Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space

In the present paper, we study a system of viscous conservation laws, which is rewritten to a symmetric hyperbolic-parabolic system, in one-dimensional half space. For this system, we derive a convergence rate of the solutions towards the corresponding stationary solution with/without the stability condition. The essential ingredient in the proof is to obtain the a priori estimate in the weighted Sobolev space. In the case that all characteristic speeds are negative, we show the solution converges to the stationary solution exponentially if an initial perturbation belongs to the exponential weighted Sobolev space. The algebraic convergence is also obtained in the similar way. In the case that one characteristic speed is zero and the other characteristic speeds are negative, we show the algebraic convergence of solution provided that the initial perturbation belongs to the algebraic weighted Sobolev space. The Hardy type inequality with the best possible constant plays an essential role in deriving the optimal upper bound of the convergence rate. Since these results hold without the stability condition, they immediately mean the asymptotic stability of the stationary solution even though the stability condition does not hold.

1. Introduction. We study the system of viscous conservation laws over the onedimensional half space R + := { x ∈ R | x > 0 }:  [12] show an existence and an asymptotic stability of a stationary solution to the system (1) under the suitable conditions, of which details are discussed later. The main purpose of the present paper is to derive a convergence rate of a time-global solution towards the stationary solution to the system (1). Following [3,8,12], we 758 TOHRU NAKAMURA, SHINYA NISHIBATA AND NAOTO USAMI rewrite the system (1) to the normal form of the symmetric hyperbolic-parabolic systems. Here we state the several assumptions on the system (1). In order to rewrite (1) into the symmetric form, we assume [A1] the system (1) has an entropy function η = η(U ) defined on O U , which satisfies three conditions: (i) η(U ) is a smooth strictly convex scalar function, that is, the Hessian matrix D 2 U η(U ) is positive definite for U ∈ O U ; (ii) there exists a smooth scalar function q(U ) defined on O U , which is called an entropy flux, such that D U q(U ) = D U η(U )D U f (U ) for U ∈ O U ; (iii) the matrix G(U )(D 2 U η(U )) −1 is real symmetric and non-negative definite for U ∈ O U .
Owing to this assumption, we see from [3,8,12] that there exists a diffeomorphism U →Û from O U onto an open set OÛ such that the system (1) is deduced to the symmetric form by setting U = U (Û ). Namely, the system (1) is rewritten tô We further assume the system (2) is deduced to the normal form of the symmetric hyperbolic-parabolic systems. Precisely, we assume there exists a diffeomorphism U → u from OÛ onto an open set O u such that lettingÛ =Û (u), the system (2) is rewritten to a system of equations for u = (v, w), v = v(t, x) ∈ R m1 , w = w(t, x) ∈ R m2 : A 0 (u)u t + A(u)u x = B(u)u xx + g(u, u x ), x ∈ R + , t > 0, where , A(u) = A 11 (u) A 12 (u) A 21 (u) A 22 (u) , , g(u, u x ) = g 1 (u, w x ) g 2 (u, u x ) .
In (3), A 0 (u) is an m × m real symmetric and positive definite matrix for u ∈ O u ; A 0 1 (u) and A 0 2 (u) are an m 1 × m 1 and an m 2 × m 2 , respectively, real symmetric and positive definite matrices for u ∈ O u ; A(u) is an m × m real symmetric matrix for u ∈ O u ; A 11 (u) and A 22 (u) are an m 1 × m 1 and an m 2 × m 2 real symmetric matrices, respectively, and A 12 (u) = A 21 (u) for u ∈ O u ; B 2 (u) is an m 2 × m 2 real symmetric and positive definite matrix for u ∈ O u ; g(u, u x ) is a smooth remainder function and g 1 depends on u and only w x . Summarizing the above, we assume [A2] the system (2) is rewritten to the normal form of the symmetric hyperbolicparabolic systems (3).
