Long-range scattering for 2D Dirac-Hartree equations

Abstract: We investigate the long-time behavior of small solutions to the Dirac-Hartree equation in two spatial dimensions. This model describes the mean-field dynamics of relativistic fermions interacting through the three-dimensional Coulomb potential $|x|<sup>{-1}$, which gives rise to long-range effects in the scattering dynamics. We prove global well-posedness and long-range scattering (modified scattering) for small initial data in weighted Sobolev spaces. In this setting, long-range scattering means that, unlike linear scattering, an additional logarithmic phase correction is required to describe the precise asymptotics of nonlinear solutions. Our approach relies on the space-time resonance method, combined with special null structures inherent in the equation. Compared to the three-dimensional case \cite{CKLY2022,cloos}, the novelty lies in overcoming the weaker time decay inherent to the two dimensional problem.

• The paper proves small-data global well-posedness and modified scattering for the massive 2D Dirac-Hartree equation, with solutions decaying at the sharp rate of ⟨t⟩⁻¹.
• It controls the critical second-order weighted norms by combining phase and Dirac null structures, space-time resonance analysis, normal forms, and Coifman-Meyer estimates.
• The result identifies an explicit logarithmic phase correction that cancels the non-integrable long-range contribution and yields convergence to a corrected linear profile, although the high regularity requirement n ≥ 1500 remains non-optimal.

The problem and the main result

The paper studies the Cauchy problem for the massive Dirac-Hartree equation in two spatial dimensions,

$$\big(-i\partial_t + \alpha^j D_j + m\beta\big)\psi = \big(|x|^{-1}*|\psi|^2\big)\psi,$$

where $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$ is a spinor, $m>0$ (normalized to $m=1$), and the self-consistent potential is the restriction of the three-dimensional Coulomb kernel to a plane. This model arises as a mean-field description of relativistic fermions confined to two-dimensional materials such as graphene with broken lattice symmetry. The equation conserves the $L^2$ charge, and the Coulomb nonlinearity is "scattering-critical": its contribution to the Duhamel formula decays like $|t|^{-1}$ in time, which is barely non-integrable. Consequently, linear scattering fails — a fact previously established for related Hartree-type Dirac systems — and one expects modified scattering: convergence to a linear profile corrected by a logarithmic phase.

The main theorem establishes small-data global well-posedness together with modified scattering. For initial data satisfying

$$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$

with $n \ge 1500$ and $k = n/100$, the solution exists globally and obeys the sharp pointwise decay $$\|\psi(t)\|_{L^\infty} \lesssim \varepsilon_0\langle t\rangle^{-1}$$, matching the linear rate. Moreover, there exists a scattering profile $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$0 such that

$\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$1

where $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$2 applies an explicit logarithmic phase correction $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$3 on each Dirac component $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$4. The phase is defined through a time integral involving the singular weight $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$5 evaluated against $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$6, restricted to low output frequencies by a cutoff $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$7. The authors do not optimize the regularity indices or the decay exponent $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$8; the large value of $\psi:\mathbb{R}^{1+2}\to\mathbb{C}^2$9 serves to absorb dyadic summation losses.

Structural reduction and null structures

Diagonalizing the square of the Dirac operator via the projections $m>0$0 reduces the system to two half Klein-Gordon equations coupled through the Hartree nonlinearity. In the interaction representation $m>0$1, the Duhamel formula involves the phase interaction $m>0$2 and the Fourier symbol of the Coulomb potential, $m>0$3 in two dimensions (versus $m>0$4 in 3D).

Two null structures are central. First, the phase null structure: since $m>0$5, one has $m>0$6 when signs coincide, while a lower bound $m>0$7 enables space-resonance integration by parts. Second, the Dirac null structure: products of opposite-sign projectors satisfy $m>0$8, which cancels the $m>0$9 singularity whenever adjacent signs differ. These are combined with Coifman-Meyer multiplier estimates throughout.

Weighted energy estimates

The bootstrap framework assumes a priori bounds in a norm $m=1$0 controlling $m=1$1, first and second moments $m=1$2 and $m=1$3 (with growth allowances $m=1$4 and $m=1$5), and the Fourier amplitude $m=1$6. Under this assumption, interpolation with charge conservation yields the decay $m=1$7.

The Sobolev and first-moment estimates follow from Hardy-Littlewood-Sobolev type inequalities and the phase/Dirac null structures applied to the differentiated Duhamel formula; the key multiplier $m=1$8 is shown to be a bounded Coifman-Meyer operator after weighting by $m=1$9.

