L P decay for general hyperbolic-parabolic systems of balance laws

We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.


1.
Introduction. We are interested in a general class of partial differential equations arising from continuum mechanics. They are hyperbolic-parabolic balance laws in the following form: [B jk (w)w x k ] xj + r(w), m ≥ 1, (1.1) where w, f j , r ∈ R n and B jk ∈ R n×n . The unknown function w = w(x, t) depends on the space variable x = (x 1 , . . . , x m ) t ∈ R m and the time variable t ∈ R + , and stands for physical densities such as mass density, momentum density, energy density, etc. As given functions of w, f j are flux functions, 1 ≤ j ≤ m, and r represents external forces, relaxation, chemical reactions and so forth. The matrices B jk , 1 ≤ j, k ≤ m, are also known functions of w. They are viscosity matrices, describing viscosity, heat conduction, species diffusion, etc. Equation (1.1) describes the balance of physical quantities, such as mass, momentum and energy, of a flow. The flux functions usually satisfy an entropy condition so that the corresponding inviscid system is completely hyperbolic [2]. We are interested in physical models (not artificial models) thus the viscosity matrices are rank deficient. This implies (1.1)
The dissipation parameters are the shear viscosity coefficient µ, the second viscosity coefficient µ , and the thermal conductivity κ. The thermodynamic equation is where v is the specific volume, and s is the entropy. Therefore, only two of the thermodynamic variables are independent. The other variables and the dissipation parameters are regarded as known functions of these two.
The Navier-Stokes equations can be derived from the Boltzmann equation by Chapman-Enskog expansion assuming the gas molecules are under translational non-equilibrium while they have no internal structure or other forms of non-equilibrium [10]. The equations are an example of (1.1) with r(w) = 0. That is, they are an example of the special case of hyperbolic-parabolic conservation laws, [B jk (w)w x k ] xj , m ≥ 1.
(1.4) Example 2. The following equations describe the motion of a polyatomic gas in vibrational non-equilibrium: ρ t + div(ρu) = 0, (ρu) t + div(ρuu t ) + ∇p = 0, (ρE) t + div(ρEu + pu) = 0, (ρe 2 ) t + div(ρe 2 u) = ρ e * 2 − e 2 τ , (1.5) where the notations are as defined in Example 1 except with e 1 and e 2 being the total of equilibrium internal energy and vibrational energy, respectively. The first three equations in (1.5) are exactly the Euler equations for inviscid, compressible flows, describing the conservation of mass, momentum and energy, respectively. The last equation in (1.5) describes the relaxation of the vibrational energy, the non-equilibrium internal energy, towards its local equilibrium value e * 2 in the time scale τ . Here τ is called the relaxation time. Because we are considering a polyatomic gas whose molecules have an internal structure, and the structure is not in dynamical equilibrium, we have to use two sets of thermodynamic variables to describe the flow: one for the total of equilibrium modes, and one for the non-equilibrium vibrational mode. We use subscript "1" for the first set and "2" for the second one, respectively. Thus with s 1 and s 2 being the equilibrium entropy and vibrational entropy, respectively. The two sets of variables obey different thermodynamic equations: (1.7) We note that the second equation is volume independent. It is clear that two variables in mode 1 and one in mode 2 are independent. With m components of the velocity, (1.5) is a system of m + 3 equations for m + 3 unknowns.
As in the case of Navier-Stokes equations, the equations for inviscid, vibrational non-equilibrium flow (1.5) can be derived from the Boltzmann equation by Chapman-Enskog expansion [10]. In this case, the only non-equilibrium mode is the vibrational mode. In particular, the flow is under translational equilibrium.
Equation (1.5) is in the form (1.1) with B jk = 0. That is, it is an example of the special case of hyperbolic balance laws: (1.8) The following example is also of this type.
Example 3. Euler equations with damping for inviscid, compressible, isentropic or isothermal flows: ρ t + div(ρu) = 0, Here since the flow is isentropic or isothermal, we regard the pressure p as a known function of the density ρ. Equation (1.9) describes the motion of a gas through a porous medium, which induces a friction force proportional to the momentum.
Other examples of (1.8) with r (w) rank deficient are the Kerr-Debye model for the propagation of electromagnetic waves in a nonlinear Kerr medium, and the equations for the motion of an unbounded, homogeneous, viscoelastic bar with fading memory when the kernel of the memory is a finite sum of exponential decay functions [14].
The next example of (1.1) is not of the special cases of (1.4) or (1.8). It has both nontrivial viscosity matrices and the lower order term.