It is shown in [8] that if the null condition [N], that is [N] the null space N := kerB(Û ) is independent ofÛ ∈ OÛ , CONVERGENCE RATE TOWARDS STATIONARY WAVE 759 holds, then (2) is deduced to (3) with m 1 := dim N , m 2 := m − m 1 , g 1 (u, w x ) ≡ 0 and g 2 (u, u x ) = B 2 (u) x w x . We discuss, in Section 4.4, the system for compressible and viscous gases, which satisfies the null condition [N] except for viscous and non-heat conductive gas. By a diffeomorphism u → U from O u onto O U , we have expression of A 0 (u), A(u) and B(u) as Here and hereafter, we often abbreviate a Fréchet derivative D U f (U ) of a function f (U ) to f U (U ). We decompose the system (3) into the symmetric hyperbolic system for v and the symmetric strongly parabolic system for w: The initial and the boundary conditions for the system (5) are prescribed as where w b ∈ R m2 is a constant. We assume a spatial asymptotic state of the initial data is a constant: lim x→∞ u 0 (x) = u + = (v + , w + ), that is, lim where u + ∈ O u is a certain constant. Moreover we impose the condition on A 11 (u).
This condition corresponds to the outflow problem for the model system of compressible viscous gases. This system is studied in [6,7,13]. Since we construct the solution to the problem (5)- (7) in a small neighborhood of u + and then the all characteristics for the hyperbolic equations are negative, the boundary condition (7) for the parabolic equations is necessary and sufficient. Hence the problem (5)- (7) is well-posed with the boundary condition only on w.
The paper [12] studies the boundary value problem (9) and (10) for the nondegenerate and the degenerate flows. We call the non-degenerate if the matrix D U f (U + ) has no zero-eigenvalue. On the other hand, if D U f (U + ) has zero-eigenvalues, we call the degenerate. For the degenerate flow, the paper [12] supposes that D U f (U + ) has only one zero-eigenvalue to show an existence of a stationary solution. Under this assumption, let µ(U ) be an eigenvalue of D U f (U ) satisfying µ(U + ) = 0 and let R(U ) be a right eigenvector of D U f (U ) corresponding to µ(U ). For a matrix A, the notation # − (A) designate the number of negative eigenvalues of the matrix A. An existence of a stationary solution for the boundary value problem (9) and (10) is summarized in Lemmas 1.1 and 1.2, which are proved in [12]. Lemmas 1.1 and 1.2 correspond to results for the non-degenerate and the degenerate flows, respectively.
holds. Then there exists a local stable manifold M s ⊂ R m2 around the equilibrium w + such that if w b ∈ M s and δ is sufficiently small, then there exists a unique smooth solutionũ to the problem (9) and (10) satisfying has only one zero-eigenvalue and the characteristic field corresponding to µ(U + ) = 0 is genuinely nonlinear, that is, Then there exists a certain region M ⊂ R m2 such that if w b ∈ M and δ is sufficiently small, then there exists a unique smooth solutionũ to the problem (9) and (10) satisfying The paper [12] also studies the asymptotic stability of the stationary solution assuming the condition [K] or equivalently [SK]. The equivalence of the conditions [K] and [SK] is proved in [15].
[K] There exists an m × m matrix K such that KA 0 (u + ) is skew-symmetric and L := [KA(u + )] + B(u + ) is real symmetric and positive definite, where [A] := (A + A)/2 is a symmetric part of a matrix A.
Here Dg(u, u x ) stands for the Fréchet derivative of g with respect to u and u x . For the non-degenerate flow, the asymptotic stability of the stationary solution is summarized in the following lemma.  (6) and (7) has a unique solution u globally in time satisfying Moreover the solution u converges to the stationary solutionũ: The paper [12] also shows asymptotic stability of the the degenerate flow under the assumption that all of the non-zero characteristics are negative, that is, the matrix A(u + ) has only one zero-eigenvalue and the other eigenvalues are negative. Note that this assumption is same as [A6] to be stated later. Lemma 1.4 ([12]). Assume the same assumptions as in Lemma 1.2, the stability condition [SK] (or [K]) and [A4] hold. Moreover, we assume that the matrix A(u + ) is non-positive definite. Then the same conclusion as in Lemma 1.3 holds true.
