Characterization of Dissipative Structures for First-Order Symmetric Hyperbolic System with General Relaxation

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.


Introduction
In this paper, we consider the first-order linear symmetric hyperbolic system with relaxation: (1) Here, n ∈ N and u = u(t, x) = (u 1 (t, x), · · · , u m (t, x)) is an unknown function with valued in C m , m ∈ N, and f = f (x) = ( f 1 (x), · · · , f m (x)) is a given function with valued in C m . We use the standard notations for derivatives; ∂ t = ∂/∂t and ∂ x j = ∂/∂x j for x = (x 1 , · · · , x n ). Each A j and L is a given m × m constant matrix with complex coefficients, and, in particular, each A j is assumed to be an Hermitian matrix, A j = A * j , where M * = M denotes the adjoint of a given matrix M. Here, M is the complex conjugate of M, and M is the transpose of M. We denote by M and M the Hermitian part and the skew-Hermitian part of M, respectively: We also use the standard notations for the kernel and the range of M as We use similar notations if the matrix M is restricted in the invariant space X other than C m .
The system of the form (1) arises as the linearization of hyperbolic systems of balance laws, and its study has a long history. One of the most active themes is the dissipation mechanism underlying (1), as a result of the effective interaction between the partial damping from the Hermitian part L and the other hyperbolic terms. To explain this in details, let us consider the system (1) in the Fourier variables with respect to x: where A(ξ) is the m × m matrix given by A(ξ) = L for ξ = 0, while, for ξ = 0, and ω = (ω 1 , · · · , ω n ) ∈ S n−1 . The functionû is the Fourier transform of u, i.e., u(t, ξ) = 1 (2π) n/2 R n u(t, x)e −ix·ξ dx .
Since each A j is Hermitian, so is A(ω) for each ω ∈ S n−1 : The one-parameter family {e −tA } t≥0 given by defines a C 0 -semigroup acting on L 2 (R n ) m with the generator −A, whose domain contains the Sobolev space H 1 (R n ) m and A f = ∑ n j=1 A j ∂ x j f + L f for f ∈ H 1 (R n ) m . Thus, by the Plancherel theorem, the estimate of the semigroup e −tA in L 2 is reduced to the analysis of e −tA(ξ) for ξ ∈ R n \ {0}.
In order for the first order ODE system (2) to be dissipative (i.e., lim t→∞ e −tA(ξ) = 0) for each ξ ∈ R n \ {0}, the necessary and sufficient condition is where λ is the real part of the complex number λ, and ρ(−A(ξ)) is the resolvent set of the matrix −A(ξ). We note that the condition (5) always implies L = O; otherwise, −A(ξ) becomes a skew-Hermitian matrix and thus must possess the eigenvalues on the imaginary axis. The key and common relaxation condition assumed in this study is the nonnegativity of L , i.e., which automatically leads to the inclusion Thus, under the nonnegativity condition (6), the condition to ensure (5) is where iR = {λ ∈ C | λ = 0}. We note that, since A(ω) is Hermitian, the identity A(ξ) = L holds. Therefore, under the condition (6) and (7) the matrix A(ξ) is an m-accretive operator for each ξ ∈ R n . We come back to this important fact below. In general, if the matrix M satisfies M ≥ 0, then it is not difficult to see Ker (iλI + M) = Ker (λI − iM ) ∩ Ker (M ) for any λ ∈ R, where I is the identity matrix. Hence, by recalling A(ξ) = irA(ω) + L with r = |ξ| and ω = ξ/|ξ| ∈ S n−1 for ξ = 0, we find that the condition (6) and (7) is equivalent with (6) and λ∈R, r>0, ω∈S n−1 The class of symmetric hyperbolic systems satisfying (8) is wide, and we call (8) the general stability condition.
In [1], it is proved that the validity of (6), (9), and (10) implies the existence of a suitable energy assumed in the work of Umeda, Kawashima, and Shizuta [2], resulting in the pointwise decay estimate of e −tA(ξ) such as where C and c are positive constants independent of ξ and t. The semigroup estimate (11) implies the condition (5), or, more strongly, λ(iξ) ≤ −c |ξ| 2 1 + |ξ| 2 , ξ ∈ R n \ {0} (12) for any eigenvalue λ(iξ) of −A(ξ). The spectral bound (12) is called the uniformly dissipative of type (1, 1) in [3]. As a further study from [1,2], Hanouzet and Natalini [4], Yong [5], Kawashima and Yong [6,7], Ruggeri and Serre [8], and Bianchini, Hanouzet, and Natalini [9] analyzed the nonlinear problems for the hyperbolic system with relaxation. Furthermore, in the case of (9), Beauchard and Zuazua [10] introduced the stability condition called Kalman rank condition which is equivalent to (10). We remark that the entropy condition and (10) guarantee the nonlinear stability of the equilibrium states for the hyperbolic balance laws, while it should be emphasized that the concrete decay rate such as (11) for the linearized system is important to achieve the nonlinear stability. When (6) holds, in general, we only have rather than (9). There are also several