Hyperbolic Elliptic Coupled System Research Articles

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

Existence of transonic shocks in a cylindrical nozzle with given exit normal momentums

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables.

Asymptotic stability of travelling wave to a hyperbolic-elliptic coupled system of the radiating gas on half line

Asymptotic stability of travelling wave to a hyperbolic-elliptic coupled system of the radiating gas on half line

Convergence rates to the planar stationary solution to a 2D model of the radiating gas on half space

This paper is concerned with the asymptotic stability of a planar stationary solution to an initial-boundary value problem for a two-dimensional hyperbolic–elliptic coupled system of the radiating gas on half space. We show that the solution to the problem converges to the corresponding planar stationary solution as time tends to infinity under small initial perturbation. This result is proved by the standard L2-energy method and the div–curl decomposition. Moreover, we prove that the solution (u, q) converges to the corresponding planar stationary solution at the rate t−α/2−1/4 for the non-degenerate case and t−1/4 for the degenerate case. The proof is based on the time and space weighted energy method.

Supersonic Euler-Poisson flows in divergent nozzles

This paper concerns the steady Euler-Poisson system in a two-dimensional divergent nozzle of finite length. The structural stability of supersonic potential flows for the Euler-Poisson system is transformed into the well-posedness of a second order quasilinear hyperbolic-elliptic coupled system with nonlinear boundary conditions. The well-posedness and the nonlinear structural stability of such supersonic flows under perturbations of boundary conditions are established by combining the iteration method with the estimates for hyperbolic-elliptic system.

Asymptotic Stability of Solutions to a Hyperbolic-Elliptic Coupled System of Radiating Gas on the Half Line

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of radiating gas, where the data on the boundary and at the far field state are defined as $u_-$ and $u_+$ satisfying $u_-<u_+$. For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara [SIAM J. Math. Anal., 29 (1998), pp. 293--308] that the solution tends toward a rarefaction wave or stationary solution or superposition of these two kinds of waves depending on the distribution of $u_\pm$. Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: $u_-<u_+=0$ such that the system has inevitable singularities. The stability of the rarefaction wave, the stationary solution, and their superposition is proved by applying the standard $L^2$-energy method.

Structural Stability of Supersonic Solutions to the Euler–Poisson System

The structural stability for supersonic solutions of the Euler–Poisson system for hydrodynamical model in semiconductor devices and plasmas in two dimensional domain is established, under the perturbation of the flow velocity and the strength of electric field in the horizontal direction at the entrance of a channel. First, the Euler–Poisson system in the supersonic region is reformulated into a second order hyperbolic–elliptic coupled system together with several transport equations. One of the key ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic–elliptic coupled system, which is achieved via a delicate choice of multiplier to gain energy estimate. The nonlinear structural stability of supersonic solution in the general situation is established by combining the iteration method with the estimate for hyperbolic–elliptic system and the transport equations together.

$L^\infty$-stability of Discontinuous Traveling Waves in a Hyperbolic-Elliptic Coupled System

In the present article, we study the $L^\infty$-stability of discontinuous shock waves or supercritical shock waves which appear in a model system of radiating gases if the shock strength is greater than a certain critical value. The author previously showed [SIAM J. Math. Anal., 46 (2014), pp. 2136--2159] that all subcritical shock waves are stable to small perturbations, while in the critical case, arbitrary small perturbations could cause blow-up in a finite time. Here we prove that supercritical shock waves recover the $L^\infty$-stability without causing blow-ups for small perturbations. In the supercritical cases, the smoothing effect of radiation is not sufficient for the stabilization, but convection contributes to stability under the presence of discontinuity in the asymptotic state.

Pointwise Estimates for Solutions to a System of Radiating Gas

In this paper we focus on the initial value problem of a hyperbolic-elliptic coupled system in multi-dimensional space of a radiating gas. By using the method of Green function combined with Fourier analysis, we obtain the pointwise decay estimates of solutions to the problem.

Asymptotic behavior of rarefaction waves for a model system of a radiating gas

This article is concerned with nonlinear stability together with the corresponding convergence rates of rarefaction waves for a hyperbolic-elliptic coupled system arising in the 1D motion of a radiating gas with large initial perturbation. Compared with former results, although we ask the L2(R)-norm of the initial perturbation to be small but the L2(R)-norm of the first- and second-order derivatives of the initial perturbation with respect to the spatial variable x can be large and consequently, the H2(R)-norm of the initial perturbation can be large. AMS Subject Classifications: 34K25; 35M10.

Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system

This paper is concerned with the large time behavior of solutions to a radiating gas model, which is represented mathematically as a Cauchy problem for a one-dimensional hyperbolic–elliptic coupled system, with suitably given far field states. Suppose the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, then we can construct a “viscous contact wave” for such a hyperbolic–elliptic system. Based on the energy methods and the ellipticity of the radiation flux equation, we prove that the “viscous contact wave” is asymptotically stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.

