We consider a hydromagnetic system with quasilinear quadratic terms. By using multiscale transform for the time and space, we obtain formally the nonlinear Schrödinger (NLS) equation. We further justify the NLS approximation rigorously based on a uniform estimate for the error R between exact solutions of the hydromagnetic system and the formal approximation solution obtained via the NLS equation in Sobolev norms and a long time scale 𝒪(ϵ ^-2) . Compared to our previous paper [Liu, Pu, Commun Math Phys 371(2): 357–398, 2019], there are more terms containing both of order 𝒪(ϵ ^2) and 𝒪(ϵ ) to be eliminated by using corresponding normal-form transforms due to the magnetic effects. In addition, we split the modified energy as two parts to simplify the process of obtaining the uniform energy estimate.

In this paper, we consider the quantum magnetohydrodynamic model for quantum plasmas. We first derive uniform estimates for the global smooth solutions in terms of the quantum coefficient ħ and the Hall coefficient ϵ . Then we establish the existence of global solutions and derive optimal convergence rates using the energy method. Next, applying the Lions-Aubin lemma, we prove that the unique smooth solution of the three-dimensional Hall-quantum-magnetohydrodynamic system converges globally in time to the smooth solution of the three-dimensional Navier-Stokes system as ħ and ϵ tend to zero. Furthermore, we provide the convergence rate estimates for any given positive time.

In this paper we justify the nonlinear Schrodinger (NLS) approximation for the one-dimensional non-isentropic Euler-Poisson (NEP) system of monatomic ionic fluids over a physically relevant timespan O(epsilon(-2)) in Sobolev space H-s. The non-isentropic condition induces two difficulties compared to the isentropic case: more terms of O(epsilon(2)) and O(epsilon) need to be eliminated, and the interaction terms for the error (R-0, R-1, R-1) induce more difficulties in obtaining the uniform estimates for the error. We obtain the NLS approximation by using a series of normal-form transformations and taking advantage of the structure of the NEP system.

The nonlinear Schrödinger (NLS) equation is known as a universal equation describing the evolution of the envelopes of slowly modulated spatially and temporarily oscillating wave packet in various dispersive systems. In this paper, we prove that under a certain multiple scale transformation, solutions to the Euler-Poisson system can be approximated by the sums of two counter-propagating waves solving the NLS equations. It extends the earlier results [Liu and Pu, Comm. Math. Phys., 371(2), (2019)357-398], which justify the unidirectional NLS approximation to the Euler-Poisson system for the ion-coustic wave. We demonstrate that the solutions could be convergent to two counter-propagating wave packets, where each wave packet involves independently as a solution of the NLS equation. We rigorously prove the validity of the NLS approximation for the one-dimensional Euler-Poisson system by obtaining uniform error estimates in Sobolev spaces. The NLS dynamics can be observed at a physically relevant timespan of order 𝒪(ε^-2). As far as we know, this result is the first construction and valid proof of the bidirectional NLS approximation.

. This paper considers the quantum magnetohydrodynamic model for quantum plasmas with potential force. We prove the optimal decay rates for the higher order derivative of the global small solution to the stationary state in the whole space. Specially, we show the optimal (center dot)Hk decay rate similar to (1 + t) 34 + k2 , which improves the work of Xu and Pu (Discrete Contin. Dyn. Syst. Ser. B, 26, 2021). The proof is based on the optimal decay of the linearized equations, multi-frequency decompositions, and nonlinear energy estimates.

This paper considers a stability result for the three-dimensional Navier-Stokes-Korteweg system uniformly in the inviscid limit. We obtain a unique global smooth solution close to the constant equilibria (1, 0), independent of the viscosity parameter , assuming that the potential part of the initial velocity is small independently of the viscosity parameter while the incompressible part of the initial velocity is small compared to . The proof is based on the parabolic energy estimates and dispersive properties involving the method of space time resonances. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

As a formal approximation, the nonlinear Schrödinger (NLS) equation can be derived to describe the evolution of the envelopes of small oscillating wave packets-like solutions to the Euler-Poisson system. In this paper we rigorously justify that the wave packets for the non-isentropic Euler-Poisson system can be approximated by solutions of the NLS equation over a physically relevant 𝒪(ε^-2) time scale. Besides the difficulties such as resonances at k=0 and k=± k_0 and loss of derivatives arising in the modulation approximation problem in the isentropic Euler-Poisson system, new difficulties arise in the non-isentropic case. In the non-isentropic Euler-Poisson system, new resonances at wave number k=± 2k_0 appear which necessitate rescaling the correction to the modulation approximation differently for different wave numbers. In addition, it is more difficult to obtain the uniform estimates for the error (R_0,R_1,R_-1) between the real solutions and the approximate solutions, due to the extra interactions with the temperature. To overcome the difficulties aroused by resonances and loss of derivatives, we find several important structural identities between the diagonalized unknowns and apply a series of normal-form transforms, to obtain uniform estimates for the error over the desired 𝒪(ε^-2) long time scale.