important examples for which (9) is not satisfied. For example, the dissipative Timoshenko system and the linearized compressible Euler-Maxwell system do not satisfy (9), and these systems were analyzed for the dissipative structure by the authors of [11][12][13][14][15]. The analysis for these physical models has revealed that, the system with the condition (6) will possess fruitful and more complicated dissipative structures. Several structural conditions have been proposed by to handle the important examples, while most of them are built upon the assumption on the existence of the matrix which provide an explicit source of the energy functional to achieve the dissipative estimate with the desired rate (see [3,16,17]).
Among others, a remarkable point of Shizuta-Kawashima theory [1] is that the condition (9) and (10) is purely algebraic; nevertheless, the quantitative estimate (11) is achieved with a concrete dependence on ξ. This is highly nontrivial. Indeed, since A(ξ) is not a normal operator, even the spectral bound (12) does not necessarily yield (11) from the abstract semigroup theory, for the constant C in (11) must be uniform in ξ. Notice that the abstract spectral mapping theorem does not give information on the prefactor constant C. Inspired by the philosophy of Shizuta [1], Ueda [18,19] tried to extend Shizuta-Kawashima theory, and partially succeeded for an extension under (13). Precisely, the author obtained the uniform dissipativity for (1) under the general stability condition (8). However, this result does not mention the optimality of the type of the uniform dissipativity, and, thus, the application to the nonlinear problem is still out of reach in this general setting. For the nonlinear problem of general hyperbolic systems, the entropy condition derived in [6,7] is not enough to cover all physical models described by the balance laws, and, thus, the theory still needs to be developed. In this context, Kawashima and Ueda [20] recently refined the entropy condition which can be applied to the compressible Euler-Maxwell system, where the key generalization is to allow the nonsymmetric relaxation. The reader is also referred to the works of Zeng [21] and Lou and Ruggeri [22] for another direction of generalization, where it is discussed even the case when the general stability condition (8) is violated but in a specific way so that the formation of shocks is prevented. However, the general theory to ensure the global existence of small smooth solutions for the nonlinear problem seems to be still open.
In this paper, we study the linear system (1) for the case of general relaxation, and our goal is to provide the algebraic characterization in achieving the uniform dissipative estimate of order 1 (see the definition in front of Theorem 1 below) without assuming (9). It is stressed here that, although the nonlinear problem is not discussed in this paper, achieving the concrete decay rate for the linear problem (1), which is nontrivial if the classical condition (9) does not hold, is a key step also for the global existence of small smooth solutions to the nonlinear problem.
As described in Theorems 1-3 in the next section, our result is optimal in the sense that the algebraic condition given in this paper is necessary for any n ≥ 1, and is sufficient for n = 1, as well as for n ≥ 2 under additional but rather mild assumptions. In particular, some important examples such as the dissipative Timoshenko system and the compressible Euler-Maxwell system are within the range of our result. We note that the finite dimensional nature of the problem (2) is the key that enables us obtaining the concrete decay rate of the semigroup only from the algebraic condition; in the infinite dimensional problem, one needs to introduce a quantitative condition at some point to achieve a concrete decay rate, as seen in the systematic work by Villani [23] in this direction. This paper is organized as follows. In Section 2, we collect some notations and state the main results. In Section 3, we briefly refer to the idea of the proof in connection with the key general assumptions (4) and (6). In Sections 4 and 5, the dissipative structure is analyzed in detail for the low frequency part and high frequency part, respectively. The proofs of the main results are stated in Section 6. In Section 7, we show how our result is applied to the well-known examples such as the dissipative Timoshenko system and the compressible Euler-Maxwell system. In Appendix A we recall the Gearhart-Prüss type theorem for the semigroup generated by the m-accretive operator on the Hilbert space, and in Appendix B we state the elementary fact about the nonnegative matrices with the spectral parameters on the imaginary axis. These results are the key in our argument.