Example 4. The following equations describe the motion of a polyatomic gas in both translational and vibrational non-equilibrium: (1.10) Here the notations are the same as in Examples 1 and 2, with a new dissipation parameter ν, which is the self-diffusion coefficient. Equation (1.10) is a system of m+3 equations for m+3 unknowns, supplemented by the thermodynamic equations 366 YANNI ZENG (1.7). In this case, mode 1 represents the combination of the translational mode and rotational mode of the gas molecules, while mode 2 is for the vibrational mode.
Similar to Examples 1 and 2, (1.10) is derived from Boltzmann equation by Chapman-Enskog expansion under the assumption of translational and vibrational non-equilibrium [1]. The translational non-equilibrium results in viscosity, heat conduction and self-diffusion, while the vibrational non-equilibrium gives rise to the relaxation. The other internal modes, such as rotation, are assumed to be in equilibrium hence share the same temperature T 1 of the translational mode. Otherwise, new relaxation equations can be added for those modes. This extends the system (1.10) but does not change its structure.
We are interested in the Cauchy problem of (1.1) with prescribed initial data: Here w 0 is assumed to be a small perturbation of a constant equilibrium statew, r(w) = 0. In a recent paper the author has proposed a set of structural conditions for (1.1), which leads to the existence of global solution of the Cauchy problem near an equilibrium state [13]. The general theorem there applies to Example 4 under physical assumptions. It also recovers the known results in the literature of the hyperbolic-parabolic conservation laws (1.4) and of the hyperbolic balance laws (1.8) as special cases.
In this paper we show that under the same set of structural conditions, we may obtain L p (p ≥ 2) convergence rates of small solutions of (1.1) to an equilibrium state. The rates are optimal, and consistent with those obtained in [3] for the hyperbolic-parabolic conservation laws (1.4), as well as those in [4] for the hyperbolic balance laws (1.8). Our result is true for all space dimensions m ≥ 1. However, the strategies used to prove the cases m ≥ 2 and m = 1 are different. (Neither applies to the other.) Therefore, we focus on multi space dimensions (m ≥ 2) here, and leave the case of one space dimension to a future paper.
We now state our basic structural conditions for (1.1). Consider a neighborhood O of the constant equilibrium statew, and define the equilibrium manifold E in O as 12) The functions f j (w), B jk (w) and r(w) are assumed to be smooth in O. In the following we use f j to denote the Jacobian matrix of f j with respect to w, etc.
1. There exists a strictly convex entropy function η, which is a scalar function of w in O, satisfying the following properties.
(iii) On E, η r is symmetric, semi-negative definite. 2. Equation (1.1) has n 1 conservation laws. That is, there is a partition n = n 1 + n 2 , n 1 , n 2 ≥ 0, such that 13) with w 1 ∈ R n1 , r 2 , w 2 ∈ R n2 , and (r 2 ) w2 ∈ R n2×n2 is nonsingular. Here (r 2 ) w2 denotes the Jacobian matrix of r 2 with respect to w 2 , etc. 3. There is a diffeomorphism ϕ → w from an open setÕ ⊂ R n to O and a constant orthogonal matrix P ∈ R n×n such that (1.14) Here n 3 , n 4 ≥ 0 are two constant integers such that n 3 + n 4 = n, and Let N 1 be the null space of B(ξ) and N 2 be the null space of r (w). Then for each ξ, N 1 ∩ N 2 contains no eigenvectors of A(ξ).
We comment that the partitions n = n 1 + n 2 and n = n 3 + n 4 in conditions 2 and 3 of Assumption 1.1 are independent. The locations of the conservation laws and rate equations are also independent of the locations of the hyperbolic equations and parabolic equations. (Matrix P is usually a permutation and is for such a purpose.) In the appendix we use Example 1.4 to illustrate the independence by different choices of dissipation parameters.
We introduce the following notations to abbreviate the norms of Sobolev spaces with respect to x: (1.16) With ϕ and P given in condition 3 of Assumption 1.1, we definẽ Letw be a constant equilibrium state, Assumption 1.1 be satisfied, s > m 2 + 1 (m ≥ 1) be an integer, and w 0 −w ∈ H s (R m ). Then there exists a constant ε > 0 such that if w 0 −w s ≤ ε, the Cauchy problem (1.1), (1.11) has a unique global solution w. The solution satisfies w −w ∈ C([0, ∞); H s (R m )), Here D x w denotes first partial derivatives of w with respect to x, etc.
Let D l x be partial derivatives (∂/∂x) α with a multi index α such that |α| = l. Our main result is the following L 2 decay estimates for w in Theorem 1.2 when m ≥ 2. Theorem 1.3. Letw be a constant equilibrium state of (1.1), and Assumption 1.1 be true. Let m ≥ 2, s > m 2 + 1 be an integer, and w 0 −w ∈ H s (R m ) ∩ L 1 (R m ).

