Isothermal Euler–Poisson System

• The isothermal Euler–Poisson system is a compressible fluid model that couples Euler dynamics with a Poisson field using the isothermal closure p(ρ)=cₛ²ρ.
• It appears in diverse formulations—including single-fluid, ion-acoustic, and two-fluid models—with distinct sign conventions affecting dispersion and stability.
• The system supports rigorous analyses ranging from global existence and Klein–Gordon-type dispersion to singular limits, asymptotic reductions, and solitary wave stability.

The isothermal Euler–Poisson system is a class of compressible fluid models in which Euler dynamics is coupled to a Poisson field and the pressure law is isothermal, $p(\rho)=c_s^2\rho$. In plasma formulations, the Poisson equation describes electrostatic self-consistency and the isothermal law produces the logarithmic force $c_s^2\nabla\log \rho$ in nonconservative variables; in astrophysical formulations, the same Euler–Poisson structure appears with the opposite, attractive Poisson sign and models self-gravitating gases. The literature therefore does not use a single canonical equation, but rather several closely related systems: single-fluid electron models with an immobile ion background, ion-acoustic models with Boltzmann electrons, two-fluid electrostatic plasma models, damped periodic models, and self-gravitating isothermal flows (Li et al., 2011, Makino, 2017).

1. Model classes, closures, and sign conventions

The basic isothermal closure is

$$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$

and this closure is explicitly identified in the two-dimensional Euler–Poisson analysis of Li and Wu, which also notes that the same global theory applies to $\gamma=1$ after normalization (Li et al., 2011). In the electron-fluid plasma model with a uniform immobile ion background of density $n_0$, the governing equations before normalization are

$$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$

together with the neutrality condition $\int_{\mathbb R^2}(n-n_0)\,dx=0$ (Li et al., 2011).

A second standard formulation is the ion-acoustic model with Boltzmann electrons. In the one-dimensional nondimensional isothermal Euler–Poisson system studied for solitary-wave stability,

$$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$

where $K=T_i/T_e>0$, $1+n$ is the ion density, and the far-field equilibrium is $c_s^2\nabla\log \rho$0 (Bae et al., 2020). Closely related one-dimensional ion-acoustic models also appear in the small-amplitude solitary-wave and KdV-limit analyses, with the nondimensional equations

$c_s^2\nabla\log \rho$1

where $c_s^2\nabla\log \rho$2 and the equilibrium is $c_s^2\nabla\log \rho$3, $c_s^2\nabla\log \rho$4, $c_s^2\nabla\log \rho$5 (Bae et al., 2018).

A third formulation is the electrostatic two-fluid model. In normalized variables around a neutral constant background, the isothermal two-fluid Euler–Poisson system in three dimensions is

$c_s^2\nabla\log \rho$6

$c_s^2\nabla\log \rho$7

with $c_s^2\nabla\log \rho$8 in the isothermal interpretation and irrotational initial data $c_s^2\nabla\log \rho$9 (Guo et al., 2013).

Boundary-value and quasineutral studies usually employ the ion formulation with massless Boltzmann electrons. In the half-space $$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$0, Gérard-Varet, Han-Kwan, and Rousset write

$$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$1

with impermeability $$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$2 and Dirichlet data $$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$3 (Gérard-Varet et al., 2011).

In astrophysical usage the sign of Poisson coupling is reversed. Makino’s coordinate derivations use

$$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$4

with the isothermal equation of state $$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$5 (Makino, 2017). In the spherically symmetric exterior-domain problem of Li, Li, and Zhu, the normalized self-gravitating system is

$$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$6

with $$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$7 (Liu, 2023). A common misconception is that all Euler–Poisson papers study the same sign convention; the literature instead separates repulsive plasma models from attractive self-gravitating ones, and their stability mechanisms differ accordingly (Li et al., 2011).

