Nonlocal Hartree Equations: Theory & Dynamics

• Nonlocal Hartree equations are defined by replacing localized nonlinearities with a convolution-based potential that captures long-range interactions.
• Scaling laws and variational methods reveal energy-critical thresholds, multi-bubble states, and solitonic dynamics in both dispersive and elliptic frameworks.
• The equations are pivotal in modeling mean-field quantum systems and semiclassical asymptotics, bridging theoretical analysis with practical applications.

Searching arXiv for recent and foundational papers on nonlocal Hartree equations to ground the article. The nonlocal Hartree equation is a class of nonlinear partial differential equations in which the effective field at a point is determined by a spatially averaged density, typically through a convolution kernel, rather than by a pointwise power of the unknown. In its dispersive form, a canonical model is

$$i\partial_t u+\Delta u+\bigl(|x|^{-4}*|u|^2\bigr)u=0,$$

while elliptic Hartree or Choquard equations take forms such as

$-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$

In both cases the nonlinearity is genuinely nonlocal: the value at $x$ depends on the full spatial distribution of $u$ through a long-range convolution operator. The literature represented here shows that this structure underlies energy-critical dispersive dynamics, variational ground-state theory, pseudo-relativistic and fractional models, and mean-field descriptions of infinite quantum systems (Miao et al., 2011, Anthal et al., 2 Feb 2026, Zelati et al., 2012).

1. Canonical forms and the meaning of nonlocality

The defining feature of a nonlocal Hartree equation is the replacement of a pointwise nonlinearity by a self-consistent potential of convolution type. For the focusing, energy-critical Hartree equation in dimension $d\ge 5$, the nonlinearity is $(|x|^{-4}*|u|^2)u$, so the interaction at $x$ is mediated by the kernel $|x-y|^{-4}$ and depends on the entire mass distribution $|u(y)|^2$ (Miao et al., 2011). Generalized Hartree models broaden this to

$\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$

thereby separating the convolution exponent from the exterior power nonlinearity and allowing intercritical regimes not present in the classical cubic Hartree setting (Arora et al., 2019).

A second family couples Hartree interaction with additional spatial structure. The inhomogeneous generalized Hartree equation in three dimensions inserts the weight $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$0 both inside and outside the convolution, breaking translation invariance and making the interaction stronger near the origin while weaker at infinity (Guzmán et al., 2023). Mixed local–nonlocal elliptic problems combine $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$1 and $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$2 on the left-hand side with a Hartree or Choquard term on the right, so that both the diffusion and the nonlinearity may be nonlocal (Anthal et al., 2 Feb 2026, Anthal et al., 2022). Pseudo-relativistic Hartree equations replace the Laplacian by $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$3, introducing nonlocality already at the kinetic level (Zelati et al., 2012).

A terminological caution is necessary. In the Gross–Pitaevskii setting, the cubic term can be rewritten as a self-consistent Hartree potential $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$4, and this is described as a nonlinear Hartree potential; however, in that formulation the potential is not nonlocal in the convolution sense, but self-consistent and nonlinear (Rawitscher, 2013). This distinction separates true convolution-mediated Hartree equations from broader Hartree-like mean-field reformulations.

2. Scaling, conserved quantities, and stationary structures

Many of the most studied nonlocal Hartree equations are organized by scaling. The focusing equation

$-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$5

is energy-critical because the natural scaling leaves the equation and conserved energy invariant (Jendrej et al., 9 Feb 2026). Its conserved energy is

$-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$6

and the associated stationary profile $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$7 solves

$-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$8

In the threshold theory, $-\Delta u+(-\Delta)^s u+u=(I_\alpha*F(u))F'(u).$9 is the ground state, its energy $x$0 is the distinguished threshold value, and its symmetry family generates the equality cases in the sharp Sobolev–Hardy–Littlewood–Sobolev inequality used in the analysis (Miao et al., 2011).

The same stationary profile controls multi-bubble dynamics. In the recent two-bubble construction, the asymptotic components are phased and rescaled copies $x$1, with one bubble remaining at scale $x$2 and the second at a much smaller scale $x$3 (Jendrej et al., 9 Feb 2026). In that setting, the bubbles are centered at the origin and asymptotically placed in a right-angle relative phase.

For generalized Hartree equations, the scaling-critical Sobolev index is

$x$4

and the intercritical regime is $x$5, equivalently

$x$6

with the standard low-dimensional modifications (Arora et al., 2019). In the three-dimensional inhomogeneous generalized Hartree equation, energy-criticality is encoded by the relation

$x$7

which tunes the weighted nonlocal interaction to the $x$8 scaling (Guzmán et al., 2023).

These scaling laws do more than classify regularity. They determine the natural threshold quantities, the correct variational inequalities, and the regimes in which coherent structures such as solitary waves, threshold solutions, and bubble towers may occur.

