is the Bessel function of the first order and θ is the Heaviside function. Finally, (2.9) and (2.10) imply where ϕ + g (x, t) := ζ 0 (g(x) − γ(x, t)).

Lemma 2.2. For any t > 0 there exists
(2.14) We prove this lemma in Sect. A. Note that the function μ(t) is continuous for t > 0, and there exists Moreover,

Reduction to integro-differential equation
Here we consider the first equation of (2.5) for ψ S with some given function We construct the solution and formulate its properties which will be proved later. Further, we substitute the constructed solution into the second equation of (2.5) and obtain an integro-differential equation for ζ.
is given by where γ is defined in (2.11), and Proof. It is easy to verify that the function ϕ S (x, t) is the unique solution to the Klein-Gordon with δ-like sourcë (2.20) In the case m = 0 this is well-known formula [14,Section 175]. Hence, In Sects. B and C , we justify the following limits E. Kopylova NoDEA Substituting these limits into the second equation of (2.5) and taking into accunt (2.14), we obtain the equation for ζ(t): In the next two sections we will solve system (2.5) in reverse order: first we solve the equation (2.24) for ζ(t) and then we solve the first equation of (2.21) for ψ S (x, t) with this ζ(t).

Local well-posedness
Here we prove the local well-posedness for the system (2.5). To do this, we modify the nonlinearity F so that it becomes Lipschitz continuous. Define where ψ(0) = ψ(·, 0) ∈ D F is the initial function from Theorem 2.1 and a, b are constants from (1.4). Then we may pick a modified potential function U (ζ) ∈ C 2 (C 4 , R), so that (i) the identity holds (iii) the functions F j (ζ j ) = ∂ ζ j U (ζ) are Lipschitz continuous: (2.28) First, we establish local well-posedness for system (2.5) with the modified nonlinearity F .
Lemma 2.7. Let conditions (2.26)-(2.28) be satisfied. Then (i) for sufficiently small τ > 0 the Cauchy problem (2.29) has a unique so- Now we define the function where ϕ S (x, t) and p S (x, t) are given by (2.18) and (2.19) with ζ(t) the solution to (2.29).

Bootstrap argument
Identity (

Solitary waves and main theorem
We assume that This assumption guarantees the bound (1.4), and it is crucial in our argument: it allow us to apply the Titchmarsh convolution theorem. Equality (3.1) implies that where the sum has a finite number of terms.
(ii) The solitary manifold is the set:
Consider the function where by (3.11). Therefore, where C k := (C k1 , . . . , C k4 ). Now we derive the explicit formulas for ψ ω k (x), using (3.12) and (3.13) only. One has Moreover, Substituting this into (3.12), we obtain by (3.13) where we denote Now we are able to find coefficients C kj . The second equation of (1.1) together with (3.4) and (3.14) imply Lemma 3.4. Let C kj be solutions to (3.19). Then for each fixed j = 1, . . . , 4 only one of the coefficients C kj is nonzero.
Proof. It suffices to consider the case j = 1 only. We should prove that may be no more than one nonzero c k := C k1 . Assume, to the contrary, with some a > 0 and b = 0. Hence, (3.3) implies where R consists of terms of the type Ce iσt with |σ| < (N 1 − 1)δ. Note that d = 0 since a 1 is a polynomial of degree The lemma and formulas (3.4) and (3.14) imply We can assume that k j = j. Then C kj j = C jj , ω kj = ω j , and ϕ ωj ,j (x) = φ ωj (x) from (3.7) with C j = C jj . Then (3.5) follows. It remains to note that equation (3.19) in the case when C jk = 0 for k = j is equation (3.8) for C j = C jj . Proposition is completely proved.
The following lemma gives a sufficient condition for the existence of nonzero solitary waves.
We prove this lemma in Appendix D. Now the solitary manifold S reads Our main result is the following theorem. It suffices to prove Theorem 3.6 for t → +∞.

Dispersive component
The following lemma states well known decay in local seminorms for the free Dirac equation.