2. Linearization, dispersion, and normal forms

For the repulsive plasma problem, linearization around a homogeneous equilibrium often produces a Klein–Gordon or Klein–Gordon-type structure. In the two-dimensional electron-fluid model, after normalization and irrotational reduction $$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$8, Li and Wu introduce

$$p(n)=c_s^2\,n,\qquad \frac{\nabla p(n)}{n}=c_s^2\,\nabla\log n,$$9

so that the linear flow is

$\gamma=1$0

The Klein–Gordon mass is normalized to $\gamma=1$1, the linear decay in two dimensions is $\gamma=1$2, and the quadratic phase

$\gamma=1$3

has no exact time resonance, which makes a Shatah-type normal form possible (Li et al., 2011). Jang, Li, and Zhang formulate the same two-dimensional problem as a quasi-linear Klein–Gordon system

$\gamma=1$4

with nonlocal quadratic terms coming from Poisson coupling and with the same nonintegrable $\gamma=1$5 decay as the principal obstruction (Jang et al., 2011).

Not all isothermal Euler–Poisson dispersions are Klein–Gordon in the strict sense. In the three-dimensional ion model with Boltzmann electrons, Guo and Pausader derive the linearized scalar equation

$\gamma=1$6

whose Fourier frequency is

$\gamma=1$7

This dispersion is wave-like at high frequency, nondegenerate at low frequency, and has an inflection point in $\gamma=1$8; the corresponding linear group satisfies $\gamma=1$9 decay $n_0$0 and $n_0$1 decay $n_0$2 in the norms used in the paper (Guo et al., 2010).

For the three-dimensional two-fluid isothermal system, Ionescu and Lie diagonalize the irrotational linearized equations into two dispersive modes $n_0$3 and $n_0$4 with dispersion relations $n_0$5 and $n_0$6. Both are Klein–Gordon-type in the sense that $n_0$7 as $n_0$8, and the linear decay is $n_0$9 in $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$0 in three dimensions. Their nonlinear analysis further isolates null-type cancellations in the ion mode and classifies bilinear resonances into several cases, including a strong low-frequency degeneracy for the ion-like dispersion that is compensated by derivative structure (Guo et al., 2013).

A recurring analytical point is therefore that the isothermal law does not determine the dispersive class by itself. In the electron-fluid model with immobile ions, Poisson coupling produces an exact Klein–Gordon mass term after normalization; in ion-acoustic and two-fluid models, the resulting dispersions are more varied, but still sufficiently structured to permit normal forms, resonance analysis, or asymptotic reductions (Li et al., 2011, Guo et al., 2010).

3. Cauchy theory: global existence, scattering, and damping

The most complete small-data result in two dimensions is Li and Wu’s solution of the full small irrotational Cauchy problem for the repulsive Euler–Poisson system. For $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$1, $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$2, and $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$3, they define a control norm $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$4 combining high Sobolev regularity, weighted dispersive bounds, and an $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$5 component. If

$$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$6

then there is a unique global smooth solution

$$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$7

with $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$8, and there exists $$\partial_t n+\nabla\!\cdot(nu)=0,\qquad m_e n(\partial_t u+u\!\cdot\nabla u)+\nabla p(n)=e n\nabla\phi,\qquad \Delta\phi=4\pi e (n-n_0),$$9 such that $\int_{\mathbb R^2}(n-n_0)\,dx=0$0 in $\int_{\mathbb R^2}(n-n_0)\,dx=0$1. The method combines diagonalization, normal forms converting quadratic interactions into cubic ones, hidden-derivative identities, phase-derivative transformations, multiscale frequency decompositions, and Coifman–Meyer control of the nonlocal Riesz multipliers. The paper states explicitly that the analysis extends to arbitrary $\int_{\mathbb R^2}(n-n_0)\,dx=0$2, including the isothermal case $\int_{\mathbb R^2}(n-n_0)\,dx=0$3 (Li et al., 2011).

An earlier two-dimensional result by Jang, Li, and Zhang constructed global smooth solutions in a final-data formulation. Their analysis subtracts a resonant correction built from the linear self-interaction, localizes the quasi-linear part in time, and then uses a modified energy scheme with an infinite-time Gronwall argument. The paper emphasizes that the same quasi-linear Klein–Gordon mechanism persists under the isothermal specialization $\int_{\mathbb R^2}(n-n_0)\,dx=0$4 (Jang et al., 2011).

