International Journal of Differential Equations

International Journal of Differential Equations / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 4363296 | https://doi.org/10.1155/2020/4363296

Ismahan Binshati, Harumi Hattori, "Global Existence and Asymptotic Behavior of Solutions for Compressible Two-Fluid Euler–Maxwell Equation", International Journal of Differential Equations, vol. 2020, Article ID 4363296, 27 pages, 2020. https://doi.org/10.1155/2020/4363296

Global Existence and Asymptotic Behavior of Solutions for Compressible Two-Fluid Euler–Maxwell Equation

Academic Editor: Wen-Xiu Ma
Received24 Nov 2019
Accepted01 Jun 2020
Published29 Jun 2020

Abstract

We study the global existence and asymptotic behavior of the solutions for two-fluid compressible isentropic Euler–Maxwell equations by the Fourier transform and energy method. We discuss the case when the pressure for two fluids is not identical, and we also add friction between the two fluids. In addition, we discuss the rates of decay of norms for a linear system. Moreover, we use the result for estimates to prove the decay rates for the nonlinear systems.

1. Introduction

We consider the Cauchy problem for the first-order nonlinear two-fluid compressible isentropic Euler–Maxwell equations in three dimensions. In the following system of equations, the first equation is the conservation of the mass. The second equation is conservation of the momentum, to which we added frictional damping besides the damping . Then, the compressible two-fluid Euler–Maxwell equations can be written aswhere denotes the density of electrons () and ions (), denotes the velocity of electrons () and ions (), denotes the electric field, and denotes the magnetic field for . The initial data are given bywith the compatibility conditions

The Euler–Maxwell system (1) is a symmetrizable hyperbolic system for , and the initial value problems (1) and (2) have a local smooth solution when the initial data are smooth. The global existence of smooth solutions to the initial boundary value problem has been given in [1] by the compensated compactness method. The authors in [2, 3] studied the existence of global smooth solutions for the three-dimensional isentropic Euler–Maxwell system with small amplitude, and the periodic problem was discussed by Uedaet al. [4]. For the special case where the solution to the Euler–Maxwell equation has asymptotic limits with small parameters, see [5, 6]. The special case of the diffusive relaxation limit of the three-dimensional nonisentropic Euler–Maxwell equation is considered in [7, 8]. Two hierarchies of models of the ionospheric plasma for two-fluid Euler–Maxwell equations were presented in [9]. The Fourier transform method was considered by Duan [2, 10] and Kawashima and Ueda [11]. Jerome [12] adapted the classical semigroup-resolvent approach of Kato [13] to the Cauchy problem in and established a local smooth solution. In [2], Duan considered the case when the pressure function depend only on density, having the expression with constants and the adiabatic exponent .

In this paper, we consider the global existence of smooth solutions for the two-fluid compressible isentropic Euler–Maxwell equation extending the results of Duan [2]. In contrast to Duan, we suppose , and we add the friction where and is a constant.

Theorem 1. Let and (3) hold. There are such that ifwhere is the norm, then the Cauchy problems (1) and (2) of the Euler–Maxwell system admit a unique global solution with

We obtain the decay rates of smooth solutions by the Fourier transform. The main results are stated as follows.

Theorem 2. There are such that ifwhere is the norm, then the solution satisfies that for any ,with . Furthermore,where is Green’s matrix for the linearized system.

The proof of Theorem 1 and Theorem 2 is based on the energy method and the Fourier transform, as in [2]. There are three key steps: the first key step is the a priori estimate to establish the global solution and has the formwhere is the perturbation of solution (1) and denote the energy functional and energy dissipation rate functional as in [2]. This differs from [4, 10] because the two-fluid system has a more complex structure than one fluid, so obtaining energy estimates for the density, velocity, and electric magnetic fields for Euler–Maxwell require a different strategy. The time decay property of solutions to the nonlinear system requires the construction of functionals, capturing the optimal energy dissipation rate. The second key step is linearizing the homogeneous form of (1) and using the Fourier transform to obtain the time decay rate and the explicit representation of the solution. The third step is combining the previous two steps and applying the Fourier transform to obtain the time decay rate of the solution to the reformulated nonlinear system to finish the proof of Theorem 1. Thus, the solutions can be represented by the solution of the linearized system and the refined energy estimates using Duhamel’s principle.

We introduce some notations that we will use later in this paper. For any integer denote the Sobolev space and the -order homogeneous Sobolev space, respectively. Set . The norm of is denoted by with . The inner product in is denoted by , i.e.,

We denote for the multi-index , and the length of is . In addition, and denote some positive constants, where both and may take different values in different places.

We organize this paper as follows. In Section 2, we reformulate the Cauchy problem and consider the proof of global existence and uniqueness of solutions. In Section 3, we discuss the time rate of decay for linearized systems, and we obtain the linearized system for . Finally, in Section 4, we discuss the time decay rate of solutions of the nonlinear system (15) and complete the proof of Theorems and 2.

2. Global Solution for the Nonlinear System

2.1. Reformulation of the Problem

Denote by a smooth solution to system (1) with initial data (2) satisfying (3). Let

Define and

Note that satisfieswith the initial data

Here, we have used the notation for the special case where is substituted into (13). Note that satisfies

Suppose is a smooth solution to the initial value problem of the original Cauchy problems (1) and (2), which satisfy (3). Now, we introduce another transformation by setting , then satisfieswith the initial datasatisfying the compatibility conditionwhere .