1)
where B R is the ball of radius R.
Now consider where ϕ g is the solution free Dirac equation with initial function ζ 0 g, given by (2.13).
In conclusion, let us show that Indeed, the energy conservation for equation (4.5) implies that Hence, Then (4.6) follows by (4.3).

Traces on the real line
It is the boundary value of the analytic function (5.2) in the following sense: where the convergence holds in S (R, L 2 ). Indeed, . Therefore, (6.1) holds by the continuity of the Fourier transform F t→ω in S (R). Similarly to (6.1), the distribution ζ(ω), ω ∈ R, is the boundary value of analytic in C + function ζ(ω): since the function θ(t)ζ(t) is bounded. The convergence holds in the space of tempered distributions S (R). Let us justify that the representation (5.6) for ψ S (x, ω) is also valid when ω ∈ R\{−m; m}. Namely, Lemma 6.1. V (x, ω) is a smooth function of ω ∈ R\{−m; m} for any fixed x ∈ R 3 \{0}, and the identity holds in the sense of distributions.

Omega-limit compactness
Proof. Theorem 2.1-iii), bound (2.8) and equation (2.24) imply that ζ ∈ C 1 b (R) ⊗ C 4 . Then (8.1) follows from the Arzelá-Ascoli theorem. Further, using the asymptotics of Bessel function [33,Formula 10.7.8], we obtain Moreover, for any t ∈ R Proof. Due to (8.1) and the continuity of the Fourier transform in S (R), we have

Spectral inclusion and the Titchmarsh theorem
Here we will prove the following identity We start with an investigation of supp F j (η j ).
Lemma 9.1. The following spectral inclusion holds: Proof. Applying the Fourier transform to (8.2), we get by the theory of quasimeasures (see [20]) that where σ j is defined in (3.9), P (ω) and Q(ω) are the Fourier transforms of the functions du. Note that P (t) and Q(t) belong to L 1 (R). Therefore, the multiplication by P (ω) and Q(ω) is welldefined in the sense of quasimeasures (see Appendix B of [20]). Finally, (9.3) implies (9.2).

Proof of global attraction
Substituting (2.13) and (2.17) into (2.4), we obtain Assume by contradiction that there exists a sequence s k → ∞ such that for some δ > 0. According to Lemma 10.1, there exist a subsequence s kn of the sequence s k , ω + j ∈ R and functions φ ω + j such that The convergence (11.3) contradict (11.2) due to (3.20). This completes the proof of Theorem 3.6.

Funding Information Open access funding provided by University of Vienna.
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A. Proof of Lemma 2.2
Note that Hence it suffices to prove that γ(x, t)).
Applying the Fourier transform f (ξ) = F x→ξ f (x), we get Then for (A.1) it suffices to justify the following permutation of limits: We will do it for each term in (A.3) separately.
Step i) Applying the Lebesgue dominated convergence theorem, we obtain Step ii) Let us prove that Indeed, for ρ := |x| < t, On the other hand, by [14,Formula 1.3. (7)]. Here K ν is the modified Bessel function, and Γ 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, ∞), and integrating by parts in the second one. Further, Step iii) It remains to check that One has  One has e ±it √ r 2 +m 2 − e ±itr = e ±itr ± itm 2 2r + R ± (r, t) , r ≥ 2m, t > 0, The last estimate implies

B. Proof of Proposition 2.4
For any t > 0, Hence, it suffices to prove that The Fourier transform of ϕ S (x, t) − ζ(t)g(x) for any t > 0 reads Therefore, because of the continuity of the Fourier transform in S , it remains to prove that Due to splitting (A.2), equality (B.2) will follow from the following three equalities Step i) For any s ∈ R and ε ≥ 0 the function T ε (ξ, s) := Moreover it is uniformly continuous in s. Therefore,  Step ii) Let us prove (B.4). Due to (A.8) where ρ = |x|, and e ±is √ r 2 +m 2 − e ±isr sin(ρr) Applying the Lebesgue dominated convergence theorem, we obtain by (A.11) and the Lebesgue theorem. Finally, which easily follows by means of integration by parts.