In three dimensions, the isothermal two-fluid problem admits a global small-data theory for localized irrotational perturbations of a neutral background. Theorem D.1 of Ionescu and Lie fixes $\int_{\mathbb R^2}(n-n_0)\,dx=0$5, $\int_{\mathbb R^2}(n-n_0)\,dx=0$6, and assumes

$\int_{\mathbb R^2}(n-n_0)\,dx=0$7

together with $\int_{\mathbb R^2}(n-n_0)\,dx=0$8. The resulting solution is global in $\int_{\mathbb R^2}(n-n_0)\,dx=0$9, remains irrotational for all time, and satisfies the decay estimate

$$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$0

together with scattering in the diagonalized variables (Guo et al., 2013).

For the three-dimensional ion-acoustic system with Boltzmann electrons, Guo and Pausader prove global smooth irrotational small-amplitude solutions in Sobolev spaces $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$1, $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$2. Their proof relies on a normal form with singular phase denominators, refined bilinear multiplier estimates, and an energy functional augmented by an $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$3-component to absorb the most singular normal-form contribution (Guo et al., 2010).

These dispersive small-data theories do not exhaust the isothermal regime. On the periodic torus $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$4, $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$5, the damped isothermal Euler–Poisson system

$$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$6

admits unique global smooth solutions for arbitrarily large $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$7 data with density bounded away from vacuum, provided $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$8 is sufficiently large. The solutions relax exponentially to $$\partial_t n + \partial_s\big((1+n)u\big)=0,\qquad \partial_t u + u\,\partial_s u + K\,\partial_s\log(1+n) = -\,\partial_s\phi,\qquad -\,\partial_s^2\phi=(1+n)-e^\phi,$$9 at rate $K=T_i/T_e>0$0, and in the large-damping regime $K=T_i/T_e>0$1 (Choi et al., 3 Jun 2026). This is a distinct, large-data, non-dispersive mechanism based on damping, hidden parabolicity in the density equation, and comparison with a drift–diffusion–Poisson system.

4. Solitary waves, embedded neutral modes, and Evans-function stability

One-dimensional isothermal Euler–Poisson equations support solitary waves in the super–ion–sonic regime. For the ion/Boltzmann-electron model with ion sound speed $K=T_i/T_e>0$2, the traveling coordinate $K=T_i/T_e>0$3, and $K=T_i/T_e>0$4, Kodama, Miki, and Nakanishi show that for each $K=T_i/T_e>0$5 there exists a unique, up to translation, even solitary wave $K=T_i/T_e>0$6 with $K=T_i/T_e>0$7, $K=T_i/T_e>0$8, and $K=T_i/T_e>0$9 on $1+n$0, and with exponential decay as $1+n$1. The profiles satisfy

$1+n$2

and admit a Hamiltonian first integral through the function $1+n$3 (Bae et al., 2020).

Linearization about such a solitary wave produces a nonlocal operator $1+n$4 whose spectral structure is delicate because $1+n$5 is embedded in the essential spectrum in unweighted $1+n$6. Translation invariance and the speed parameter generate the Jordan chain

$1+n$7

so the geometric multiplicity is one and the algebraic multiplicity is at least two (Bae et al., 2020). In unweighted $1+n$8, the essential spectrum fills the imaginary axis; in exponentially weighted spaces $1+n$9, conjugation by the weight moves the essential spectrum strictly into the left half-plane when $c_s^2\nabla\log \rho$00. On the weighted spectral domain $c_s^2\nabla\log \rho$01, the Evans function is analytic and vanishes only at $c_s^2\nabla\log \rho$02, with order two, for sufficiently small amplitude. The same paper proves spectral stability in $c_s^2\nabla\log \rho$03, and in weighted space it obtains asymptotic linear stability modulo the two neutral modes: $c_s^2\nabla\log \rho$04 for data orthogonal to the spectral projection onto $c_s^2\nabla\log \rho$05 (Bae et al., 2020).

