Global convergence of the Euler‐Poisson system for ion dynamics

We consider smooth solutions of the Euler‐Poisson system for ion dynamics in which the electron density is replaced by a Boltzmann relation. The system arises in the modeling of plasmas, where appear two small parameters, the relaxation time and the Debye length. When the initial data are sufficiently close to constant equilibrium states, we prove the convergence of the system for all time, as each of the parameters goes to zero. The limit systems are drift‐diffusion equations and compressible Euler equations. The proof is based on uniform energy estimates and compactness arguments.     

Stability of non‐constant steady‐state solutions for non‐isentropic Euler–Maxwell system with a temperature damping term

This work is concerned with the periodic problem for compressible non‐isentropic Euler–Maxwell systems with a temperature damping term arising in plasmas. For this problem, we prove the global in time existence of a smooth solution around a given non‐constant steady state with the help of an induction argument on the order of the mixed time‐space derivatives of solutions in energy estimates. Moreover, we also show the convergence of the solution to this steady state as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation of the systems with the non‐isentropic Euler–Maxwell and the isentropic Euler–Maxwell systems. Copyright © 2015 John Wiley & Sons, Ltd.     

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. For some velocity fields and entropy functions that solve the conservation of mass and energy, we consider the existence of stationary solutions of Euler–Poisson equations. Under various restriction to the strength of velocity field, different assumptions on the isentropic function and adiabatic exponent, we get the existence, multiplicity and uniqueness of the stationary solutions to the Euler–Poisson system, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions p^ν in place of the unknown variables densities n^ν.     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

Initial layers and zero‐relaxation limits of multidimensional Euler–Poisson equations

In this paper, we consider zero‐relaxation limits for periodic smooth solutions of the time‐dependent Euler–Poisson system. For well‐prepared initial data, we construct an approximate solution by an asymptotic expansion up to any order. For ill‐prepared initial data, we construct initial layer corrections in an explicit way. In both cases, the asymptotic expansions are valid in a time interval independent of the relaxation time, and their convergence is justified by establishing uniform energy estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods

We prove local existence and uniqueness for the two‐dimensional Prandtl system in weighted Sobolev spaces under Oleinik's monotonicity assumption. In particular we do not use the Crocco transform or any change of variables. Our proof is based on a new nonlinear energy estimate for the Prandtl system. This new energy estimate is based on a cancellation property that is valid under the monotonicity assumption. To construct the solution, we use a regularization of the system that preserves this nonlinear structure. This new nonlinear structure may give some insight into the convergence properties from the Navier‐Stokes system to the Euler system when the viscosity goes to 0. © 2015 Wiley Periodicals, Inc.     

On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.     

On the hydrodynamic limit of Ginzburg‐Landau wave vortices

In this paper, we study the hydrodynamic limit of the finite Ginzburg‐Landau wave vortices, which was established in [16]. Unlike the classical vortex method for incompressible Euler equations, we prove here that the densities approximated by the vortex blob method associated with the Ginzburg‐Landau wave vortices tend to the solutions of the pressure‐less compressible Euler‐Poisson equations. The convergence of such approximation is proved before the formation of singularities in the limit system as the blob sizes and the grid sizes tend to zero in appropriate rates. © 2002 John Wiley & Sons, Inc.     

Nonlinear Stability of Expanding Star Solutions of the Radially Symmetric Mass‐Critical Euler‐Poisson System

We prove nonlinear stability of compactly supported expanding star solutions of the mass‐critical gravitational Euler‐Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expansion rate of such solutions can be either self‐similar or non‐self‐similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global‐in‐time radially symmetric solutions, which are not homologous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free‐boundary problem as a degenerate quasilinear wave equation on a compact spatial domain. The problem is mass‐critical with respect to an invariant rescaling and the analysis is carried out in similarity variables. © 2017 Wiley Periodicals, Inc.     

The ultra‐relativistic Euler equations

We study the ultra‐relativistic Euler equations for an ideal gas, which is a system of nonlinear hyperbolic conservation laws. We first analyze the single shocks and rarefaction waves and solve the Riemann problem in a constructive way. Especially, we develop an own parametrization for single shocks, which will be used to derive a new explicit shock interaction formula. This shock interaction formula plays an important role in the study of the ultra‐relativistic Euler equations. One application will be presented in this paper, namely, the construction of explicit solutions including shock fronts, which gives an interesting example for the non‐backward uniqueness of our hyperbolic system. Copyright © 2014 John Wiley & Sons, Ltd.     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

Solitary Waves and N‐Particle Algorithms for a Class of Euler–Poincaré Equations

We study a class of partial differential equations (PDEs) in the family of the so‐called Euler–Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses nonsmooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particle system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when nonhomogeneous “constant stream” boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic‐like to dispersive with a nontrivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two‐particle dynamics under the resulting lack of Lipschitz continuity. Curiously, we find that for the conically shaped kernels the motion of the related two‐dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two‐dimensional system to the one‐dimensional completely integrable Shallow‐Water equation [1], while retaining the full dependence on two spatial dimensions for the single channel solutions. Finally, by comparing with an operator‐splitting pseudospectral method we illustrate the performance of the particle algorithms with respect to their Eulerian counterpart for this class of nonsmooth kernels.     

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Minimal geodesics on groups of volume‐preserving maps and generalized solutions of the Euler equations

The three‐dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations but can also be seen, following Arnold [1], as a geodesic on a group of volume‐preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden [16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as shown by Shnirelman [26]. For such data, we show that the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, are sharp measure‐valued solutions to the Euler equations in the sense of DiPerna and Majda [14]. © 1999 John Wiley & Sons, Inc.     

On the stabilizing effect of convection in three‐dimensional incompressible flows

We investigate the stabilizing effect of convection in three‐dimensional incompressible Euler and Navier‐Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three‐dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three‐dimensional Euler or Navier‐Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier‐Stokes equations. We will present numerical evidence that seems to support that the three‐dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three‐dimensional model and how the convection term in the full Euler and Navier‐Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time. © 2008 Wiley Periodicals, Inc.     

Klein‐Gordon‐Zakharov system in energy space: Blow‐up profile and subsonic limit

In this paper, we prove finite‐time blowup in energy space for the three‐dimensional Klein‐Gordon‐Zakharov (KGZ) system by modified concavity method. We obtain the blow‐up rates of solutions in local and global space, respectively. In addition, by using the energy convergence, we study the subsonic limit of the Cauchy problem for KGZ system and prove that any finite energy solution converges to the corresponding solution of Klein‐Gordon equation in energy space.     

Finite time blowup for Klein‐Gordon‐Schrödinger system

We investigate the blowup solutions to the Klein‐Gordon‐Schrödinger (KGS) system with power nonlinearity in spatial dimensions (N ≥ 2). Relying on a Lyapunov functional, we establish a perturbed virial‐type identity and prove the existence of blowup solutions for the system with a negative energy and small mass. Moreover, we obtain a new finite‐time blowup result of solutions to KGS system in the energy space by constructing a differential inequality.