The genuinely new difficulty is the second moment, where differentiating twice produces a term carrying a factor $L^2$0. In three dimensions the $L^2$1 decay compensates this quadratic loss; in two dimensions it does not. The authors classify sign configurations into three cases:

• Case 1 ($L^2$2): the phase null structure $L^2$3 yields an extra gain of the input frequency scale $L^2$4, reducing the analysis to the scalar semi-relativistic Hartree case treated previously by Kwon, Lee, and Yang.
• Case 2 ($L^2$5): at least three phase terms share a sign, giving the time-resonance lower bound $L^2$6 (or stronger); normal-form integration by parts in time closes the estimate despite the $L^2$7 factor.
• Case 3 ($L^2$8): neither time resonance nor phase null structure is available. Here the paper introduces its principal algebraic novelty: the identity expressing $L^2$9 as a combination of squares and products of derivatives of the full four-wave phase $|t|^{-1}$0, namely $|t|^{-1}$1 plus terms involving $|t|^{-1}$2. The $|t|^{-1}$3 part gains time decay via integration by parts in $|t|^{-1}$4; the $|t|^{-1}$5 part is handled by up to three integrations by parts in $|t|^{-1}$6, exploiting the space-resonance lower bound on $|t|^{-1}$7, with the induced singularities $|t|^{-1}$8 canceled by the Dirac null structure and by a further null structure hidden in the inner product $|t|^{-1}$9.

A high-frequency regime $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$0 is closed directly by Hölder and Sobolev regularity, uniformly over sign configurations.

Modified scattering

The asymptotic analysis proceeds by comparing $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$1 across dyadic time intervals $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$2. The dominant contribution comes from the low-frequency region $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$3 of the nonlinear term, and only for the sign configurations in $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$4, where no Dirac null structure is available. There, the phase is expanded as $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$5, and successive approximations reduce the integral to an oscillatory integral whose limit is computed explicitly:

$$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$6

with error $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$7 obtained by interpolating two complementary bounds. This limiting kernel is precisely what defines $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$8, so the leading logarithmic divergence cancels against the phase correction, leaving an $$\|\psi_0\|_{H^n} + \|\langle x\rangle^2\psi_0\|_{L^2} + \|\langle\xi\rangle^k\widehat{\psi_0}\|_{L^\infty_\xi} \le \varepsilon_0,$$9 remainder. All other sign configurations are time-integrable: those with differing adjacent signs benefit from the Dirac null structure (giving $n \ge 1500$0 smallness), and the high-frequency contributions are controlled by space-resonance integration by parts (Cases 1 and 3 of the scattering analysis) or by time non-resonance normal forms (Case 2), where the decay bound $n \ge 1500$1 handles the boundary terms created by the correction itself. Summing these estimates over dyadic times yields the Cauchy property of $n \ge 1500$2 and hence the scattering profile.

Limitations and open questions

Several restrictions are explicit. The regularity requirement $n \ge 1500$3 and the angular weight $n \ge 1500$4 are far from optimal, and the proof makes no attempt at optimization; whether the result holds for data near the natural threshold is open. The argument relies essentially on both null structures, so it does not extend to the Chern-Simons-Dirac system in the Coulomb gauge, where one bilinear component lacks the Dirac null structure even though its Fourier symbols exhibit the same order-$n \ge 1500$5 singularity; global existence itself remains unknown there. The massless case $n \ge 1500$6 is also out of reach because linear solutions then decay only like $n \ge 1500$7, though the scaling symmetry suggests vector-field methods could be combined with the present techniques. Finally, for generalized potentials $n \ge 1500$8, the short-range regime $n \ge 1500$9 presents the open problem of establishing small-data linear scattering for Dirac-Hartree systems in both two and three dimensions.

Conclusion

The paper completes, for the massive two-dimensional Dirac-Hartree equation with Coulomb potential, the program of proving small-data global well-posedness and modified scattering that had previously been carried out only in three dimensions. The central technical achievement is the control of second-order weighted norms under the weaker $k = n/100$0 decay, achieved by balancing space-time resonance analysis against the singularities it induces, with the new algebraic identity relating single-pair phases to the four-wave phase serving as the bridge between phase null structures and space-resonance methods. The explicit logarithmic phase correction identified here is intrinsic to the ODE dynamics of the profile and provides the precise asymptotic description of relativistic mean-field dynamics in planar geometries.