The small-amplitude structure of solitary waves is tied rigorously to KdV. Chae and Hwang study the stretched moving frame

$c_s^2\nabla\log \rho$06

and prove existence of smooth solitary waves for the one-dimensional Euler–Poisson system that converge to the KdV soliton

$c_s^2\nabla\log \rho$07

For every integer $c_s^2\nabla\log \rho$08, the remainder satisfies an exponentially weighted $c_s^2\nabla\log \rho$09 bound of order $c_s^2\nabla\log \rho$10 (Bae et al., 2018). In the Evans-function language of the stability paper, the long-wave/small-amplitude scaling also leads to uniform convergence of the Euler–Poisson Evans function to the explicit KdV Evans function, which vanishes only at the origin with order two (Bae et al., 2020). The isothermal solitary-wave problem is thus one of the clearest settings in which profile construction, spectral stability, and asymptotic reduction to KdV are simultaneously rigorous.

5. Singular limits, asymptotic reductions, and boundary layers

A major branch of the theory concerns singular limits in which the isothermal Euler–Poisson system approaches a reduced dispersive, diffusive, or quasineutral model. In the quasineutral limit on $c_s^2\nabla\log \rho$11, letting $c_s^2\nabla\log \rho$12 in

$c_s^2\nabla\log \rho$13

formally yields neutrality $c_s^2\nabla\log \rho$14 and the limiting isothermal Euler system

$c_s^2\nabla\log \rho$15

with only the impermeability condition $c_s^2\nabla\log \rho$16 at the boundary (Gérard-Varet et al., 2011). Because the Dirichlet datum $c_s^2\nabla\log \rho$17 is generally incompatible with neutrality, Gérard-Varet, Han-Kwan, and Rousset construct a Debye sheath of thickness $c_s^2\nabla\log \rho$18 through a high-order boundary-layer expansion. The leading profile $c_s^2\nabla\log \rho$19 solves

$c_s^2\nabla\log \rho$20

with decay as $c_s^2\nabla\log \rho$21. They prove $c_s^2\nabla\log \rho$22-convergence to the limiting Euler flow with rate $c_s^2\nabla\log \rho$23, and sharper convergence to the full boundary-layer corrected approximation (Gérard-Varet et al., 2011).

The supersonic outflow case requires a different stabilization mechanism. In the half-space formulation with

$c_s^2\nabla\log \rho$24

Gérard-Varet, Han-Kwan, and Rousset analyze the Bohm regime $c_s^2\nabla\log \rho$25, where the limiting Euler characteristics are all outgoing and the boundary layer includes an $c_s^2\nabla\log \rho$26 normal-velocity correction. The Debye sheath is again of thickness $c_s^2\nabla\log \rho$27, but its stability is obtained through Goodman-type weights adapted to outward convection, yielding convergence to the Euler solution plus explicit layer correctors (Gérard-Varet et al., 2014).

In one space dimension, the long-wave/small-amplitude scaling

$c_s^2\nabla\log \rho$28

reduces the isothermal ion-acoustic Euler–Poisson system to KdV. Wang proves that the leading amplitude $c_s^2\nabla\log \rho$29 satisfies

$c_s^2\nabla\log \rho$30

with $c_s^2\nabla\log \rho$31 and $c_s^2\nabla\log \rho$32, and establishes global-in-time convergence of the Euler–Poisson solution to the KdV asymptotics in $c_s^2\nabla\log \rho$33, with an $c_s^2\nabla\log \rho$34 remainder in the slow variables (Guo et al., 2012). Under a uniform magnetic field, Colin and Lannes derive the Zakharov–Kuznetsov equation from the magnetized Euler–Poisson system, first for cold plasma and then with the isothermal pressure term, while also proving local well-posedness for the Cauchy problem in dimensions two and three (Lannes et al., 2012).

