Kobe : New Stability Criterion for the Dissipative Linear System and Analysis of Bresse System

: In this article, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called in this article Classical Stability Condition for the corresponding eigenvalue problem of the system, and derived the detailed relation between the coefﬁcient matrices of the system and the eigenvalues. However, there are some complicated physical models which possess a non-symmetric viscosity or relaxation term and we cannot apply Classical Stability Condition to these models. Under this situation, our purpose in this article is to extend Classical Stability Condition for complicated models and to make the relation between the coefﬁcient matrices and the corresponding eigenvalues clear. Furthermore, we shall explain the new dissipative structure through the several concrete examples.


Introduction
We are interested in the profile of solutions for a system of differential equations. To investigate the profile, our first step is to analyze the eigenvalue of the corresponding linearized system. If the coefficient matrices of our system have a good property, it might be easy to analyze the eigenvalue problem. However, there are a lot of physical models which do not have enough properties to analyze the corresponding eigenvalue problem. (We will study several problems in Sections 3 and 4). Under this situation, we focus on a general linear system with weak dissipation and try to construct the useful condition which induces the notable property of eigenvalues in this article.
Precisely, we consider a general linear system Here, u = u(t, x) over t > 0, x = (x 1 , · · · , x n ) ∈ R n is an unknown vector function, and A 0 , A j , B jk and L are m × m constant matrices for 1 ≤ j, k ≤ n and m ≥ 2. Here and hereafter, we use notations that where ω = (ω 1 , · · · , ω n ) is a unit vector in R n , which means ω ∈ S n−1 . Then, throughout this paper, we assume the following condition for the coefficient matrices of (1).
Reλ(r,ω) 0 r -c Under the symmetric property for B(ω) and L, Umeda et al. [2] and Shizuta and Kawashima [3] introduced the useful stability condition called Kawashima-Shizuta condition or Classical Stability Condition in this article. Precisely, they introduced the following conditions.
On the other hand, Kalman et al. [4], Coron [5] and Beauchard and Zuazua [6] discussed the different condition called Kalman Rank Condition for the system (1), that is as follows. Under this situation, the following theorem is obtained.
Furthermore, if B jk (1 ≤ j, k ≤ n) is zero matrix, the above four conditions are equivalent to the following. (v) Classical Kalman Rank Condition (CR) holds. [6] considered the system (1) with B jk ≡ O for 1 ≤ j, k, ≤ n, and assumed that L satisfies

Remark 3. Beauchard and Zuazua
We note that the assumption (5) is the sufficient condition for L ≥ 0 and Ker(L) = Ker(L ). Thus, we regard the assumption (5) as the essentially symmetric property. We will discuss in detail in Lemma 1. Emphasize that the physical examples in Section 4 do not satisfy (4) (and (5)).
We remark that the typical feature of the type (1, 1) is that the high-frequency part decays exponentially while the low-frequency part decays polynomially with the rate of the heat kernel (see Figure 1). A lot of physical models satisfy these conditions and can be treated by applying Theorem 1. For example, the model system of the compressible fluid gas and the discrete Boltzmann equation is studied by Kawashima [7] and Shizuta and Kawashima [3], respectively.
In recent 10 years, some complicated physical models which possess the weak dissipative structure called the regularity-loss structure was studied. For example, the dissipative Timoshenko system was discussed in [8][9][10], the Euler-Maxwell system was studied in [11,12], and the hybrid problem of plate equations is in [13][14][15][16]. We would like to emphasize that these physical models do not satisfy (4) but Condition (A). Namely, we can no longer apply Theorem 1 to these models. Under this situation, Ueda et al. [1] introduced the new condition called Condition (S) for the system (1) with B jk ≡ O (1 ≤ j, k ≤ n) as follows.