In the remainder of this section, we formulate an initial boundary value problem for the perturbation from the stationary solution. The perturbation is defined by We use a notation ξ(t, The system (15) is rewritten to a system for ϕ and ψ as where h = (h 1 , h 2 ), We solve the system (16) with the initial and the boundary conditions, derived from (6), (7) and (10a), as Since the existence of solutions to (16)-(18) is shown in H 2 Sobolev space, the initial data is assumed to be compatible with the boundary condition: ψ 0 (0) = 0.
[A5] The matrix D U f (U + ) is negative definite.
[A6] The matrix D U f (U + ) has only one zero-eigenvalue and the other eigenvalues of the matrixD U f (U + ) are negative.
The condition [A5] corresponds to the non-degenerate flow; [A6] is for the degenerate flow. We have the relation between D U f (U ) and A(u): Here, for matrices A and B of which all eigenvalues are real number, a notation A ∼ B means that the numbers of positive eigenvalues, zero eigenvalues and negative values of A coincide with those of B. For the proof of (19), see [12]. Owing to (19), the assumption [A5] is equivalent to that A(u + ) is negative definite; [A6] is equivalent to that A(u + ) has only one zero-eigenvalue and the other eigenvalues are negative.
The results concerning the non-degenerate flows are summarized in Theorems 1.5 and 1.6. (i) (Exponential decay) We assume u 0 −ũ ∈ H 2 (R + ) and e αx/2 (u 0 −ũ) ∈ L 2 (R + ) hold for a certain positive constant α. Then, for a certain constant β ∈ (0, α], there exists a positive constant ε 0 such that if then the problem (5), (6) and (7) has a unique solution u globally in time satisfying Moreover there exists a certain constant ν ∈ (0, β) such that the solution u verifies the decay estimate for t > 0. (ii) (Algebraic decay) We assume u 0 −ũ ∈ H 2 (R + ) and (1 + x) α/2 (u 0 −ũ) ∈ L 2 (R + ) hold for a certain positive constant α. Then there exists a positive constant ε 0 such that if then the problem (5), (6) and (7) has a unique solution u globally in time satisfying Moreover the solution u verifies the decay estimate for t > 0.
Related results. The method of symmetrization for the hyperbolic equations by the entropy function are first proposed by well-known researches such as Godunov [2] and Friedrichs-Lax [1]. Then Kawashima and Shizuta in [8] generalize this method to the hyperbolic-parabolic equations. For the system (1) in the full space R n , Umeda, Kawashima and Shizuta in [17] show the asymptotic stability of the constant state under the condition [K]. The equivalence of conditions [K] and [SK] is proved by Shizuta and Kawashima in [15]. For the system (1) in the half space R + , Nakamura and Nishibata in [12] construct a general theory for the existence and the asymptotic stability of the stationary solution under the stability condition. Their results are summarized in Lemmas 1.1-1.4. Our main purpose is to show the asymptotic stability of the stationary solution and obtain the convergence rate. We utilize the weighted energy method and derive these results with/without the stability condition [SK] (or equivalency [K]).
This kind of problems are first studied by Kawashima, Nishibata and Zhu in [7] for the isothermal/isentropic model of compressible viscous gases in the half space R + . They prove the existence and the asymptotic stability of the stationary solution for the outflow problem. Then the convergence rate towards its stationary solution is studied by Nakamura, Nishibata and Yuge in [13].
For the scalar viscous conservation law in the half space R + , Kawashima and Kurata in [4] study the asymptotic stability of the degenerate stationary solution and derive the algebraic convergence rate under the assumption that the initial perturbation is in the weighted space L 2 ω (R + ) with ω(x) = (1 + x) α for α < 3 + 2/q, where q is the degenerate exponent. The upper bound of α is best possible in the sense that the spectrum of the corresponding linearized operator is positive for α > 3 + 2/q. We note that the upper bound of α in Theorems 1.7 and 1.8 is best possible too. In the paper [10], the system of viscous conservation laws with Burgers-type flux term is studied. This system is simplified system for the case of q = 1. Then the algebraic convergence rate toward the degenerate stationary solution is considered under the assumption α < 5.