Nondegenerate Condition and Main Results
Let X ω be a nontrivial subspace of C m and P ω be the orthogonal projection to X ω , which depend on ω ∈ S n−1 . Then, we also introduce a family of nontrivial subspaces {X ω,r } r>0 such that X ω,r ⊂ X ω , and P ω,r denotes the orthogonal projection from C m to X ω,r . The spaces X ω and X ω,r , rather than C m , are introduced in order for the application to the system with the constraint condition such as the compressible Euler-Maxwell system.
Definition 1 (Orthogonal projections). Below the notation F : Y → Z denotes that F is the orthogonal projection from the subspace Y of C m to the subspace Z of Y. We also denote by F ⊥ the orthogonal projection I| Y − F, where I| Y is the identity map on Y. Each s j is a given real number and ω ∈ S n−1 .
Next, we define the singular sets, which consist of the parameters such that the resolvent can be singular at the limit of low/high frequencies.
Definition 2 (Singular sets in the limit).

Remark 2.
The singular sets can be empty. The following statements are verified in virtue of L(ω) ≥ 0 and Lemma A1.
The first inclusion in Assertion (4) above is apparently nontrivial, but it follows from the formula given in Remark 7. The above inclusions imply that (I) π 0Slow,1 ⊂ S low,0 . (II) π 1 S low,1 ⊂S low,1 .
The singular sets defined above characterize the dissipation rate for the semigroup.
To give a precise statement, let us introduce some terminology about the semigroup bound. Set Let α, β ≥ 0. We say that {e −tA(ξ) } t≥0 has the uniform dissipative bound of order α at low frequency if there exist C, c > 0 such that e −tA(rω) X ω,r →X ω,r ≤ Ce −cr 2α t holds for any t > 0, ω ∈ S n−1 , and 0 < r ≤ 1. Similarly, we say that {e −tA(rω) } t≥0 has the uniform dissipative bound of order β at high frequency if there exist C, c > 0 such that e −tA(rω) X ω,r →X ω,r ≤ Ce −cr −2β t holds for any t > 0, ω ∈ S n−1 , and r ≥ 1. When n = 1, we have the complete characterization of these dissipative structures in terms of the singular sets, as follows.
(1) {e −tA(rω) } t≥0 has the uniform dissipative bound of order α at low frequency if and only if S low,α = ∅.
(2) {e −tA(rω) } t≥0 has the uniform dissipative bound of order β at high frequency if and only if S high,β = ∅.
For the higher dimensional case n ≥ 2, the following necessary condition holds.
Even when n ≥ 2, the absence of the singular sets is almost sufficient to achieve the uniform dissipative estimate, but we need a technical assumption to get rid of the difficulty related to a spectral bifurcation coming from the dependence on ω ∈ S n−1 . This situation is very similar to the result in [19]. To this end, we introduce the following condition.
Definition 3 (no-splitting condition on real eigenvalues). Let {Z ω } ω∈S n−1 be a family of the subspaces of C m , and let {M ω } ω∈S n−1 , M ω : Z ω → Z ω , be a family of linear operators. We say that {(M ω , Z ω )} ω∈S n−1 has no-splitting real eigenvalues if the following two conditions are satisfied.
is the set of the eigenvalues of M ω , are independent of ω ∈ S n−1 .

