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

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 L − L norms for a linear system. Moreover, we use the result for L − L estimates to prove the decay rates for the nonlinear systems.


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. e second equation is conservation of the momentum, to which we added frictional damping α ± (u ± − u ∓ ) besides the damping ] ± u ± . en, the compressible two-fluid Euler-Maxwell equations can be written as z t n ± + ∇. n ± u ± � 0, where n ± � n ± (t, x) ∈ R 3 denotes the density of electrons (n − ) and ions (n + ), u ± � u ± (t, x) ∈ R 3 denotes the velocity of electrons (u − ) and ions (u + ), E � E(t, x) ∈ R 3 denotes the electric field, and B � B(t, x) ∈ R 3 denotes the magnetic field for t > 0, x ∈ R 3 . e initial data are given by with the compatibility conditions e Euler-Maxwell system (1) is a symmetrizable hyperbolic system for n > 0, and the initial value problems (1) and (2) have a local smooth solution when the initial data are smooth.
e global existence of smooth solutions to the initial boundary value problem has been given in [1] by the compensated compactness method. e 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]. e 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 twofluid Euler-Maxwell equations were presented in [9]. e 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 R 3 and established a local smooth solution. In [2], Duan considered the case when the pressure function p ± (·) depend only on density, having the expression p ± (n ± ) � A ± n c ± with constants A + � A − > 0 and the adiabatic exponent c > 1.
where ‖.‖ N is the H N norm, then the Cauchy problems (1) and (2) of the Euler-Maxwell system admit a unique global so- We obtain the decay rates of smooth solutions by the Fourier transform. e main results are stated as follows.

Theorem 2.
ere are δ 1 > 0 and C 1 such that if where ‖.‖ 13 is the H 13 norm, then the solution [n ± (t, x), u ± (t, x), E(t, x), B(t, x)] satisfies that for any t ≥ 0, with 2 ≤ q ≤ ∞. Furthermore, where G(t, x) is Green's matrix for the linearized system. e proof of eorem 1 and eorem 2 is based on the energy method and the Fourier transform, as in [2]. ere are three key steps: the first key step is the a priori estimate to establish the global solution and has the form where V(t) is the perturbation of solution (1) and Ε N (.)and D N (.) denote the energy functional and energy dissipation rate functional as in [2]. is 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. e time decay property of solutions to the nonlinear system requires the construction of functionals, capturing the optimal energy dissipation rate. e second key step is linearizing the homogeneous form of (1) and using the Fourier transform to obtain the L p − L q time decay rate and the explicit representation of the solution. e 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 eorem 1. us, 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 N ≥ 0, H N and _ H N denote the Sobolev space H N (R 3 ) and the N th -order homogeneous Sobolev space, respectively. Set L 2 � H 0 . e norm of H N is denoted by ‖ · ‖ N with ‖ · ‖ � ‖ · ‖ 0 . e inner product in International Journal of Differential Equations We denote z α � z α 1 , and the length of α is |α| � α 1 + α 2 + α 3 . In addition, C and λ denote some positive constants, where both C and λ 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 9 × 9. Finally, in Section 4, we discuss the time decay rate of solutions of the nonlinear system (15) and complete the proof of eorems and 2.

Global Solution for the Nonlinear System
2.1. Reformulation of the Problem. Denote by [n ± , u ± , E, B] a smooth solution to system (1) with initial data (2) satisfying (3). Let Note that V satisfies with the initial data Here, we have used the notation V 0 � [σ ±0 , υ ±0 , E 0 , B 0 ] for the special case where [n ±0 , u ±0 , E 0 , B 0 ] is substituted into (13). Note that V 0 satisfies International Journal of Differential Equations Suppose U ≔ [n ± , u ± , E, B] 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 ρ ± (t, with the initial data satisfying the compatibility condition where ρ ±0 � n ±0 − 1. We will assume N ≥ 4 is an integer. In addition to V � [σ ± , υ ± , E, B], define the full instant energy functional Ε N (V(t)) and the high-order instant energy functional where 0 < k 3 ≪ k 2 ≪ k 1 ≪ 1 are constants to be chosen later in the proof such that k i , (i � 1, 2, 3) are small enough compared to 1 and satisfy Define the dissipation rates D N (V(t)) and D h N (V(t)) by for any t ≥ 0.
for the integer m ≥ 4.

