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

Binshati, Ismahan; Hattori, Harumi

doi:https://doi.org/10.1155/2020/4363296

International Journal of Differential Equations

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Research Article | Open Access

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

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

Ismahan Binshati¹and Harumi Hattori¹

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 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: 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 get where Step 3. We subtract equation (35) from equation (33) to get Now, we apply to (38), multiply by , integrate by parts in , and replace with the third equation of (16). Then, we have where Step 4. We apply to the third equation of (15), multiply by , integrate by parts in , and use the relation for each . Then, we obtain where Step 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 as Thus, the approximation of the eigenvalue when is Therefore, if we define then the Green matrix for (61) is given by , and the solution is Thus, after a series of calculation using symbolic manipulator, we have where is the sum of real parts of . Thus, In fact, we know that Therefore, plugging (73) in the second and the fourth equations of (72), we obtain(ii)When has the following asymptotic expansion: where each coefficient is given by direct computation as The approximation of the eigenvalue when is So, is the Green matrix for (61), and the solution is Thus, after a series of calculation using the symbolic manipulator, we obtain Therefore, after plugging (72) into the second and fourth equations of (79), we obtain(iii)When , we consider the Routh–Hurwitz stability condition of the characteristic polynomial (64). That is, if we write (64) in the form

The 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. Therefore, the stability conditions are satisfied.

3.2.2. Perpendicular Part

Now, we considerwhen and . Taking the curl of the second, third, and fourth equations in system (51), we get

Now, taking the Fourier Transform and multiplying by , we obtain

Subtracting the first equation from the second equation in (86), we obtainand we simplify the above computation by letting .

So,

Thus,where we write the initial data in the formwith

Taking the time derivative of and substituting and as given in (89), we get that

From , we find ; thus,

Now, taking the time derivative of (93) and replacing by the first equation of (89) and taking the sum with (93), we getwith the initial data

Note that the characteristic equation of (94) is

Set

Then, equation (94) is written aswhere

Asymptotic Expansion of Eigenvalues. Let be the eigenvalues of the matrix . We will find the asymptotic expansion of the eigenvalues .

The eigenvalues , are the solutions of the characteristic equation which can be written as

The eigenvector for is(i)When has the following asymptotic expansion: where each coefficient is given by the direct computation as Thus, the approximation of the eigenvalue when is Therefore, if we set the Green matrix for (98) is given by , and the solution is represented as where each component of this solution is itself a diagonal matrix. We integrate the first and the third equations of (89), and we get the following expressions: where

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 and substituting the result into equation (112), we obtain Now, since , we have Moreover, taking the sum of the first and the second equations of (86), we obtain Thus, 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 has the following asymptotic expansion:

Each coefficient is given by direct computation as

The approximation of the eigenvalue when is

Hence, the Green matrix for (98) is given by , and the solution is represented aswhere each component of this solution is itself a 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

Furthermore, we have

Therefore, multiplying the first and the third equation of (79) by and substituting into the above equation, we obtain

Since , we get

Substituting the first equation of (124) into equation (116) results in

Next, substituting equation (80) into equation (129), we obtain

Now, taking the sum and difference, respectively, of equations (125) and (130), we obtain

Theorem 4. Let and let be an integer. Definewhere denotes the integer part of the argument. Suppose satisfies (53). Then, for any , satisfies the following time decay property:where .

Theorem 5. Let and let be an integer. Assume is the solution of the initial value problems (51) and (52) with initial data which satisfies (53). Then, for any , satisfieswhere and is defined in (132).

Proof. Let , and be a nonnegative integer. Let satisfy . Using the Hausdorff–Young inequality, we prove (134) as follows:We estimate the first term of (138) using the Hölder inequality with and fixing sufficiently small; that is,Now, we estimate the second term; by taking sufficiently small and using the Hölder inequality with , we findWe prove (135) similarly. That is,whereWe estimate of the first term of (142) using the Hölder inequality with and fixing sufficiently small:We estimate the second term of (142) byWe estimate the third term of (142) in two parts. First, take sufficiently small and use the Hölder inequality with :whereTo estimate the second part of the third term, we havewhere we usedThe proofs of (136) and (137) are similar.
Note that we transformed to the Fourier space to find the solution representation. Now, we go back to the physical space and obtain the decay rates for the solutions to the linearized system. Fundamental properties of the system are given by the following corollary based on Theorem 5.