Condition (S):
There is a real compensating matrix S with the following properties: (SA 0 ) T = SA 0 and for each ω ∈ S n−1 .
Then they derived the sufficient condition which is a combination of Condition (K) and (S) to get the uniformly dissipativity of the type (1, 2), which is the regularity-loss type. We remark that the dissipative structure of the regularity-loss type is weaker than the one of the standard type. Precisely, Reλ(r, ω) may tend to zero as r → ∞ (see Figure 2). This structure requires more regularity for the initial data when we derive the decay estimate of solutions. This is the reason why this structure is called the regularity-loss type. Indeed, the dissipative Timoshenko system, the Euler-Maxwell system and the thermoelastic plate equation with Cattaneo's law has the weak dissipative structure of type (1,2). For the detail, we refer the reader to [8,9,11,12,16].
However, the stability condition constructed in [1] is not enough to understand the regularity-loss structure. In fact, some physical models which possess the regularity-loss structure do not satisfy the stability condition in [1] (e.g., [16][17][18]). Moreover, we can construct artificial models which have the several kinds of the regularity-loss structure (in detail, see [19]). Furthermore, in recent, Ueda et al. in [20] succeeded to extend Condition (K) and (S), and analyzed the more complicated dissipative structure.
This situation tells us that it is difficult to characterize the dissipative structure for the regularity-loss type. In fact, there is no related result. Under this situation, we try to extend Classical Stability Condition (CSC) and Classical Kalman Rank Condition (CR), and derive the sufficient and necessary conditions to get the strict dissipativity for (1) in Section 2. Furthermore, we will extend our main theorem to apply to a system under constraint conditions in Section 3. In Section 4, we introduce several physical models and apply our main theorems to them. Finally, we focus on the Bresse system as an interesting application of our main theorems in Section 5.

New Stability Criterion
We introduce the new stability condition for (1) in this section. The following conditions are important to characterize the dissipative structure for (1).
Here and hereafter, we use notations that R + := (0, ∞) and Indeed, (4) and the second property of (6) give us ϕ ∈ Ker(B(ω) ) ∩ Ker(L ) for each ω ∈ S n−1 . (ii) It is easy to check that the system (1) under Condition (A) satisfies Condition (CSC) if the system is strictly dissipative. Namely, Condition (SC) is sufficient condition for Condition (CSC).
To prove Theorem 2, we shall reduce our system. We introduce the new functionũ := (A 0 ) 1/2 u. Then (1) is rewritten asũ where we defineÃ j : Similarly as before, we use notations that Remark that the matrices of (7) satisfy Condition (A) if the matrices of (1) satisfy Condition (A). In this situation, the eigenvalue problem (2) is equivalent to For the problem (8), we consider the contraposition for Theorem 2. More precisely, we introduce the complement condition of Condition (SC) and (R), and prove the contraposition of Theorem 2.
Theorem 3. Suppose that the system (7) satisfies Condition (A). Then, for the system (7), the following conditions are equivalent.

Condition (R) :
Then we show that Condition (R) is equivalent to Condition (R) . Indeed, Condition (R) means In the rest of this section, we study the relations between the assumption in Theorem 1 and (5).

Lemma 1.
Let X be m × m matrix and m 1 ≤ m. Then, is sufficient condition for X ≥ 0, Ker(X) = Ker(X ).

New Stability Criterion under Constraint Condition
In this section, we consider the system (1) under the constraint condition where P jk , Q j and R arem × m real constant matrices. In fact, a lot of physical models are described as (1) under (14). For example, the linearized system of the electro-magneto-fluid dynamics and Euler-Maxwell system are described as (1) under (14). For the detail, we refer [2,12] to the reader.
Similarly as before, we study the corresponding eigenvalue problem for the system (1) under the constraint condition (14). Namely, we look for the eigenvalue and the eigenvector of the eigenvalue problem (2) under the condition for r ≥ 0 and ω ∈ S n−1 , where Here, we introduce a notation that for r ≥ 0 and ω ∈ S n−1 . From this notation, (15) can be expressed as ϕ ∈ X r,ω . Then, the strict dissipativity and the uniform dissipativity under the constraint condition are defined as follows.