Asymptotic behavior of solutions to a hyperbolic–elliptic coupled system in multi-dimensional radiating gas

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem of a hyperbolic–elliptic coupled system in the multi-dimensional radiating gas u t + a ⋅ ∇ u 2 + div q = 0 , − ∇ div q + q + ∇ u = 0 , with initial data u ( x 1 , … , x n , 0 ) = u 0 ( x 1 , … , x n ) → u ± , x 1 → ± ∞ . First, for the case with the same end states u − = u + = 0 , we prove the existence and uniqueness of the global solutions to the above Cauchy problem by combining some a priori estimates and the local existence based on the continuity argument. Then L p -convergence rates of solutions are respectively obtained by applying L 2 -energy method for n = 1 , 2 , 3 and L p -energy method for 3 < n < 8 and interpolation inequality. Furthermore, by semigroup argument, we obtain the decay rates to the diffusion waves for 1 ⩽ n < 8 . Secondly, for the case with the different end states u − < u + , our main concern is that the corresponding Cauchy problem in n-dimensional space ( n = 1 , 2 , 3 ) behaviors like planar rarefaction waves. Its convergence rate is also obtained by L 2 -energy method and L 1 -estimate.

The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions

In this paper, we study the time-asymptotic behavior of solutions for a hyperbolic–elliptic coupled system known as a model system of the radiating gas in R n . When the initial perturbation corresponding to a positive constant state is sufficiently small in H s ( R n ) , the global existence and pointwise estimates of the solution are obtained by using the method of the Green function combined with some energy estimates. Furthermore, we obtain the L p , 1 ≤ p ≤ + ∞ , convergence rate of the solution.

ASYMPTOTIC DECAY TOWARD THE PLANAR RAREFACTION WAVES FOR A MODEL SYSTEM OF THE RADIATING GAS IN TWO DIMENSIONS

In this paper, we consider the asymptotic decay rate towards the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in two dimensions. The analysis based on the standard L2-energy method, L1-estimate and the monotonicity of profile obtained by the maximum principle.

Decay rates to the planar rarefaction waves for a model system of the radiating gas in n dimensions

In this paper, we study the asymptotic decay rates to the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in R n ( n = 3 , 4 , 5 ) if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in ( H 2 ∩ L 1 ∩ W 2 , 6 ) ( R n ). The analysis is based on the L p -energy method and several special interpolation inequalities.

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.

Asymptotic stability of rarefaction wave for hyperbolic-elliptic coupled system in radiating gas

In this article, authors study the Cauch problem for a model of hyperbolic-elliptic coupled system derived from the one–dimensional system of the radiating gas. By considering the initial data as a small disturbances of rarefaction wave of inviscid Burgers equation, the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of rarefaction wave is proved. The analysis is based on a priori estimates and L 2 -energy method.

Shock waves for radiative hyperbolic-elliptic systems

The present paper deals with the following hyperbolicelliptic coupled system, modelling dynamics of a gas in presence of radiation, u t + f(u) x + Lq x = 0, { x ∈ R, t > 0, -q xx + Rq + G · u x = 0, where u ∈ R n , q ∈ R and R > 0, G, L ∈ R n . The flux function f: R n → R n is smooth and such that Vf has n distinct real eigenvalues for any u. The problem of existence of admissible radiative shock wave is considered, i.e., existence of a solution of the form (u,q)(x,t):= (U,Q)(x - st), such that (U,Q)(±∞) = (u±, 0), and u ± ∈ R n , s ∈ R define a shock wave for the reduced hyperbolic system, obtained by formally putting L = 0. It is proved that, if u_ is such that ∇λ k (u_)·r k (u_) ≠ 0 (where λ k denotes the k-th eigenvalue of ∇ f and r k a corresponding right eigenvector), and (l k (u_) ·L) (G ·r k (u_)) > 0, then there exists a neighborhood U of u_ such that for any u + ∈ U, s ∈ R such that the triple (u_, u + ; s) defines a shock wave for the reduced hyperbolic system, there exists a (unique up to shift) admissible radiative shock wave for the complete hyperbolic-elliptic system. The proof is based on reducing the system case to the scalar case, hence the problem of existence for the scalar case with general strictly convex fluxes is considered, generalizing existing results for the Burgers' flux f(u) = u 2 /2. Additionally, we are able to prove that the profile (U, Q) gains smoothness when the size of the shock |u + - u_ | is small enough, as previously proved for the Burgers' flux case. Finally, the general case of nonconvex fluxes is also treated, showing similar results of existence and regularity for the profiles.

Large BV solutions to the compressible isothermal Euler–Poisson equations with spherical symmetry

The Euler–Poisson equations for isothermal flows in charge transport are studied. The global weak entropy solution with spherical symmetry is constructed for arbitrarily large initial data in BV. The global solution is in BV and has large total variations. One difficulty is that the system of Euler–Poisson equations is a hyperbolic–elliptic coupled system, and the property of finite propagation speed does not hold. A modified Glimm scheme is used to construct the approximate solution of the Euler equations while the Poisson equation is approximated by an integral of the solution to the Euler equations. Other difficulties include the geometrical structure, the damping and some additional source terms, which make the analysis more complicated and difficult. To overcome these difficulties, the solution of a discrete ordinary differential equation is used to construct the approximate solution. It is proved that vacuum will not occur in the solution, which is important to obtain the compactness of the approximate solution in BV.

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic–elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L 1-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic–parabolic and the hyperbolic–hyperbolic relaxation limits for that system.