Remark 3.
(1) Assume that {(M ω , Z ω )} ω∈S n−1 has no-splitting real eigenvalues. Then, we have from the continuity of the eigenvalues about ω and, from Condition (ii), We also have the continuity of the spectral projection where γ is a small circle centered at µ j (ω) oriented counter clockwise, in the sense that T µ j (ω) P Z ω : C m → C m is continuous about ω. These facts, which are valid from (i) and (ii) in the definitions, are frequently used in this paper.
(2) If n = 1, then S n−1 = {±1}, which is a finite set. Thus, the concept of the no-splitting condition on real eigenvalues is needed only when n ≥ 2.
(2) IfS low,1 = ∅, then Condition (i) in Definition 5 holds. Similarly, ifS high,0 = ∅, then Condition high,1 is important in actual applications, as, if Condition (iii') holds, then one can skip checking Condition (iii), that would need lengthy computation. For the (linearized) dissipative Timoshenko system and the compressible Euler-Maxwell equations, which are well known examples for the nonclassical case, we can indeed show that (iii') holds. We note that the classical stability condition, in which Ker (L) = Ker (L ) holds, impliesS low,1 =S high,0 = ∅ (see, e.g., Remark 6 forS low,1 = ∅; the conditionS high,0 = ∅ is trivial in the classical case), resulting in (NDC) low,1 and (NDC) high,0 . Thus, our result covers the classical theory by Shizuta and Kawashima [1] and Umeda, Kawashima, and Shizuta [2].
(3) Note that S n−1 = {±1} when n = 1. Thus, if n = 1, then the condition of the no-splitting real eigenvalues is automatically satisfied. Even when n ≥ 2, for actual applications, the condition of the no-splitting real eigenvalues is widely satisfied and is not a 'real' obstacle for the range of applications.
The nondegenerate conditions stated above are sufficient to obtain the uniform dissipative estimate, as follows.
Note that, if (NDC) high,β holds with β = 0, then the solution decays exponentially in the high frequency region. From Theorem 3 combined with the Plancherel theorem and the Hausdorff-Young inequality for the Fourier transform, we have the following corollary. For s ≥ 0, we denote by H s the closed subspace of the usual Sobolev space H s (R n ) m defined as To simplify the notation, we also write L q instead of the Lebesgue space L q (R n ) m , 1 ≤ q ≤ ∞.
Corollary 1. Assume that (NDC) low,α and (NDC) high,β hold. Then, it follows that, for any t > 0 and f ∈ H s ∩ L 1 with k, l ≥ 0 and 0 ≤ k + l ≤ s, Here, C and c are independent of t and f .
The proof of Corollary 1 is omitted in this paper, as the derivation from the pointwise estimate in Theorem 3 is rather standard. We refer readers to [3,19] for the details.

Remark on the General Strategy and the Role of
Before going into the details of the proof for our main result, let us give some comments on the general strategy. Our proof for the semigroup estimate relies on the resolvent analysis studying the quantity called the pseudospectral bound. In the technical level, the argument is closely related to the reduction argument, systematically described by Kato [24] in the case of perturbations with one parameter. In essence, it is applied to investigate the asymptotic expansion of the eigenvalues or the resolvent for the operator A(rω) = irA(ω) + L(ω), in both the low frequency limit r = |ξ| → 0 and the high frequency limit r = |ξ| → ∞. One difficulty here is the additional parameter ω in the higher dimensional case n ≥ 2, as the general theory is not available for perturbations with multi-parameters. This is the reason we have to introduce the condition of the no-splitting real eigenvalues, which enables us to avoid the unpleasant complexity coming from the possible bifurcation due to the continuous dependence on ω for n ≥ 2.
In principle, under the suitable assumption on the no-splitting of the eigenvalues about ω, the reduction argument works for general couple (A(ω), L(ω)) without even symmetry of A(ω) or the nonnegativity L(ω) ≥ 0. The problem here is that, however, if such structures of A(ω) and L(ω) are absent, the reduction argument becomes rather complicated in general, even under the assumption of the no-splitting eigenvalues about ω. This is indeed a serious problem for actual applications with concrete operators.
One important observation of this paper is that the symmetry of A(ω) and the nonnegativity of L(ω) drastically simplify the reduction process, which would not be possible without these structures. To clarify this point, let us give a list of benefits brought by the conditions A(ω) = A(ω) and L(ω) ≥ 0.

1.
The operator A(rω) = irA(ω) + L(ω) becomes m-accretive, which enables us to obtain the semigroup bound directly from the pseudospectral bound, the resolvent estimate with resolvent parameters only along the imaginary axis, in virtue of the Gearhart-Püss type theorem by Wei [25] (see also [26]) (see Theorem A1). The pseudospectral bound was discussed by Gallagher, Gallay, and Nier [27], who studied the harmonic oscillator with some class of large skew-symmetric perturbations, which was also discussed, for example, by Li, Wei, and Zhang [28] and and Ibrahim, Maekawa, and Masmoudi [29] in order to study the semigroup estimate for the linearization around the stationary flows for the Navier-Stokes equations such as the Burgers vortex and the Kolmogorov flow.

