MULTIPLE STATIONARY SOLUTIONS OF EULER-POISSON EQUATIONS FOR NON-ISENTROPIC GASEOUS STARS

The motion of the self-gravitational gaseous stars can be described by the Euler-Poisson equations. The main purpose of this paper is concerned with the existence of stationary solutions of Euler-Poisson equations for some velocity fields and entropy functions that solve the conservation of mass and energy. Under different restriction to the strength of velocity field, we get the existence and multiplicity of the stationary solutions of Euler-Poisson system.

UNIQUENESS OF STATIONARY SOLUTIONS WITH VACUUM OF EULER-POISSON EQUATIONS

In this paper, the uniqueness of stationary solutions with vacuum of Euler-Poisson equations is considered. Through a nonlinear transformation which is a function of density and entropy, the corresponding problem can be reduced to a semilinear elliptic equation with a nonlinear source term consisting of a power function, for which the classical theory of the elliptic equations leads the authors to the uniqueness result under some assumptions on the entropy function S(x). As an example, the authors get the uniqueness of stationary solutions with vacuum of Euler-Poisson equations for S(x) =|x|θandθ∈{0}∪[2(N-2),+∞).

A NOTE ON THE EXISTENCE OF STATIONARY SOLUTIONS OF THE COMPRESSIBLE EULER-POISSON EQUATIONS WITH 6/5<γ<2

This paper is concerned with the system of Euler-Poisson equations as a model to describe the motion of the self-induced gravitational gaseous stars. When ~ 〈 7 〈 2, under the weak smoothness of entropy function, we find a sufficient condition to guarantee the existence of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy.

SOLUTIONS OF EULER-POISSON EQUATIONS IN R^n

In this article, the authors study the structure of the solutions for the EuierPoisson equations in a bounded domain of Rn with the given angular velocity and n is an odd number. For a ball domain and a constant angular velocity, both existence and nonexistence theorem are obtained depending on the adiabatic gas constant 7. In addition, they obtain the monotonicity of the radius of the star with both angular velocity and center density. They also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is different to the case of the non-rotating star.

GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS

Global in time weak solutions to the α-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to α-model regularization for the three dimension compressible EulerPoisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies γ >4/3.

GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SMOOTH SOLUTIONS TO A MULTIDIMENSIONAL NONISENTROPIC EULER-POISSON EQUATIONS

The global existence and large time behavior of smooth solutions to multidimensional nonisentropic Euler-Poisson equations are established.

BIOLOGICAL INVASION PROBLEM WITH FREE BOUNDARY NONLOCAL DIFFUSION EQUATION

In order to better describe the phenomenon of biological invasion,this paper introduces a free boundary model of biological invasion.Firstly,the right free boundary is added to the equation with logistic terms.Secondly,the existence and uniqueness of local solutions are proved by the Sobolev embedding theorem and the comparison principle.Finally,according to the relevant research data and contents of red fire ants,the diffusion area and nest number of red fire ants were simulated without external disturbance.This paper mainly simulates the early diffusion process of red fire ants.In the early diffusion stage,red fire ants grow slowly and then spread over a large area after reaching a certain number.

Normalized Solutions of Nonlinear Choquard Equations with Nonconstant Potential

In this paper,we mainly focus on a type of nonlinear Choquard equations with nonconstant potential.Under appropriate hypotheses on potential function and nonlinear terms,we prove that the above Choquard equation with prescribed 2-norm has some normalized solutions by introducing variational methods.

Two Second-Order Ecient Numerical Schemes for the Boussinesq Equations

In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.

WELL-POSEDNESS AND PEAKON SOLUTIONS FOR A HIGHER ORDER CAMASSA-HOLM TYPE EQUATION

In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.

DEEP NEURAL NETWORKS COMBINING MULTI-TASK LEARNING FOR SOLVING DELAY INTEGRO-DIFFERENTIAL EQUATIONS

Deep neural networks(DNNs)are effective in solving both forward and inverse problems for nonlinear partial differential equations(PDEs).However,conventional DNNs are not effective in handling problems such as delay differential equations(DDEs)and delay integrodifferential equations(DIDEs)with constant delays,primarily due to their low regularity at delayinduced breaking points.In this paper,a DNN method that combines multi-task learning(MTL)which is proposed to solve both the forward and inverse problems of DIDEs.The core idea of this approach is to divide the original equation into multiple tasks based on the delay,using auxiliary outputs to represent the integral terms,followed by the use of MTL to seamlessly incorporate the properties at the breaking points into the loss function.Furthermore,given the increased training dificulty associated with multiple tasks and outputs,we employ a sequential training scheme to reduce training complexity and provide reference solutions for subsequent tasks.This approach significantly enhances the approximation accuracy of solving DIDEs with DNNs,as demonstrated by comparisons with traditional DNN methods.We validate the effectiveness of this method through several numerical experiments,test various parameter sharing structures in MTL and compare the testing results of these structures.Finally,this method is implemented to solve the inverse problem of nonlinear DIDE and the results show that the unknown parameters of DIDE can be discovered with sparse or noisy data.

Convergence of Compressible Euler-Maxwell Equations to Compressible Euler-Poisson Equations

In this paper, the convergence compressible Euler-Poisson equations in a of time-dependent Euler-Maxwell equations to torus via the non-relativistic limit is studied. The local existence of smooth solutions to both systems is proved by using energy estimates for first order symmetrizable hyperbolic systems. For well prepared initial data the convergence of solutions is rigorously justified by an analysis of asymptotic expansions up to any order. The authors perform also an initial layer analysis for general initial data and prove the convergence of asymptotic expansions up to first order.

Global Smooth Solutions to Relativistic Euler-Poisson Equations with Repulsive Force

In this paper, we are concerned with the global existence of smooth solutions for the one dimen- sional relativistic Euler-Poisson equations： Combining certain physical background, the relativistic Euler-Poisson model is derived mathematically. By using an invariant of Lax＇s method, we will give a sufficient condition for the existence of a global smooth solution to the one-dimensional Euler-Poisson equations with repulsive force.

Structure-Preserving Finite-Element Schemes for the Euler-Poisson Equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and selfgravitation modeling.The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law,as well as hyperbolic invariant domain properties,such as positivity of the density and a minimum principle of the specific entropy.A detailed discussion of algorithmic details is given,as well as proofs of the claimed properties.We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme.

Nonlinear Schrödinger Approximation for the Electron Euler-Poisson Equation

The nonlinear Schr?dinger(NLS for short)equation plays an important role in describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet.In this paper,the authors study the NLS approximation by providing rigorous error estimates in Sobolev spaces for the electron Euler-Poisson equation,an important model to describe Langmuir waves in a plasma.They derive an approximate wave packet-like solution to the evolution equations by the multiscale analysis,then they construct the modified energy functional based on the quadratic terms and use the rotating coordinate transform to obtain uniform estimates of the error between the true and approximate solutions.

SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION

This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.

A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study,new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas.Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls.However,in industrial plasma reactors for semiconductor manufacturing,the plasma is partially confined by internal reactor structures.We predict the effect of the open boundary area(A′_(L,eff))and ion escape velocity(u_(i))on electron temperature and density by developing new particle and energy balance equations.Theoretically,we found a low ion escape velocity(u_(i)/u_(B)≈0.2)and high open boundary area(A′_(L,eff)/A_(T,eff)≈0.6)to result in an approximately 38%increase in electron density and an 8%decrease in electron temperature compared to values in a fully bounded reactor.Additionally,we suggest that the velocity of ions passing through the open boundary should exceedω_(pi)λ_(De)under the condition E^(2)_(0)?(Φ/λ_(De))^(2).