The large-friction limit produces a different reduced model. On $c_s^2\nabla\log \rho$35, $c_s^2\nabla\log \rho$36, the damped isothermal Euler–Poisson equations admit the slow-time rescaling

$c_s^2\nabla\log \rho$37

under which the limit $c_s^2\nabla\log \rho$38 is the drift–diffusion–Poisson system

$c_s^2\nabla\log \rho$39

The density converges with rate $c_s^2\nabla\log \rho$40, and after subtracting a damped-heat initial layer the rescaled flux converges with the same rate to the drift–diffusion flux $c_s^2\nabla\log \rho$41 (Choi et al., 3 Jun 2026).

6. Singularity formation and self-gravitating isothermal flows

The isothermal closure does not imply global smoothness in every geometry or sign regime. For the one-dimensional ion/Boltzmann-electron model

$c_s^2\nabla\log \rho$42

Hou, Huang, and Li give a constructive proof of finite-time cusp formation from smooth data near a Burgers self-similar blow-up profile. In Riemann variables $c_s^2\nabla\log \rho$43, $c_s^2\nabla\log \rho$44, they build a renormalized dynamics around the Burgers profile $c_s^2\nabla\log \rho$45 and show that the solution can develop a $c_s^2\nabla\log \rho$46 singularity in finite time while remaining $c_s^2\nabla\log \rho$47 at the blow-up time. More precisely, for any $c_s^2\nabla\log \rho$48,

$c_s^2\nabla\log \rho$49

whereas for any $c_s^2\nabla\log \rho$50,

$c_s^2\nabla\log \rho$51

as $c_s^2\nabla\log \rho$52 (Bae et al., 2024). This result is specific to a one-dimensional ion-acoustic regime, but it shows that electrostatic coupling does not universally suppress finite-time singularity formation.

For attractive self-gravity, the mathematical emphasis shifts from dispersive stabilization to geometric structure, entropy solutions, and equilibrium configurations. Makino gives coordinate-system derivations of the self-gravitating isothermal Euler–Poisson equations in Cartesian, cylindrical, and spherical coordinates, together with the enthalpy $c_s^2\nabla\log \rho$53 and the hydrostatic reduction leading to the isothermal Lane–Emden equation

$c_s^2\nabla\log \rho$54

under the scaling $c_s^2\nabla\log \rho$55 (Makino, 2017). In the exterior of a ball, Li, Li, and Zhu prove global existence of spherically symmetric entropy solutions for the isothermal self-gravitating problem by a fractional-step Lax–Friedrichs scheme and compensated compactness. In weighted radial variables $c_s^2\nabla\log \rho$56, $c_s^2\nabla\log \rho$57, they obtain global bounds

$c_s^2\nabla\log \rho$58

for almost every $c_s^2\nabla\log \rho$59, allowing vacuum and shocks (Liu, 2023).

Rotation can also regularize the attractive isothermal flow. Kwong and Yuen construct exact periodic solutions of the two-dimensional self-gravitating isothermal Euler–Poisson equations with

$c_s^2\nabla\log \rho$60

where $c_s^2\nabla\log \rho$61 solves the Emden-type ODE

$c_s^2\nabla\log \rho$62

For $c_s^2\nabla\log \rho$63 and $c_s^2\nabla\log \rho$64, the solutions are nontrivially periodic unless $c_s^2\nabla\log \rho$65 and $c_s^2\nabla\log \rho$66, in which case they are steady. The paper interprets the $c_s^2\nabla\log \rho$67 term as the rotational mechanism preventing the blowup found earlier in non-rotating two-dimensional isothermal solutions (Kwong et al., 2014).

Taken together, these results show that the phrase “isothermal Euler–Poisson system” names a structurally coherent but mathematically diverse family. In repulsive plasma settings it supports Klein–Gordon dispersion, scattering, Evans-function stability, KdV and Zakharov–Kuznetsov asymptotics, and overdamped drift–diffusion limits; in attractive settings it supports boundary layers, hydrostatic Lane–Emden equilibria, entropy solutions with vacuum, periodic rotating states, and, in some one-dimensional plasma regimes, constructive cusp singularities (Li et al., 2011, Bae et al., 2020, Gérard-Varet et al., 2011, Bae et al., 2024).