YANNI ZENG
Then there exists a constant ε > 0 such that if w 0 −w s + w 0 −w L 1 ≤ ε, the solution of (1.1), (1.11) given in Theorem 1.2 has the following estimates for t ≥ 0: To obtain L p decay rates with p ≥ 2 we recall Gagliardo-Nirenberg inequality [8]: There is a constant C > 0 such that for g ∈ H k (R m ), . Applying (1.21) to g = w −w with k = s − 2, and to g = r 2 (w) with k = s − 4, we have the following corollary of Theorem 1.3: Corollary 1.4. Under the assumptions of Theorem 1.3, the solution of (1.1), (1.11) has the following L p estimates with p ≥ 2: For t ≥ 0, As applications, in the appendix we give the reduced versions of Assumption 1. For hyperbolic-parabolic conservation laws (1.4), L 2 -decay rates similar to those in Theorem A.2 and convergence rates to the solution of the corresponding linear system have been obtained in [3] under similar assumptions. For hyperbolic balance laws (1.8), a parallel result of Theorem A.4 has been obtained in [4]. In fact, Theorem A.4 simplifies and slightly weakens the assumptions in [4] for the same decay rates, see [11] for a discussion of the two sets of hypotheses. Here we extend the study of L p decay rates to the general hyperbolic-parabolic system of balance laws (1.1), which leads to our main results, Theorem 1.3 and Corollary 1.4.
The plan of the paper is as follows. Section 2 is for preliminaries needed in our analysis. Section 3 is devoted to estimates of the linearized system. In Section 4 we give a weighted energy estimate. In Section 5 we carry out the proof of Theorem 1.3. Finally, in the appendix we discuss applications of the main results to Example 4 for the thermal non-equilibrium flow, and to the two special cases of hyperbolicparabolic conservation laws (1.4) and hyperbolic balance laws (1.8).
Throughout this paper we use C to denote a universal positive constant. Also, we use the bar accent for the value of a variable taken at the constant equilibrium statew, e.g.,φ = ϕ(w), etc.