3. Elliptic existence theory, symmetry, and uniqueness

The elliptic side of nonlocal Hartree theory is dominated by variational methods. For the mixed local–nonlocal equation

$x$9

ground states exist under Berestycki–Lions type assumptions on $u$0, regularity is upgraded to $u$1 for all $u$2, and weak solutions satisfy the Pohožaev identity

$u$3

Positive ground states are shown to be radially symmetric by polarization, with the Hartree term improving rather than obstructing the symmetry argument (Anthal et al., 2 Feb 2026).

On bounded domains, the mixed operator $u$4 combined with the critical Choquard nonlinearity leads to a mixed Hardy–Littlewood–Sobolev variational theory. The optimal constant in the corresponding inequality equals the classical Choquard constant, yet is never attained on any open set. The same framework yields a Pohožaev identity adapted to the mixed operator and nonexistence on strictly star-shaped domains for certain parameter ranges, alongside existence results for subcritical perturbations (Anthal et al., 2022).

Pseudo-relativistic Hartree equations admit an extension-based treatment. By passing from $u$5 to a local elliptic problem in one higher dimension with nonlinear Neumann boundary condition, one obtains constrained minimizers under a fixed mass, including critical cases with $u$6 or weak $u$7, and then recovers smooth, positive, radially symmetric ground states when the interaction kernel is radial and nonincreasing (Zelati et al., 2012).

Several recent works push the functional framework further. A superposition of Hartree-type nonlinearities indexed by a measure on Riesz orders leads to the superposed Coulomb space and the superposed Coulomb–Sobolev space, from which variational existence and multiplicity for radial solutions follow, as well as Brezis–Nirenberg-type multiplicity near nonlinear eigenvalues (Marinho et al., 4 Jun 2026). At the same time, a generalized defocusing Hartree equation with a nonlocal exchange operator admits a distinct uniqueness mechanism: the inverse optimal problem approach proves existence and uniqueness of ground states, existence of principal solutions, and a dual variational formulation, with reconstruction formulas of the form $u$8 (Il'yasov et al., 21 Jan 2026).

Symmetry theory also extends to singular solutions. For

$u$9

positive singular solutions are symmetric with respect to the hyperplane containing the compact zero-capacity singular set $d\ge 5$0, strictly monotone away from it, and become radially symmetric when the singularity is isolated at the origin (Cai et al., 11 Aug 2025). This places moving-plane methods firmly within nonlocal Hartree analysis.

4. Global dynamics, threshold behavior, and multi-bubble solutions

For the focusing, energy-critical Hartree equation, the threshold dynamics at energy $d\ge 5$1 are highly structured. In the radial setting, there are two special solutions $d\ge 5$2 and $d\ge 5$3 in addition to the stationary ground state $d\ge 5$4. The classification states that radial data with $d\ge 5$5 and $d\ge 5$6 either scatter or coincide with $d\ge 5$7 up to symmetry; data with equal gradient norm are exactly $d\ge 5$8; and supercritical radial data in $d\ge 5$9 either coincide with $(|x|^{-4}*|u|^2)u$0 up to symmetry or blow up in finite time (Miao et al., 2011). The proof combines uniqueness and regularity of the nonlocal elliptic ground state, spectral analysis of the linearized operator, modulation around $(|x|^{-4}*|u|^2)u$1, and localized virial arguments.

Recent work extends this threshold picture from single-bubble to pure two-bubble dynamics. In dimensions $(|x|^{-4}*|u|^2)u$2, a radial solution has been constructed on $(|x|^{-4}*|u|^2)u$3 such that

$(|x|^{-4}*|u|^2)u$4

It is pure in the sense that no radiation term appears asymptotically, and necessarily satisfies $(|x|^{-4}*|u|^2)u$5 (Jendrej et al., 9 Feb 2026). The novelty is precisely the nonlocal interaction: the convolution kernel couples the two profiles globally, so the construction relies on refined nonlocal interaction bounds, modulation analysis, a bootstrap argument, and a topological shooting argument.

Away from the energy-critical model, generalized Hartree equations admit threshold theories of Kenig–Merle type. In the intercritical regime, under the mass–energy assumption $(|x|^{-4}*|u|^2)u$6, the sign of the renormalized gradient quantity $(|x|^{-4}*|u|^2)u$7 separates global scattering from finite-time blow-up or divergence of the gradient along a time sequence, with radial and finite-variance hypotheses sharpening the blow-up conclusions (Arora et al., 2019). For the three-dimensional energy-critical inhomogeneous generalized Hartree equation, global well-posedness and scattering hold below the ground-state threshold even for non-radial data, and in the defocusing case scattering holds for general data (Guzmán et al., 2023).