2.
In each reduction process for the uniform dissipativity, the leading operator M is either the skew-Hermitian (i.e., iM is Hermitian) or M ≥ 0. As a result, when is ∈ iR, which is compatible with Statement 1 above, the orthogonal projection to Ker (isI + M) coincides with the spectral projection to −M. That is, any eigenvalue of −M located on the imaginary axis must be semisimple, and furthermore, the associated eigenprojection is the orthogonal projection; see Lemma A1. This makes the reduction process much simpler, as the eigennilpotent (which would yield a serious complexity of the reduction formula) does not appear in the reduction process, and the orthogonal projection is easier to compute. This is also the reason we can describe the nondegenerate condition only through the orthogonal projections listed in Definition 1. It should be stressed that this remarkable feature is available only from the conditions A(ω) = A(ω) and L(ω) ≥ 0.

3.
We can derive the sufficient condition by making use of Ker (L(ω) ), which gives a chance to stop the reduction process before making the full reduction. To be precise, as in the classical case of Sizuta and Kawashima [1], there are fruitful examples such that the investigation ofS low,1 andS high,0 is enough to achieve a uniform dissipation estimate, and the study of the full singular sets such as S loiw,1 and S high,0 is not always required. This is a great advantage in actual applications, which is why we introducẽ S low,1 ,S high,0 , S high,1 , and S (2) high,1 .

Analysis of Low Frequency
In this section, we study the case for 0 < |ξ| ≤ 1.

Resolvent Analysis
In this subsection, our interest is the quantitative estimate of the resolvent (iλI + irA(ω) + L(ω))| −1 X ω,r with λ ∈ R and 0 < r ≤ 1; in particular, we aim to obtain the estimate with the concrete dependence on the parameter r uniformly in λ and ω. Let Ψ(r, ω) be the pseudospectral bound of the matrix irA(ω) The main result of this section is as follows.
Before going into the details of the proof, let us state a useful consequence of Theorem 4.

Remark 6.
Let us consider the case of the classical stability condition, where X ω = X ω,r = C m , L is independent of ω, and Ker (L) = Ker (L ). In this case, we have Ker (L ) = Ran (P s 0 ,ω ) with s 0 = 0, and, hence, Corollary 2 is applied.

Semigroup Estimate
The estimate of the semigroup is a consequence of the pseudospectral bound obtained in the previous subsection and the Gearhard-Prüss type theorem by Wei [25], stated in Theorem A1. Let us recall that the semigroup considered here is {e −tA(ξ) } t≥0 in X ξ/|ξ|,|ξ| , where A(ξ) is defined by (14). The main result of this subsection is stated as follows.

Optimality
In this subsection, we show the optimality stated in Theorem 2 for the low frequency.

Analysis of High Frequency
In this section, we study the case for |ξ| ≥ 1.
Next, we decompose u N as u N = w N + w ⊥ N , where w N := Q 0,N u N , w ⊥ N := Q ⊥ 0,N u N , and Q 0,N := Q s 0 (ω N ),ω N . Then, we have w N , w ⊥ N ∈ X ω N by the invariance (Inv). Furthermore, g N is also decomposed by We notice that, for large N, the operator is invertible on Q ⊥ 0,N X ω N with the uniform bound in N for its inverse. Thus, using (31), we get Furthermore, since (30) and (32) with lim N→∞ w N = u * and lim N→∞ µ which are bounded uniformly in N. Thus, by taking a subsequence if necessary, we may assume that lim N→∞μN =μ * and lim N→∞w ⊥ N =w ⊥ * . Then, we have u * ∈ Ker(iμ * I + Q µ * ,ω * L(ω * )) andw As a summary, we have u * = 0 and u * ∈ V high,0 (μ * , µ * , ω * ). In particular, V high,0 (μ * , µ * , ω * ) must be nontrivial.
Case β = 0: Assume that (NDC) high,0 holds. Then, we reach the contradiction to S high,0 = ∅ in (NDC) high,0 , and the proof is complete in this case.