Outline of the paper. The remainder of the present paper is organized as follows. To discuss the convergence rate towards the stationary solution, we state the summary of results in [12] for the existence of the stationary solution in Section 2. The essential ingredient for deriving convergence rate is to obtain the a priori estimate in the suitable weighted Sobolev space. To this end, in Section 3, we obtain several estimates with the general weight function: basic estimates and higher order estimates up to second order derivatives. To derive the basic estimates, we employ an energy form, which is defined from the entropy function. Especially, for the degenerate flow, we also utilize the Hardy type inequality with the best possible constant in [4]. The higher order estimates are obtained by applying the energy method on the symmetric system. In the section 4, we get the estimates in the suitable weighted Sobolev space by using the results in Section 3 to derive the convergence rate towards the stationary solution. In Section 4.4, we discuss the heat-conductive model for compressible and viscous gases as the application of the main results in the present paper. For viscous and non-heat conductive gas, the stability condition does not hold. The stability of the stationary solution for this case has been open problem in [12] as it assumes the stability condition. Note that the results in the present paper solves this open problem. Notation. The norm | · | denotes the Euclidean norm on vectors and the operator norm on matrices, and the notation · , · denotes the standard inner product on pairs of vectors. For 1 ≤ p ≤ ∞, L p = L p (R + ) represents the Lebesgue space over R + equipped with the norm · L p . For a non-negative integer s, H s = H s (R + ) represents the s-th Sobolev space over R + in the L 2 sense with the norm · H s . We notice the relations H 0 (R + ) = L 2 (R + ) and Let ω = ω(x) > 0 be a smooth scalar function defined on R + . For 1 ≤ p < ∞, we define the weighted L p space over R + by For a non-negative integer s, we define the weighted H s space over R + by

2.
Summary for existence of stationary solution. The existence of the stationary solution to the system (1) is proved in [12]. This result is summarized in Lemmas 1.1 and 1.2. In this section, we state the outline for the proofs of Lemmas 1.1 and 1.2.
2.1. Non-degenerate flow. Integrating (9) over [x, ∞) and multiplying the resultant equality by U u (ũ)D 2 U η(Ũ ) on the left, we have the equality Here we have utilized (4c), owing to Taylor's Theorem and (4b). Due to the assumption [A3], we solve (21a) with respect tov by the implicit function theorem. Thusv is represented as the function ofwv Substituting (22) in (21b), we have an m 2 × m 2 autonomous system of first order ordinary differential equations forw Since the condition [A5] implies that the matrixÃ is negative definite, the existence and the decay property of the solution to the boundary value problem (23) and (10) follow from the stable manifold theorem.
| is sufficiently small, then the problem (23) and (10) has a unique smooth solutionw satisfying To study the degenerate flow, we have to calculate the right-hand side of (21) up to quadratic terms inū = (v,w). Precisely, by using Taylor's Theorem and (4b), we rewrite the equation (20) as where Substituting (25) in (24b), we have an m 2 × m 2 autonomous system of first order ordinary differential equations forw The existence and the decay property of the solution to the boundary value problem (26) and (10)  holds. Here we define the m-vector r as Then there exists a certain region M ⊂ R m2 such that if w b ∈ M and δ := |w + −w b | is sufficiently small, then the problem (26) and (10) has a unique smooth solution w satisfying To derive the optimal convergence rate towards the stationary solution, we have to utilize its properties in detail. Since (B 2 (u + )r)r = B 2 (u + )r,r > 0, we may normalizer as˜ r = 1,˜ := (B 2 (u + )r).