2.
Preliminaries. In this section we assume that condition 2 of Assumption 1.1 holds. In (1.1) the lower order term r(w) represents the part of solution with faster decay rate, as evidenced by (1.18). We separate this part from the leading term by introducing a new variable ψ using the notations in (1.13): where ψ 1 = w 1 ∈ R n1 and ψ 2 = r 2 ∈ R n2 . Under condition 2 of Assumption 1.1, ψ is a diffeomorphism, with the Jacobian matrices Next we linearize (1.1) around the constant equilibrium statew using the new variable ψ. Letψ be the perturbation. Multiplying from the left by ψ w , (1.1) can be written as Multiplying the equation by a constant matrixÃ 0 to be defined in (2.6), (1.1) can be further written as (2.7) Without the nonlinear sourceR, (2.5) would be a linear system ofψ with constant coefficients. The coefficients possess important properties, which we cite from Lemma 2.8 of [13]: Under conditions 1, 2 and 4 of Assumption 1.1, we have the following.
ThenÃ(ξ) is real, symmetric, andB(ξ) is real, symmetric and semi-positive definite. They satisfyÃ When (i) and (ii) of Lemma 2.1 hold, there are several equivalent forms of (iii), see Theorem 1.1 and Remark 1.2 in [9]. Among them there is the existence of a so-called compensating function. As it is needed in our analysis, we state the following lemma as a consequence of Lemma 2.1 above, together with Theorem 1.1 and Remark 1.2 in [9]: Lemma 2.2. Let conditions 1, 2, and 4 of Assumption 1.1 be true. Then there exists a smooth function K(ξ) ∈ R n×n , defined on S m−1 and called a compensating function, with the following properties: is real, symmetric, positive definite.
We may use Fourier transform to study a linear system with constant coefficients. We use the hat accent to denote the Fourier transform of a function in x: (2.10) Taking Fourier transform of (2.5) we havẽ which can be further written as using (2.8).
We take (n 1 , n 2 ) partition ofÃ 0 in rows and columns. From (2.2), (2.6) and (2.22), its (2, 1) block is zero. By definition,Ã 0 is symmetric thus block diagonal. That is, under the assumption of Lemma 2.3 we havẽ Together with (2.21),R is simplified as In our analysis of (1.1) in later sections, the treatment of the lower order term r relies on the following crucial estimate obtained in [11]: Under conditions 1(iii) and 2 of Assumption 1.1, in a small neighborhood ofw we have To handle the viscosity term in (1.1) we need the diffeomorphism ϕ defined in condition 3 of Assumption 1.1. The direct use of ϕ is the main difference in our approach to handle the general system (1.1). This is to compare to the traditional approach of converting the system under study into a "normal form", commonly used in the study of the special cases (1.4) and (1.8), see [3,4] and references therein. For the application in later sections we cite the following lemma describing properties related to ϕ from [13]: Lemma 2.5. [13] Conditions 1(ii) and 3 in Assumption 1.1 imply that P t w t ϕ η P is block-upper triangular, and P t w t ϕ η B jk w ϕ P is block diagonal in the partition n = n 3 + n 4 : Our study in later sections needs some tools from analysis. We summarize them as Lemmas 2.6-2.8 as follows. The lemmas are either special cases of Gagliardo-Nirenberg inequality or its applications (Moser-type calculus inequalities). (Here Lemma 2.6 is a special case of (1.21).) They can be found in many papers dealing with energy estimates in multi space dimensions, e.g. [3]. Here our formulation follows those in [11].

29)
where α = m/(2s) < 1 and C > 0 is a constant depending only on m and s.
Lemma 2.7. Let g be a given smooth function of w in a neighborhood ofw. If w −w ∈ H s (R m ) with w −w s ≤ ε and s > m/2, then where C > 0 is a constant depending only on m, s and ε.
where C > 0 is a constant depending only on m and l.
3. Estimates for linear system. Motivated by the linear system (2.11) (in the Fourier space) and the expression forR in (2.24), we consider the following linear system with sources: whereÃ 0 ,Ã,B andL are defined by (2.6) and (2.8) Similar systems have been studied before. In fact, the case without iĤξ in the source was considered in [3], see Lemma 3.A.1 therein. As evidenced by (2.24), the term iĤξ in (3.1) is to accommodate conserved quantities. The case without the viscosityB was considered in [4] for hyperbolic balance laws. Here we extend the previous works to our general system of hyperbolic-parabolic balance laws. The first result of this section is the following lemma.
where C and c 1 are positive constants,

4)
and the matrix/vector norms are the Euclidean norms.
Our next result is to apply Plancherel's theorem to obtain estimates in the physical space.
. Under conditions 1, 2 and 4 of Assumption 1.1, the solution of (3.1) with h satisfying (3.2) has the following estimate in the physical space: For 0 ≤ t ≤ T ,

17)
where C and c 1 are positive constants.
Proof. For a multi index α with |α| = l, by Plancherel's theorem and (3.3) we have where is defined in (3.4). Noting (r) ≥ r 2 /2 for 0 ≤ r ≤ 1 and (r) ≥ 1/2 for r ≥ 1, the first term on the right-hand side of (3.18) is bounded by This gives the first term on the right-hand side of (3.17). Similarly, we bound the second term on the right-hand side of (3.18) by This gives the last three terms on the right-hand side of (3.17).