Corollary 1. Assume is the solution of the initial value problems (51) and (52) with initial data which satisfies (53). Then, satisfies

Proof. The results of Corollary 1 are particular cases of Theorem 5.
For example, we explain how we get the decay rates for . From (135) for , take , and :Thus,Now, for , take , and :Thus,

4. Decay Rates for Nonlinear Systems

4.1. Decay Rates for System (15)

In this section, we apply the linear time decay property of the homogeneous system (51) to the nonlinear case. Throughout this section, we suppose that is the solution to the Cauchy problem (15) satisfying (16).

By Duhamel’s principle, the solution can be formally written aswhere is as defined in Section 3.2 and

Remark 3. Note that in (156), satisfies the compatibility condition (53). Thus, acts on for all .

Proposition 2. Suppose initial data satisfies (17). If is small enough, then the solution satisfiesfor any . Moreover, being sufficiently small implies that the solution satisfiesfor any .

Lemma 2. Suppose is the solution of the initial value problems (15) and (16) with initial data satisfying (17) as described in Proposition 1. If is sufficiently small, then

Proof. Let . Multiplying equation (160) by , we getIntegrating over , we findwhich impliesRecallingwe obtainSimilarly,for . Thus,To estimate the term on the right-hand side of (167), defineApplying Duhamel’s principle for along with the linear estimate on B from (149) to (156), we getfor all .
Sincewe can apply the Sobolev imbedding theorem along with the Hölder and Cauchy inequalities to obtainThus,Therefore, we getThus,Thus,Therefore,Notice thatFrom (168),for any .
Then,Substituting (180) in (169), we getwhich impliesFinally, we need to show the uniform-in-time bound of which implies the decay rates of the energy functional . Using in (172) with sufficiently small and using (181), we getwhich impliesNow, from the definition (28) and since is sufficiently small, holds for any , and we getwhich is the proof of (158) in Proposition 2.

4.2. Time Rate for the High-Order Energy Functional

In this section, we determine the time decay estimates of the high-order energy functional , that is, the proof (159) of Proposition 2. To do so, we investigate the time decay estimates on and using the following lemma.

Lemma 3. Suppose is the solution to the Cauchy problems (15) and (16) with initial condition satisfying (17) as obtained in Proposition 1. If is sufficiently small, then there exists the high-order energy functional and the high-order dissipation rate such that for all ,

By comparing the definitions of (22) and (25) and based on Lemma 3 equation (186), we havewhich implies

Now, to estimate the time integral term on the right-hand side of inequality (189), we have the next lemma.

Lemma 4. Suppose is the solution of the initial value problems (15) and (16) with initial condition satisfying (17) as obtained in Proposition 1. If is sufficiently small, where defined in (28), then for all ,

The proofs of Lemmas 3 and 4 are analogous to those of Lemmas 5.1 and 5.2 in [10].

Now, we suppose that the above lemma is true. Then, by putting (188) into (186), we getin which (159) in Proposition 2 holds.

4.3. Decay Rate in

In this section, we determine the decay rates in , , of (7)–(9) for solutions to the Cauchy problems (51) and (52. Suppose that is sufficiently small. In addition, for , Proposition 2 shows that if is sufficiently small, thenand if is sufficiently small, then

We establish the estimates on and as follows.

To estimate the decay rate on , note that from (191), we obtain

To estimate the decay rate, note that the estimates on in (150) to (156) give

From (191), we obtainand thus

Moreover, by interpolation,for .

To estimate , note that for the decay rate, we can utilize the estimate on and in (149) to (156). That is,

From (191),and it therefore holds that

For the decay rate, we can utilize the estimates on and in (150) to (156). That is,

Sinceit holds that

Moreover, by interpolation,for .

To estimate the decay rate on , we utilize the estimate on in (149) to (156). We get

Since

(205) implies the slower decay estimate:

Moreover, after estimating and utilizing the previous slower decay estimate, we obtainand it follows from (205) that

For the decay rate, we utilize the estimates on in (150) to (156). We haveand it is simple to check that

Applying (210), we obtain

Thus, by interpolation,for . Note that (197), (204), and (213) correspond to (9), (8), and (7), respectively. This completes the proof of Theorem 1.

Now, to obtain the decay rates of smooth solutions from (156) let . For the decay rate, by applying (156) and (157), we havesince

Therefore,

For the decay rate, by applying (156) and (157), we get

Thus,

So, by interpolation,for . This completes the proof of Theorem 2.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Copyright © 2020 Ismahan Binshati and Harumi Hattori. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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