Note that A
(1)

N is a Hermitian and L
(1) N is a nonnegative definite. Indeed, we compute Here, | · | ,ω N is the weighted seminorm defined by | f | 2 N has a similar structure as the one discussed in Theorem 4 for the analysis of the low frequency. The only difference is that the operators (1) N depend on not only ω N but also τ N . The argument in the proof of Theorem 4 for the case α = 1 but with (36) and (37) imply that Furthermore, (38) also leads to and these facts together with L(ω N ) In the rest of the proof, we consider two cases and derive the contradiction in each case.
Case 1: Suppose that for any N. Then, using (40) and given by the orthogonality, we obtain lim N→∞ |K N L(ω N ) w N | ,ω N = 0. Hence, we have where Hence, if (iii') of (NDC) high,1 holds, then we reach the contradiction.
Case 2: Next, we consider the case (iii) of (NDC) high,1 holds. If (41) is not necessary satisfied, then one cannot derive (42). Thus, we should analyze (36) in more detail.
and (36) is written as Then, we just follow the argument in the proof of Theorem 4 for the case α = 1; below, only a sketch of the proof is given. Firstly, we decompose w N as w N = y N + y ⊥ N , where y N := Q 1,N w N , y ⊥ N := Q ⊥ 1,N w N , and Q 1,N := Q s 1 (ω N ),s 0 (ω N ),ω N . Here, s 1 (·) : S n−1 → R is a continuous map associated with the no-splitting real eigenvalues and thus satisfying s 1 (ω * ) =μ * . Similar to above, y N and y ⊥ N satisfy which comes from (43). Then, we set and this gives from A (1) where A N − K N L(ω N ) * . Thus, letting N → ∞ and setting lim N→∞ σ N = σ * , lim N→∞ỹ ⊥ N =ỹ ⊥ * , and

Semigroup Estimate
As in Theorem 5, by applying the result of Wei [25], we obtain from Theorem 7 the following semigroup estimate for the high frequency. Theorem 8. There exist positive constants C and c such that the following statements hold. Let β ∈ {0, 1} and assume that (NDC) high,β holds. Then, for any ξ ∈ R n with |ξ| ≥ 1, Proof. Theorem 7 implies the pseudospectral bound Ψ(r, ω) ≥ r −2β /C, where the positive constant C is independent of r ≥ 1 and ω ∈ S n−1 . Then, the estimate (49) follows from Theorem A1. The proof is complete.

Optimality
The optimality for the high frequency is stated as follows.

Proof of Main Theorems
The results of the previous sections imply the theorems stated in Section 2. Indeed, Theorem 1 follows from Theorems 4, 6, 7, and 9; Theorem 2 follows from Theorems 6 and 9; and Theorem 3 follows from Theorems 4 and 7. The proof is complete.

Application
In this section, we apply our main theorems to some models.

Classical Case
We recall the known results obtained by Shizuta and Kawashima [1] and Umeda, Kawashima, and Shizuta [2].

Dissipative Timoshenko System
We consider the linear dissipative Timoshenko system described as Here, a and γ are positive constants and φ = φ(t, x) and ψ = ψ(t, x) are unknown scalar functions of t > 0 and x ∈ R. The Timoshenko system is a model system describing the vibration of the beam called the Timoshenko beam, and φ and ψ denote the transversal displacement and the rotation angle of the beam, respectively.

Thus, this gives
Ker for all s 2 ∈ R and ω ∈ {±1}. This implies S low,1 = ∅ and therefore Condition (i-b) of (NDC) low,1 is satisfied. Hence, the proof is complete for the low frequency part.
Case a = 1: In this case, we have Q ±1,ω = ·, e ±1 e ±1 + ·, e ±a e ±a , where e ±a is defined as (56). As in the case a = 1, we havẽ we must have ics 1 e ±1 ∓ ce ±a /2 = 0, which is possible only when c = 0. Thus, we conclude that S high,0 = ∅ when a = 1, and the condition (NDC) high,0 is proved for the case a = 1. The proof is complete.

Compressible Euler-Maxwell System
As an application of our theorems, we deal with the compressible Euler-Maxwell system Here, the density ρ > 0, the velocity v ∈ R 3 , the electric field E ∈ R 3 , and the magnetic induction B ∈ R 3 are unknown functions of t > 0 and x ∈ R 3 . Assume that the pressure p(ρ) is a given smooth function of ρ satisfying p (ρ) > 0 for ρ > 0, and ρ ∞ is a positive constant.
It is easy to see that the invariant condition (Inv) and the general stability condition (SC) hold true. Therefore, we can apply our theorems to the linearized Euler-Maxwell system and derive the pointwise estimate of solutions and optimality of its estimate as follows.