Weighted energy estimates.
In this section we use weighted energy estimate to derive decay rates for the nonlinear system. Although the rates are not optimal, they help to obtain the optimal ones in next section. This is possible because these rates are needed in the nonlinear source when performing a priori estimate, and can be enhanced by the other decay factors in the nonlinear terms. Similar techniques have been used for obtaining both pointwise estimates and L 2 -decay rates, e.g., see [5,6,4,7].
Theorem 4.1. Letw be a constant equilibrium state of (1.1), and Assumption 1.1 be true. Let m ≥ 2, s > m 2 + 1 be an integer, and w 0 −w ∈ H s (R m ). Then there exists a constant ε > 0 such that if w 0 −w s ≤ ε, the solution of (1.1), (1.11) given in Theorem 1.2 has the following estimates:  Our goal is to prove M 2 (t) ≤ C w 0 −w 2 s (4.4) for some constant C > 0. Equations (4.1) and (4.2) are then direct consequence of (4.3) and (4.4). Below we assume that M (t) is small.
We apply D l x to (1.1) and multiply the result by (D l x w) t η (w). This gives us We replace the time variable t by τ , multiply the equation by (τ +1) k , and integrate the result over R m × [0, t]. Then we have, for 1 ≤ k ≤ l ≤ s, where Note that we have used the symmetry of (η f j )(w) in Assumption 1.1 to obtain I 3 , and (1.13) to derive the second expression of I 5 .
To estimate I 5 in (4.6) we need the key estimate (2.26), which implies (4.31) where we have applied (2.1) and (2.2). Substituting (4.32) into the right-hand side of (4.31), we simplify it as (4.33) We replace the right-hand side of (4.31) by (4.33) and substitute the result into I 5 in (4.6). Noting the leading term in (4.33) is η w2w2 D l−1 x [(r 2 ) −1 w2 r 2 (w) xj ], which can be further linearized, we have (4.34) Here we note that for l = 1 the terms involving D l−1 x on the right-hand side of (4.34) disappear.
We change t to τ in (4.39), multiply the equation by (τ + 1) k |ξ| 2l for 1 ≤ k ≤ l ≤ s − 1, and integrate the result over R m × [0, t]. From Lemma 2.2, S is real, symmetric, and positive definite. Thus we have, for some constant c 5 > 0, with (4.41) Noting I 6 is real and by direct calculation, we have where in the last step we have applied Plancherel theorem.

5.
A priori estimates. In this section we prove our main result, Theorem 1.3. Let for t ≥ 0 and 0 ≤ k ≤ s, where w is the solution given by Theorem 1.2 andw is the constant equilibrium state. By a standard continuity argument, to prove (1.19) in Theorem 1.3 under smallness assumption on the initial data, we only need to prove the following proposition.
where C > 0 is a constant independent of T .
Proof. We recall from (2.1) and (2.3), where w 1 and r 2 are from condition 2 of Assumption 1.1, and the partition of the vectors are n = n 1 +n 2 . The unknownψ satisfies (2.5), with coefficient matrices defined in (2.6), and the nonlinear sourceR given by (2.21). Taking Fourier transform with respect to x, we have found thatψ satisfies (2.11).
Next we consider higher derivatives. From (2.16), (2.30) and (5.9), (5.11) Here we have derived (5.11) for 1 ≤ l ≤ s. However, it is easy to verify that (5.11) is true for 0 ≤ l ≤ s. Similarly, from (2.19) we have for 0 ≤ l ≤ s − 1. Now we apply (5.9) and (4.1) to the first term on the right-hand side and (5.9) to the other terms. This gives us, for 0 ≤ l ≤ s − 1, (5.14) These further imply that for 0 ≤ l ≤ s − 2,  To estimate R 22 we recall (4.7). Thus from (2.17) and similar to (5.21)-(5.23) we have 24) where the terms containing D l x w do not exist when l = 0. Here we have used (4.7) to obtain D 2 xw2 s−1 . This is necessary since onlyw 2 has better regularity. Applying  The rest of this section is to obtain (1.20) using (1.19), hence complete the proof of Theorem 1.3. Note that r 2 =ψ 2 from (5.3). Recall (4.14), which gives us where c 2 > 0 is a constant and R is defined in (4.13).
To estimate D l x r 2 we apply (2.30) and note that r 2 (w) = 0. This gives us D l x r 2 (w 0 ) ≤ C D l x (w 0 −w) , 0 ≤ l ≤ s.