This paper presents the derivation of a (3+1)-dimensional quantum Zakharov-Kuznetsov (QZK) equation for ion acoustic waves. Using a singular perturbation method within the long-wavelength limit of the (3+1)-dimensional quantum Euler-Poisson system, we demonstrate that the QZK equation can be systematically derived through the Gardner-Morikawa transformation. The derived equation is valid over a time interval of order O(epsilon-3/2)$O(\varepsilon <^>{-3/2})$.

In this paper, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is established under the assumption of initial data (v(0),T-0) is an element of H-1 with additional regularity partial derivative(z)v(0 )is an element of L-4. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, with the convergence rate O(lambda(min{2,beta-2,gamma-2}/2))(2<beta,gamma < infinity), in the cases of initial data (v(0),T-0)is an element of H(1 )with partial derivative(z)v(0 )is an element of L-4 and initial data (v(0),T-0)is an element of H-2, respectively, as the aspect ratio lambda goes to zero.

This paper is concerned with the linear stability analysis for the Couette flow of the Euler-Poisson system for both ionic fluid and electronic fluid in the domain TxR. We establish upper and lower bounds of the linearized solutions of the Euler-Poisson system near Couette flow. In particular, inviscid damping for the solenoidal component of the velocity is obtained.

This paper considers the stability of the 3D incompressible MHD-Boussinesq system with mixed dissipation in . The main purpose of this paper is to prove the global stability of perturbations near the hydrostatic equilibrium for the 3D incompressible MHD-Boussinesq system with mixed dissipation and damping. By using the energy methods, we obtain that this system possesses a global solution for initial data in .

The Maxwell-Landau-Lifshitz equation with spin accumulation is studied in the paper. We prove the existence and uniqueness of global solutions using energy estimates method in two-dimensional space.

In this paper, we study the linear stability of Couette flow for 2D compressible Navier-Stokes-Poisson system at high Reynolds number in the domain 𝕋×ℝ with initial perturbation in Sobolev spaces. We establish the upper bounds for the solutions of linearized system near Couette flow. In particular, we show that the irrotational component of the perturbation may have a transient growth, after which it decays exponentially.

In the present paper, we study the linear stability of perturbations around the Couette flow for a two dimensional compressible non-isentropic Euler-Poisson system in the domain I x I[8. A Lyapunov type instability result for the density and the temperature is obtained, and in particular, the inviscid damping for the solenoidal component of the velocity field is proved.

In this paper, the Landau-Lifshitz-Gilbert equation with helicity is considered. In R3 or a bounded regular domain Ω of R3, we establish the global existence of a weak solution. In Rn, a global existence criterion and uniqueness of the smooth solution are given. In R1, the local smooth solution is indeed global with large initial data. In R2, we prove the existence of a global weak solution, which is smooth with the exception of at most finite singular points.

In this paper, we investigate the three-dimensional axisymmetric incompressible magnetohydrodynamics (MHD) system. Under the slip boundary conditions for both the velocity and the magnetic field, we prove that the MHD system has a unique global solution in the exterior of a cylinder Π={x=(x1,x2,x3)∈R3||xh|>1,xh=(x1,x2)} under initial axisymmetric data.

In this paper, we show the existence of a steady-state solution [rho 0, u0 equivalent to 0, Phi 0] of the three-dimensional Euler-Poisson-Korteweg system in a bounded domain with physical boundary conditions using calculus of variations. Based on the existence result, the asymptotic stability for the EulerPoisson-Korteweg system is established near the given steady state.

This paper is concerned with the linear stability analysis for the Couette flow of the Euler-Poisson system for both ionic fluid and electronic fluid in the domain T×R. We establish the upper and lower bounds of the linearized solutions of the Euler-Poisson system near Couette flow. In particular, the inviscid damping for the solenoidal component of the velocity is obtained.

In the previous paper Liu and Pu (2019) [17], we proved the nonlinear Schrödinger (NLS) approximation for the Euler-Poisson system for a hot ion-acoustic plasma, where the appearance of resonances and the loss of derivatives of quadratic terms are the main difficulties. Note that when the ion-acoustic plasma is hot, the Euler-Poisson system is Friedrich symmetrizable, and the linear term can provide a derivative to compensate the loss of derivative induced by quadratic terms after diagonalizing the linearized system. When the ion-acoustic plasma is cold, as considered in the present paper, the situation is very different from that in the previous paper. The Euler-Poisson system becomes a pressureless system, so the linear operator has no regularity, and the quadratic terms still lose a derivative in the diagonalized system. This fact makes it more difficult to prove the NLS approximation of Euler-Poisson system for a cold ion-acoustic plasma. In this paper, we take advantage of the special structure of the pressureless Euler-Poisson system and the normal-form transformation to deal with the difficulties caused by resonances, especially the difficulties caused by derivative loss, in order to prove the NLS approximation.

This paper is concerned with the initial–boundary value problem of the three-dimensional primitive equations for oceanic dynamics. The global existence of strong solutions to the primitive equations without vertical diffusivity is proved under the assumption of initial data (v0,T0)∈H1.