Definition 2.
(Strict dissipativity and uniform dissipativity under constraint) (i) The system (1) under the constraint condition (14) is called strictly dissipative under constraint if the real parts of the eigenvalues of (2), which eigenvectors are in X r,ω , are negative for each r > 0 and ω ∈ S n−1 . (ii) The system (1) under the constraint condition (14) is called uniformly dissipative under constraint of the type (α, β) if the eigenvalues λ(r, ω) of (2), which eigenvectors are in X r,ω , satisfy for each r ≥ 0 and ω ∈ S n−1 , where c is a certain positive constant and (α, β) is a pair of non-negative integers.
Under the constraint condition (15), we introduce the modified stability condition and modified Kalman rank condition as follows.
Theorem 4. Suppose that the system (1) satisfies Condition (A). Then, for the system (1) under the constraint condition (14), the following conditions are equivalent.
The strategy of proof is almost the same as before. Namely, we consider the contraposition for (7) under (14) as follows.
Theorem 5. Suppose that the system (7) satisfies Condition (A). Then, for the system (7) under the constraint condition (14), the following conditions are equivalent.
then X r,ω is equivalent to C m . Thus Condition (SCC) is equivalent to Condition (SC), and Theorem 4 is also equivalent to Theorem 2.
In the rest of this section, we discuss a relation for the constrain condition and the initial data. More precisely, we introduce the following condition.

Condition (C):
The matrices P(ω), Q(ω) and R satisfy Condition (C) implies the fact that (14) holds at an arbitrary time t > 0 for the solution of (14) if it holds initially. For the detail, we refer the reader to [1]. Therefore, it is reasonable for the Cauchy problem to assign the constraint condition (14) which satisfies Condition (C). If we suppose that Condition (C) for the system (1) under (14), we can relax Condition (SCC).

Remark 6.
Theorem 6 tells us that if the system does not satisfy Condition (SC) for some µ ∈ R\{0}, then it is difficult to find the useful constraint condition and apply Condition (SCC). On the other hand, if the system satisfies Condition (SC) for µ = 0, it might be possible to find the useful constraint condition and apply Condition (SCC)(or (MSCC)) to the system. We will explain the situation by using concrete examples in Sections 4.3, 4.4, 5.2 and 5.3.

Application to Physical Models
In this section, we introduce the several physical models for the application of Theorem 2, 4 and 6.

Timoshenko System
In this subsection, as an application of Theorems 2, we consider the following dissipative Timoshenko system where 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 above 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. Here we only mention [8,9] for related mathematical results.
As in [8,9], we introduce the vector function u = (φ x + ψ, φ t , aψ x , ψ t ) T . Then the Timoshenko system (18) is written in the form of (1) with coefficient matrices where I is the 4 × 4 identity matrix and O is the 4 × 4 zero matrix. Here the space dimension is n = 1 and the size of the system is m = 4. Notice that the relaxation matrix L is not symmetric. From the above matrices, we have for ω ∈ {−1, 1}, and the relaxation matrix L is decomposed L = L + L with It is obvious that these matrices satisfy Condition (A), and we can apply Theorem 2 to the dissipative Timoshenko system. Corollary 1. The dissipative Timoshenko system (18) satisfies Condition (SC). Therefore, this system is strictly dissipative.

Thermoelastic Plate Equation with Cattaneo's Law
In this subsection, we consider the following linear thermoplastic plate equation in R n , where heat conduction is modeled by Cattaneo's (Maxwell's, Vernotte's) law Here, v describes the elongation of a plate, while θ and q denote the temperature and the heat flux, respectively. For Cattaneo's law, the relaxation parameter τ is a positive constant. We have a lot of known results for the system (19). Especially, the system (19) is analyzed in detail by [16].
The authors of [16] obtained the sharp dissipative structure for the system (19), which is also regularity-loss structure.
We can rewrite (19) to a general system (1). To this end, we introduce new functions z and w as z = ∆v and w = v t . Then our equation (19) can be rewritten as Now, we introduce an unknown vector function u = (z, w, θ, q) T and n + 3 dimensional coefficient matrices A j , B jk and L such that where I is the n × n identity matrix and δ jk denotes Kronecker's delta. Then the problem (20) can be rewritten as (1). Remark that the matrices A j and L are symmetric but B jk is skew-symmetric. From the above matrices, we get for ω ∈ S n−1 . Under this situation, it is easy to check that our system satisfies Condition (A), and we can get the following property.