(30) Hence r and r are right and left eigenvectors of A(u + ) corresponding to the zeroeigenvalue, respectively. We also have from (4b) and (30). Therefore U u (u + )r and r U u (u + )D 2 U η(U + ) are right and left eigenvectors of f U (U + ) corresponding to the zero-eigenvalue, respectively.
AsÃ is transformed to the Jordan normal form, we see that there exists an where the ( Hereλ j , j = 2, . . . , m 2 , are non-zero-eigenvalues ofÃ. Setting z(x) := Q −1w (x), we deduce (26) to the system for z as Letting z = (z 1 , z 2 , . . . , z m2 ) = (z 1 , z ), we decompose the system (33) into that for z 1 and z : The system (34) has a local center manifold z = Φ c (z 1 ) = O(|z 1 | 2 ) corresponding to the zero-eigenvalue. Let σ be a solution to the system (34) restricted on the local center manifold satisfying Owing to (27), we see the problem (35) has a solution. Precisely, if r, F[r] > 0 holds and |σ(0)| is sufficiently small, we have the monotone increasing solution σ satisfying σ(x) < 0 for x ∈ R + and σ(x) → 0 as x → ∞; on the other hand, we have the monotone decreasing solution σ satisfying σ(x) > 0 for x ∈ R + and σ(x) → 0 as x → ∞ if r, F[r] < 0 holds and |σ(0)| is sufficiently small. Hence we also have By virtue of the center manifold theorem again, the solution z = (z 1 , z ) to the system (34) is given by The solution σ to the problem (35) exists if the boundary data z(0) = z b := Q −1 (w b − w + ) corresponding to the system (34) belongs to a certain region M ⊂ R m2 related to the local stable manifold and the local center manifold. If |z b | is sufficiently small, the solution σ also satisfies From (14) and (38), we have From (31), we see U u (u + )r is a right eigenvector of f U (U + ) corresponding to the zero-eigenvalue. Hence we may take R(U + ) in (13) by Moreover, due to (31) again, we see is a left eigenvector of f U (U + ) corresponding to the zero-eigenvalue. As D 2 U η is positive definite, the inequality L(U + )R(U + ) > 0 holds. Then we have the equalities where κ := L(U + )R(U + ) > 0. The properties (35), (37) and (42) yield that for |σ| 1. See [12] for the details.
3. Weighted energy estimates. In the remainder of the present paper, we show the asymptotic stability and obtain the convergence rate towards the stationary solution, which is summarized in Theorems 1.5-1.8. We first introduce several function spaces. Letting ω c (x) = e αx or (1 + x) α , we define function spaces X(0, T ) and X ωc (0, T ) for T > 0 by Then we obtain the following lemma for the existence of the time-local solution.
The essential ingredient in the proof of main results is to obtain the a priori estimates in the suitable weighted Sobolev space. To this end, we derive, in this section, the several estimates with the general weight function W (t, x) = χ(t)ω(x), satisfying Basic L 2 estimate for non-degenerate flow. To derive the basic L 2 estimate, we make use of an energy form E defined by following [12]. The energy norm N (T ) is defined by If N (T ) is sufficiently small, the energy form E is equivalent to |(ϕ, ψ)| 2 owing to the positivity of the Hessian matrix D 2 U η(U ). Lemma 3.2. Assume [A5] holds. Let (ϕ, ψ) be a solution to the problem (16), (17) and (18) such that (ϕ, ψ) ∈ C([0, T ]; H 2 (R + )) and ω 1/2 (ϕ, ψ) ∈ C([0, T ]; L 2 (R + )) for a certain positive constant T . Then there exists a positive constant ε 1 such that if N (T ) + δ ≤ ε 1 , then the solution (ϕ, ψ) satisfies Proof. Straightforward computations yield the equation for E: where where Ξ := U −Ũ . By using (18), (54), [A5] and taking N (T ) and δ sufficiently small, the integration of the second and third terms in the left-hand side of (53) are estimated as Owing to the positivity of B 2 (u), we have the estimate for the fourth term Since E is equivalent to |(ϕ, ψ)| 2 , the first term in the right-hand side is estimated as Owing to the boundary condition (18) and (55), we have for the second term. Here we have also used the fact that |ũ x | ≤ Cδ and |ξ x | ≤ N (T ), which follows from (12). Estimating (55) gives where is an arbitrary positive constant. Then the integration of W x B over R + is handled as Consequently, integrating (53) with respect to x and t over R + × (0, t), substituting the above estimates (58)-(63) in the resultant equality and letting N (T ), δ and suitably small, we obtain the desired estimate (51).