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

A Priori Estimates.
In this section, we obtain uniformin-time a priori estimates for smooth solutions to the Cauchy problems (15) and (16) by using the classical energy method.
and that V solves system (15) for t ∈ (0, T). en, there exist E N (.) and D N (.) having the forms (21) and (24) such that for all 0 ≤ t ≤ T, Proof. Performing the energy estimate, we obtain the following results: Step 1. We apply z α to the first equation of (15) and then multiply that equation by z α σ ± ; also, we apply z α to the second equation of (15) and then multiply that equation by z α υ ± ; after many steps, we get Step 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: If we now let It follows that International Journal of Differential Equations us, and this concludes the proof.
We consider the global existence of the smooth solution to the isentropic Euler-Maxwell system for a quasilinear symmetric hyperbolic system (15). erefore, 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 E N (V(t)).

and (16) admit a unique solution on
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 eorem 3 and the continuity argument show that E N (V(t)) is bounded uniformly in time under the assumption that E N (V 0 ) > 0 is sufficiently small. erefore, global solutions satisfying (26) and (27)

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 data which satisfies the compatibility conditions roughout this section, we let U � [ρ ± , u ± , E, B] be the solution to system (51). Moreover, in this section, we introduce some notation about Fourier transform f: where i 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 |U(t, k)| 2 and moreover its dissipation rate can be represented by itself.

3.2.
Representation of Solution. Denote by U � [ρ ± , u ± , E, B] � e tL U 0 the explicit solution to the Cauchy problems (51) and (52), satisfying (53). In this section, we study the representation of U.
First, we take the time derivative for the first equation and the divergence of the second equation of system (51) and substitute ∇ · E � ρ + − ρ − . So, By combining the two equations (55) and (56), we have with the initial data given by International Journal of Differential Equations en, taking the Fourier transform of the second-order ODE (57) with (58), we get Now, set en, where Note that the eigenvector of the matrix A(k) is given by In the next two sections, we provide an estimate for u ± , ρ ± , E, and B. In Section 3.2.1, we estimate for u ± and ρ ± , and in Section 3.2.2, we estimate for u ± , E, and B. To do so, we set k � k/|k|, and we use the relation u ± � kk · u ± − k × (k × u ± ) where we refer to kk · u ± as the "parallel part" and k × (k × u ± ) as the "perpendicular part."

Parallel Part.
We proceed with the asymptotic expansion of eigenvalues: let λ j (k), j � 1, 2, 3, 4, be the eigenvalues of the matrix A(k). Taking the determinant, we see the eigenvalues satisfy λ j (k) has the following asymptotic expansion: where each coefficient λ (n) j is given by direct computation as us, the approximation of the eigenvalue when |k| ⟶ 0 is International Journal of Differential Equations erefore, if we define then the Green matrix G for (61) is given by , and the solution is us, after a series of calculation using symbolic manipulator, we have where θ is the sum of real parts of λ j , j � 1, 2, 3, 4. us, In fact, we know that erefore, plugging (73) in the second and the fourth equations of (72), we obtain (ii) When |k| ⟶ ∞, λ j (k) has the following asymptotic expansion: where each coefficient μ (n) j is given by direct computation as e approximation of the eigenvalue when |k| ⟶ ∞ is So, G � B(t)B − 1 (0) is the Green matrix for (61), and the solution is us, after a series of calculation using the symbolic manipulator, we obtain International Journal of Differential Equations erefore, after plugging (72) into the second and fourth equations of (79), we obtain (iii) When 0 < |k| < ∞, we consider the Routh-Hurwitz stability condition of the characteristic polynomial (64). at is, if we write (64) in the form e system stability requires It can be shown that, in our case, the conditions corresponding to (82) are given, respectively, by It is not difficult to show that the above inequalities are satisfied, and this implies that all roots of the characteristic equation have negative real parts.
Although the eigenvalues may coalesce, the computations in (i), (ii), and (iii) show that coalescence occurs when the real parts of the eigenvalues are negative. erefore, the stability conditions are satisfied. B(t, k)),