The defocusing energy-critical Hartree equation in $(|x|^{-4}*|u|^2)u$8 exhibits a different type of large-data theory under randomization. With narrowed Wiener randomization, almost sure scattering is proved for initial data $(|x|^{-4}*|u|^2)u$9 for any $x$0, using a modified interaction Morawetz estimate, stability theory, and probabilistic Strichartz bounds (Tao et al., 2023). Across these works, the recurring technical difficulty is that the Hartree term is not pointwise: Morawetz identities, virial estimates, profile decompositions, and energy increments all acquire extra convolution terms.

5. Semiclassical asymptotics, coherent states, and WKB theory

Semiclassical analysis reveals a distinctive separation between classical transport and nonlocal self-interaction. For a Hartree equation with smooth even kernel $x$1 and external potential $x$2, a coherent state initially concentrated near $x$3 remains close in $x$4 to a modulated coherent state whose center follows the Hamiltonian flow of

$x$5

and the error is bounded by $x$6 (Athanassoulis et al., 2010). Because $x$7 is even, the Hartree self-interaction affects the leading classical motion only through the constant $x$8; the first nontrivial spatial effect appears at second order through $x$9, which enters the linearized profile equation as a time-dependent harmonic term.

A complementary wave-packet theory distinguishes smooth kernels from homogeneous singular kernels. For

$|x-y|^{-4}$0

the critical exponent is $|x-y|^{-4}$1 when $|x-y|^{-4}$2 is smooth near the origin, but becomes

$|x-y|^{-4}$3

when $|x-y|^{-4}$4, $|x-y|^{-4}$5 (Cao et al., 2011). In the critical regime, the envelope equation is genuinely nonlinear and Hartree-type; in subcritical regimes the nonlinearity is asymptotically negligible at leading order. The approximation remains valid up to Ehrenfest time $|x-y|^{-4}$6, and a nonlinear superposition principle holds for two separated wave packets.

In the weakly nonlinear high-frequency regime,

$|x-y|^{-4}$7

with $|x-y|^{-4}$8, $|x-y|^{-4}$9, multiphase WKB analysis shows that the Hartree term changes the amplitude but not the leading oscillatory phases. No new resonant wave is created; the nonlocal interaction suppresses such resonances through high-frequency averaging, and the approximation error is $|u(y)|^2$0 in $|u(y)|^2$1, where $|u(y)|^2$2 is the Wiener algebra (Mouzaoui, 2012).

These semiclassical results indicate a recurring structural fact: nonlocal Hartree effects can be decisive at the level of envelope modulation while leaving the leading phase-space trajectory unchanged.

6. Infinite-particle formulations, random fields, and atypical kernels

Nonlocal Hartree equations also arise as effective dynamics for infinite systems. In the density-matrix formulation,

$|u(y)|^2$3

translation-invariant states of the form $|u(y)|^2$4 are stationary, and in two dimensions small Schatten-class perturbations scatter back to the homogeneous state under an explicit invertibility condition on the linear response multiplier $|u(y)|^2$5 (Lewin et al., 2013). The multiplier $|u(y)|^2$6 is the rigorous counterpart of the Lindhard function, and its analysis drives the asymptotic stability theory.

A closely related random-field formulation replaces the density matrix by a random field $|u(y)|^2$7 satisfying

$|u(y)|^2$8

For defocusing interactions in dimensions $|u(y)|^2$9, spatially nonlocalized equilibria are stable under localized perturbations, and the perturbation scatters to a free wave (Collot et al., 2018). This framework extends to a quintic Hartree equation for random fields with three-body interaction,

$\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$0

where small perturbations around a nonlocalized equilibrium scatter in dimensions $\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$1 for a large class of interaction measures including the Dirac delta (Malézé, 2023).

The theory also encompasses kernels that grow rather than decay at spatial infinity. For

$\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$2

mass conservation permits an effective decomposition of the nonlinearity into a leading external-potential term plus a remainder. For $\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$3 this yields global well-posedness in the weighted energy space $\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$4, while in the exactly quadratic case $\bigl(|x|^{-(N-\gamma)}*|u|^p\bigr)|u|^{p-2}u,$5 the solution can be written explicitly in terms of harmonic-oscillator propagators, translations, modulations, and a scalar phase (Masaki, 2011). This shows that “nonlocal Hartree” does not require a decaying kernel; the essential feature is self-consistent convolution, not the sign of spatial growth.

Taken together, these developments portray the nonlocal Hartree equation not as a single PDE but as a broad analytical framework. Its central mechanism is always the same—a density-generated field acting back on the wave function through a convolution operator—but its manifestations range from threshold soliton dynamics and semiclassical coherent states to variational elliptic theory and infinite-particle mean-field limits.