3.2.
Basic L 2 estimate for degenerate flow. To derive the basic L 2 estimate for the degenerate flow, we introduce the constant matrices P and P as follows.
Let r be the m-vector defined in (28), and Q and Λ be the m 2 × m 2 matrices in (32). We represent Λ by (0, Λ 0 ), where Λ 0 = (0, Λ ) is the m 2 × (m 2 − 1) constant matrix. Then let an m × (m − 1) constant matrix P , and also an m 1 × (m − 1) and an m 2 × (m − 1) constant matrices P 1 and P 2 , be Moreover, let an m × m constant matrix P be given by Here I m and O m,n denote the identity matrix of size n and the m × n zero matrix, respectively. From straightforward calculations with (29)-(30) and (32), we see that [C] The matrix P defined in (65) satisfies the conditions (i) P is invertible; (ii) the m × m constant matrix P A(u + )P satisfies the equality where the (m − 1) × (m − 1) constant matrixĀ is real symmetric and negative definite; (iii) P 2 B 2 (u + )r = 0.
Moreover we utilize the following Hardy type inequality with the best possible constant, which is proved in [4].
To show the following lemma, we utilize the weight function   (52) and (78).
Remark 1. The upper bound "5" for the index α is optimal. This fact is proved in [4] for the scalar viscous conservation laws, which is the special case of (1).

3.3.
Estimate for the first order derivatives. We next derive the estimate for the first order derivatives. Note that the computations in Subsections 3.3 and 3.4 hold for both the non-degenerate and the degenerate flows. We first show the estimate for ϕ x and ψ x .  Proof. Differentiating (16a) with respect to x and then taking an inner product of the resultant equality with ϕ x yield Taking an inner product of (16b) with −ψ xx , we have Here we have used A 21 = A 12 . Adding (89) to (90), we obtain 778 TOHRU NAKAMURA, SHINYA NISHIBATA AND NAOTO USAMI Multiplying (91) by the weight function W (t, x) = χ(t)ω(x), we have Since ψ t (t, 0) = 0 due to the boundary condition (18), the integration of the second term in the left-hand side satisfies Here we have also used [A3]. The third term is also estimated by [A3] as The fourth term in the left-hand side and the second term in the right-hand side of (92) are handled as where ε and are arbitrary positive constants. The integral of the first term in the right-hand side is estimated as Consequently, integrating (92) with respect to x and t over R + × (0, t) and then using the above estimates (93)-(97) with sufficiently small constant , we obtain the desired estimate (88).
In Lemma 3.5, we have the estimate for the dissipation ϕ x with the weight function ω x . However, this estimate disappears if ω ≡ 1. The stability condition [SK] allows us to get the estimate |ϕ x | L 2 ,ω , which equals ϕ x L 2 if ω ≡ 1. This estimate is necessary in the proof of Theorem 1.5.  Proof. LettingĀ 0 := A 0 (u + ),Ā := A(u + ) andB := B(u + ), we rewrite the system (15) asĀ Multiplying (99) by the compensation matrix K and taking an inner product of the resultant equality with ξ x yield since KĀ 0 is skew-symmetric. Multiplying (100) by the weight function W (t, x) = χ(t)ω(x), we have From a direct computation with (16a), the estimate holds. Owing to the boundary condition ψ(t, 0) = ψ t (t, 0) = 0, (102) and Sobolev's lemma, the second term in the left-hand side is estimated as The third term is estimated as Due to the positivity of the matrix L in the condition [K], we have The first term in the right-hand side of (101) is estimated as The second and third terms are estimated as where is an arbitrary positive constant. Integrate (101) with respect to x and t over R + × (0, t), substitute the above estimate (103)-(108) and then take sufficiently small. These procedures yield the desired estimate (98).