We will assume is an integer. In addition to , define the full instant energy functional and the high-order instant energy functional viawhere are constants to be chosen later in the proof such that are small enough compared to 1 and satisfy

Define the dissipation rates by

Proposition 1. Suppose initial data satisfies (17). Then, there exist and having the forms (21) and (24), respectively, such that if is sufficiently small, the Cauchy problems (15) and (16) admit a unique, global, nonzero solution , satisfyingfor any .

Remark 1. The solutions obtained in Proposition 1 indeed represent the decay rates in time under some regularity and integrability conditions on initial data and setfor the integer .

Remark 2. Note that the existence result in Theorem 1 follows from Proposition 1, the derivation of rates of (7) and (9) in Theorem 1, and Proposition 2. The proof of Proposition 2 is analogous to that of Lemma 5.2 in [10].

2.2. A Priori Estimates

In this section, we obtain uniform-in-time a priori estimates for smooth solutions to the Cauchy problems (15) and (16) by using the classical energy method.

Theorem 3. Let be given. Suppose that is smooth, that satisfiesand that solves system (15) for . Then, there exist and having the forms (21) and (24) such that for all ,

Proof.   Performing the energy estimate, we obtain the following results:Step 1. We apply to the first equation of (15) and then multiply that equation by ; also, we apply to the second equation of (15) and then multiply that equation by ; after many steps, we getStep 2. We rewrite the first and second equations of (15) by putting the linear terms on the left-hand sides and the nonlinear terms on the right-hand sides:Let . If we apply to (34), multiply by , integrate in , and then combine the result with an application of to (35) after which we multiply by and integrate in , then we getwhereStep 3. We subtract equation (35) from equation (33) to getNow, we apply to (38), multiply by , integrate by parts in , and replace with the third equation of (16). Then, we havewhereStep 4. We apply to the third equation of (15), multiply by , integrate by parts in , and use the relationfor each . Then, we obtainwhereStep 5. Utilizing steps (1)–(4) above, we can now prove (30). Defineand note that constants are to be determined. We observe that if for , are sufficiently small, then holds. Furthermore, by letting be sufficiently small, taking , and taking the sum of (31), (36), (42), and (42), we find that there exists such that (30) is satisfied:If we now letthenIt follows thatThus,and this concludes the proof.

2.3. Proof of Global Existence

We consider the global existence of the smooth solution to the isentropic Euler–Maxwell system for a quasilinear symmetric hyperbolic system (15). Therefore, we combine those a priori estimates with the local existence of solutions to extend the local solution up to infinite time by using the continuity of .

Lemma 1 (local existence of smooth solution, see [2, 13, 14]). Assume satisfies (17). Then, there exists such that the Cauchy problems (15) and (16) admit a unique solution on with

Proof of Proposition. 1. Since (15) is a quasilinear symmetric hyperbolic system, the global existence of smooth solutions follows from the local existence result in Lemma 1 (see also Section 16 of [14]). In addition, the a priori estimate (30) in Theorem 3 and the continuity argument show that is bounded uniformly in time under the assumption that is sufficiently small. Therefore, global solutions satisfying (26) and (27) exist. This concludes the proof of Proposition 1.

3. Linearized Homogeneous System

3.1. Linearized Equations

To obtain the time decay rates of a solution to the nonlinear system (15) or (18), we consider the linearized homogeneous equations of system (18):with the given initial datawhich satisfies the compatibility conditions

Throughout this section, we let be the solution to system (51). Moreover, in this section, we introduce some notation about Fourier transform , defined bywhere is the complex number, and we use the energy method to the initial value problems (51) and (53) in Fourier space to show that there is a time-frequency Lyapunov inequality, which leads to the pointwise time-frequency upper-bound of the solution.

We will use the energy method to the initial value problems (51) and (53) in the Fourier transform to show that there is a time-frequency Lyapunov functional which is equivalent to and moreover its dissipation rate can be represented by itself.

3.2. Representation of Solution

Denote by the explicit solution to the Cauchy problems (51) and (52), satisfying (53). In this section, we study the representation of .

First, we take the time derivative for the first equation and the divergence of the second equation of system (51) and substitute . So,

By combining the two equations (55) and (56), we havewith the initial data given by

Then, taking the Fourier transform of the second-order ODE (57) with (58), we get

Now, set

Then,where

Note that the eigenvector of the matrix is given by

In the next two sections, we provide an estimate for . In Section 3.2.1, we estimate for , and in Section 3.2.2, we estimate for . To do so, we set , and we use the relation where we refer to as the “parallel part” and as the “perpendicular part.”

3.2.1. Parallel Part

We proceed with the asymptotic expansion of eigenvalues: let , be the eigenvalues of the matrix . Taking the determinant, we see the eigenvalues satisfy(i)First we consider when , since satisfies has the following asymptotic expansion:where each coefficient is given by direct computation asThus, the approximation of the eigenvalue when isTherefore, if we definethen the Green matrix for (61) is given by , and the solution isThus, after a series of calculation using symbolic manipulator, we have