Coupled System of Wave and Heat Equations
We treat a coupled system of wave and heat equations as one of concrete examples in this subsection.
v tt − ∆v + aθ = 0, Here v = v(t, x) and θ = θ(t, x) over t > 0, x ∈ R n are unknown scalar functions, and a and γ denote constants which satisfy a ∈ R\{0} and γ > 0. The system (21) is one of the typical examples of the regularity-loss type equations. Indeed this system was concerned in [21] and the authors derived the weak dissipative structure in a bounded domain. Moreover, Liu and Rao in [22] analyzed this equation to derive the stability criterion for the regularity-loss type problems in a bounded domain. Recently, the author of [23] also considered this problem in R n and obtained the detailed dissipative structure.
To employ our main theorem, we rewrite (21) to a general system. Introduce new functions z and w as z = ∇v and w = v t . Then (21) can be rewritten as Here we remark that by the fact that z = ∇v, the solution z should satisfy for an arbitrary j and k with 1 ≤ j, k ≤ n, where z j denotes the jth component of the vector z. Thus, we assign the constraint condition (23) for the system (22). We remark that the constraint condition (23) is trivial in R, and is same as rot z = 0 in R 3 . We introduce an unknown vector function u = (z, w, θ) T and n + 2 dimensional coefficient matrices A j , B jk and L such that A 0 = I and where I is the (n + 2) × (n + 2) identity matrix and δ jk denotes Kronecker's delta. Then the problem (22) can be rewritten as (1). We note that the matrices A j and B jk are symmetric. However, the matrix L is skew-symmetric. From these matrices, we have On the other hand, the constraint condition (23) can be expressed (14) with P jk = O, R = O and Q(ω) = Q n (ω) such that whereQ n (ω) is defined byQ 2 (ω) = (−ω 2 ω 1 ) and for ω ∈ S n−1 and n ≥ 3. Here,Q n (ω) is a n(n − 1)/2 × n matrix. For example, there arẽ We can check that these matrices satisfy Condition (A). Moreover, it is not difficult to check that Q n (ω) satisfies Condition (C). Therefore, we can also apply our main theorems to this problem. Namely, we obtain the following corollary.

Euler-Maxwell System
As a next application of Theorem 4, we deal with the following 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.
The Euler-Maxwell system above arises from the study of plasma physics. The authors of [11,12] derived the asymptotic stability of the equilibrium state and the corresponding decay estimate. Furthermore, they analyzed the dissipative structure and concluded that the Euler-Maxwell system is a regularity-loss type which is of type (1,2). To get the structure of uniform dissipativity, they applied the complicated energy estimate. On the other hand, we suggest the different approach to get the information of the dissipative structure for Euler-Maxwell system in this subsection.
From the analysis in [11,12], we had already known that the system (26) can be written in the form of a symmetric hyperbolic system. Precisely, we introduce that u = (ρ, v, E, B) T , u ∞ = (ρ ∞ , 0, 0, B ∞ ) T , which are regarded as column vectors in R 10 , where B ∞ ∈ R 3 is an arbitrarily fixed constant. Then the Euler-Maxwell system (26) is rewritten as where the coefficient matrices are given explicitly as Here I denotes the 3 × 3 identity matrix, ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ R 3 , and Ω ξ is the skew-symmetric matrix defined by for ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ R 3 , so that we have Ω ξ E T = (ξ × E) T (as a column vector in R 3 ) for E = (E 1 , E 2 , E 3 ) ∈ R 3 . We note that (28) is a symmetric hyperbolic system because A 0 (u) is real symmetric and positive definite and A j (u) with j = 1, 2, 3 are real symmetric. Also, the matrix L(u) is non-negative definite, so that it is regarded as a relaxation matrix. Moreover, we have L(u)u ∞ = 0 for each u so that the constant state u ∞ lies in the kernel of L(u). However, the matrix L(u) or L(u ∞ ) has skew-symmetric part and is not real symmetric. Consequently, our system is not included in a class of systems considered in Theorem 1. Next, we consider the linearization of (28) with (27) around the equilibrium state u ∞ . If we denote u − u ∞ by u again, then the linearization of the system (28) with (27) can be written in the form of (1) with (14), where the coefficient matrices are given by B jk = O and and P jk = O and where a ∞ = p (ρ ∞ )/ρ ∞ and b ∞ = p (ρ ∞ ) are positive constants. Here the space dimension is n = 3 and the sizes of the systems are m = 10 andm = 2. For this linearized system it is easy to check that the system satisfies Condition (A). Furthermore, using the expression (30), we can also check Condition (C) for the constraint condition. Therefore we can apply Theorem 4 and 6 for (1), (14) with (29), (30), and get the following result.