3.4.
Estimate for the second order derivatives. The main purpose of this subsection is to derive the estimate for the second order derivatives. Precisely, we obtain the estimates for ϕ t , ψ t and ϕ xx and the dissipative estimate for ϕ xx . In deriving these estimates, the regularity of the time local solution is insufficient. Hence we need to employ a difference approximation with respect to x as in [7] and a mollifier with respect to t (see the paper [12] for the details). We, however, assume the perturbation (ϕ, ψ) is sufficiently smooth and omit these arguments since these procedures are standard. We first show the estimate for the time derivatives ϕ t and ψ t in Lemma 3.7. In fact, these computations are executed with the mollifier with respect to the time variable.
Proof. Differentiating (15) with respect to t and taking an inner product of the resultant equality with ξ t , we have (110) Multiplying (110) by the weight function W (t, x) = χ(t)ω(x) yields Owing to [A3] and the boundary condition ψ t (t, 0) = 0, the second term in the left-hand side is estimated as The fourth term is estimated as where is an arbitrary positive constant. Integrating (111) with (112), (113) and taking sufficiently small, we derive the desired estimate (109).
The estimate for ϕ xx is summarized in Lemma 3.8. To prove Lemmas 3.8 and 3.9, the regularity of the time-local solution is not enough. To cover this problem, we employ the difference quotient with respect to the spatial variable. However, this procedure is same as the proof of the corresponding estimates in [12]. Hence, we omit this discussion and proceed to estimate as if the solution has enough regularity.  (ϕ, ψ) satisfies ω 1/2 (ϕ, ψ) ∈ C([0, T ]; H 2 (R + )), then we have for t ∈ [0, T ], where H 4 and h 1 2 are defined in (115) and (116). Proof. Applying ∂ 2 x and ∂ x to (16a) and (16b) and then taking an inner product of the resultant equalities with ϕ xx and ψ xxx , respectively, we have Here we have also used A 21 = A 12 . Adding (115) to (116) yields Multiplying (117) by the weight function W (t, x) = χ(t)ω(x), the following equality holds: The second and the third terms in the right-hand side are estimated as 782 TOHRU NAKAMURA, SHINYA NISHIBATA AND NAOTO USAMI where and are arbitrary positive constants. Consequently, integrating (118) and substituting (119) and (120) in the resultant equality and then taking and sufficiently small, we obtain the desired estimate (114).
We derive the dissipative estimate for ϕ xx with respect to the norm |·| L 2 ,ω , which is equal to the norm · L 2 if ω ≡ 1, under the stability condition.  Proof. Differentiating (99) with respect to x and multiplying the resultant equality by the compensation matrix K on the left yield The skew-symmetric matrix KĀ 0 is represented as where K 1 and K 3 are m 1 × m 1 and m 2 × m 2 skew-symmetric matrices, respectively; K 2 is an m 1 × m 2 matrix. Then the following equalities hold: Taking an inner product of (122) with ξ xx and utilizing the above equalities, we have 1 2 Multiplying (123) by the weight function W (t, x) = χ(t)ω(x) yields From straightforward calculations with (16a), the estimate holds. Then the second term in the left-hand side is estimated as by Sobolev's lemma and (125). The third terms in both sides are estimated as where ε is an arbitrary positive constant. The terms in the right-hand side except the first, the third and the last ones are estimated as where is an arbitrary positive constant. Consequently, integrating (124), utilizing the above estimates (126)-(129) and letting sufficiently small, we obtain the desired estimate (121).

4.3.
Degenerate flow with stability condition. To prove Theorem 1.7, we derive the following estimate.