Perpendicular Part. Now, we consider
when t > 0 and |k| ≠ 0. Taking the curl of the second, third, and fourth equations in system (51), we get Now, taking the Fourier Transform and multiplying by − ik, we obtain Subtracting the first equation from the second equation in (86), we obtain and we simplify the above computation by letting where we write the initial data in the form with Taking the time derivative of z t M 3 and substituting z t M 5 and z t M 4 as given in (89), we get that Now, taking the time derivative of (93) and replacing z t M 5 by the first equation of (89) and taking the sum with (93), we get with the initial data Note that the characteristic equation of (94) is (97) en, equation (94) is written as where Asymptotic Expansion of Eigenvalues. Let λ j (k) be the eigenvalues of the matrix Φ(k). We will find the asymptotic expansion of the eigenvalues λ j (k) for|k| ⟶ 0 and|k| ⟶ ∞. e eigenvalues λ j (k), j � 1, 2, 3, are the solutions of the characteristic equation which can be written as det(Φ(k) − λI) � λ 3 + 1 + α + + α − λ 2 + 2 +|k| 2 λ (i) When |k| ⟶ 0, λ j (k) has the following asymptotic expansion: where each coefficient λ (n) j is given by the direct computation as International Journal of Differential Equations (103) us, the approximation of the eigenvalue when |k| ⟶ 0 is erefore, if we set the Green matrix G for (98) is given by G(t) � D(t)D − 1 (0), and the solution is represented as where each component of this solution is itself a 3 × 3 diagonal matrix. We integrate the first and the third equations of (89), and we get the following expressions: where

International Journal of Differential Equations
After a series of calculation using the symbolic manipulator and definition (84), the solution can be written as By using (73) and substituting (74) in the first equation of (109), we obtain Furthermore, we know from (51) that and thus Multiplying the first and the third equation of (72) by k and substituting the result into equation (112), we obtain Now, since kk · B � 0, we have Moreover, taking the sum of the first and the second equations of (86), we obtain us, Substituting the first and the second equations of (84) into equation (116), we get that Next, substituting (74) into the above computations, we obtain Now, taking the sum and difference, respectively, of (110) and (118), we obtain (ii) When |k| ⟶ ∞, λ j (k) has the following asymptotic expansion: International Journal of Differential Equations Each coefficient μ (n) j is given by direct computation as μ (1) 1 � 0, (121) e approximation of the eigenvalue when |k| ⟶ ∞ is (122) Hence, the Green matrix G for (98) is given by G(t) � D(t)D − 1 (0), and the solution is represented as where each component of this solution is itself a 3 × 3 diagonal matrix. After a series of calculation using the symbolic manipulator and equation (84), the solution can be written as By using (73) and substituting (80) into the first equation of the above computation, we obtain International Journal of Differential Equations Furthermore, we have erefore, multiplying the first and the third equation of (79) by k and substituting into the above equation, we obtain Since kk · B � 0, we get Substituting the first equation of (124) into equation Next, substituting equation (80) into equation (129), we obtain 18 International Journal of Differential Equations Now, taking the sum and difference, respectively, of equations (125) and (130), we obtain Theorem 4. Let 1 ≤ p, r ≤ 2 ≤ q ≤ ∞, and l ≥ 0 and let m ≥ 0 be an integer. Define , when r ≠ 2 or q ≠ 2 or l is not an integer, l, when r � q � 2 and l is an integer, where [·] − denotes the integer part of the argument. Suppose U 0 satisfies (53). en, for any t ≥ 0, ∇ m e tL U 0 satisfies the following time decay property: where C � C(p, q, r, l, m).