Remark 7.
When we check Condition (CSC) for the linearized Euler-Maxwell system, we do not need to use the first condition in (32).

Bresse System
In the last section, we introduce the important application of Condition (SC). The Bresse system is a one of good examples that Condition (CSC) is not enough to check what the physical model is strictly dissipative.

Dissipative Bresse System
We consider the dissipative Bresse system where a, γ, κ 1 and κ 2 are positive constants, is a non-zero constant, and φ = φ(t, x), ψ = ψ(t, x) and w = w(t, x) are unknown scalar functions of t > 0 and x ∈ R. If we put = 0, the dissipative Bresse system (35) is equivalent to the dissipative Timoshenko system (18) and the simple wave equation. Now, we introduce new functions such that v := κ 1 (φ x + ψ + w), s := φ t , z := aψ x , then (35) is rewritten as Symmetry 2018, 10, 542 19 of 25 Namely the system (36) is described as (1), where u = (v, s, z, y, q, p) T , and the matrices A 0 , A 1 , B 11 and L are defined by A 0 = I, B 11 = O and The space dimension is n = 1 and the size of the system is m = 6. Notice that the relaxation matrix L is not symmetric. Then, we obtain It is clear that these matrices satisfy Condition (A). Thus we can apply Theorem 2 and get the following result. Theorem 7. The dissipative Bresse system (35) does not satisfy Condition (SC). Therefore, this system is not strictly dissipative.
The proof of Theorem 7 tells us that the real part of some eigenvalue for (2) which comes from the dissipative Bresse system (36) contacts the imaginary axis at r = | |. Namely, we can expect that the real parts of the eigenvalues are located in the gray region in Compare with the Corollary 1 and Theorem 7, we can predict that the difficulty of the analysis for (36) comes from the terms related with . Therefore we focus on the effect of the terms of and analyze the structure of strict dissipativity in the next subsections.

Reduced Bresse System (I)
Inspired by the analysis in the previous subsection, we regard that p ≡ 0 in (36) and study the reduced system. Namely, we treat the system Then the problem (39) can be rewritten as (1), where u = (v, s, z, y, q) T , and the matrices are defined by A 0 = I, B 11 = O and Hence, we get It is obvious that the system (39) satisfies Condition (A). Under this situation, we obtain the following result which comes from Theorem 2.
Theorem 8. The reduced Bresse system (39) does not satisfy Condition (SC). Therefore, this system is not strictly dissipative.

Reduced Bresse System (II)
Based on the similar motivation as in Section 5.2, we also regard q ≡ 0 in (36). Then this yields Here, we note that our problem (1) with (46) satisfies Condition (A). Therefore, we can apply Theorem 2 and get the following result.
Theorem 10. The reduced Bresse system (45) does not satisfy Condition (SC). Therefore, this system is not strictly dissipative.

Conclusions
In this article, we succeeded in introducing new stability conditions. By virtue of Stability Condition (SC), it is easy to check the dissipative structure for the general system (1), and there are a lot of applications. However, if the system has the symmetric property (4), Classical Stability Condition (CSC) is equivalent to the uniform dissipativity. Inspired by this situation, we predict that the system (1) is uniformly dissipative under Stability Condition (SC). If we can get the positive answer for this conjecture, Stability Condition (SC) is applicable to nonlinear problems.