1 Introduction

ASYMPTOTIC LIMIT AND DECAY ESTIMATES FOR A CLASS OF

DISSIPATIVE LINEAR HYPERBOLIC SYSTEMS

IN SEVERAL DIMENSIONS

August 4, 2019Thinh Tien NGUYEN111Department of Mathematics, Gran Sasso Science Institute – Istituto Nazionale di Fisica Nucleare, Viale Francesco Crispi, 7 - 67100 L’Aquila (ITALY), nguyen.tienthinh@gssi.infn.it

Abstract. In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers , let be a matrix-vector, where , and let be not required to be symmetric but have one single eigenvalue zero, we consider the Cauchy problem for linear systems having the form

 ∂tu+A⋅∇xu+Bu=0,(x,t)∈Rd×R+.

Under appropriate assumptions, we show that the solution is decomposed into , where has the asymptotic profile which is the solution, denoted by , of a parabolic equation and decays at the rate as in any -norm, and decays exponentially in -norm, provided for . Moreover, decays at the optimal rate as if the system satisfies a symmetry property. The main proofs are based on asymptotic expansions of the solution in the frequency space and the Fourier analysis.
Keywords: Large-time behavior, Dissipative linear hyperbolic systems, Asymptotic expansions.
MSC2010: 35L45, 35C20.

## 1. Introduction

Consider the Cauchy problem for partially dissipative linear hyperbolic systems

 (1.1) {∂tu+A⋅∇xu+Bu=0,(x,t)∈Rd×R+,u(x,0)=u0(x),

where and , not required to be symmetric. The system (1.1) can be regarded as discrete-velocity models where determines the velocities of moving particles and gives the transition rates of the velocities after collisions among the particles in the system. For instance, this type of dissipative linear systems arises in the Goldstein–Kac model [6, 8] and the model of neurofilament transport in axons [5].

The large-time behavior of the solution to (1.1) in terms of decay estimates has been established for years. It follows from [18] that under appropriate assumptions on and , if is the solution to (1.1) with the initial data , then one has

 (1.2) ∥u∥L2≤C(1+t)−d4∥u0∥L1+Ce−ct∥u0∥L2,∀t>0,

for some positive constants and . Moreover, the estimate (1.2) was generalized in [3], where can be written in the conservative-dissipative form with , a positive definite matrix not required to be symmetric. The authors in [3] also showed that the conservative part , describing the low dynamics of to the equilibrium manifold , satisfies the estimate

 (1.3) ∥u(1)−U∥Lp≤Ct−d2(1−1p)−12∥u0∥L1,∀p≥min{d,2},t≥1,

while the dissipative part of decays exponentially in , where is the solution to a parabolic system given by applying the Chapman–Enskog expansion method. A more exact asymptotic parabolic-limit of was established for a class of the generalized Goldstein–Kac system in [12], where the matrix is symmetric. The solution to (1.1) then satisfies the estimate

 (1.4) ∥u−U∥L2≤Ct−d4−12∥u0∥L1∩L2,∀t≥1,

where is obtained based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchoff’s matrix tree theorem from the graph theory. Recently, [13] has optimized the above decay estimates for the solution to (1.1) in one dimension , with an explicit parabolic-limit and a corrector , namely, one has

 (1.5) ∥u−U−V∥Lp≤Ct−12(1q−1p)−δ∥u0∥Lq,∀1≤q≤p≤∞,t≥1,

where , solving a parabolic system arising in the low-frequency analysis decays diffusively, and solving a hyperbolic system arising in the high-frequency analysis decays exponentially. The decay estimate (1.5) is remarkable since it holds for general and ranging over . Such kind of decay estimates is very well-known e.g. the - decay estimate for the linear damped wave equation as in [11, 15, 7, 14].

To obtain (1.5) in one dimension , one primarily considers the asymptotic expansions of the fundamental solution to the system (1.1) in the Fourier space, divided into the low frequency, the intermediate frequency and the high frequency which naturally produce the time-asymptotic profile. Then, by an interpolation argument once the - estimate and the - estimate for are accomplished, one obtains the desired - estimate for any . The same strategy will be applied to the system (1.1) in several dimensions in this paper. Nevertheless, difficulties occur as the dimension increases. For instance, as mentioned in [3], one cannot expect the estimate

 (1.6) ∥u∥L1≤C∥u0∥L1

hold in general since for large time, , where is the left eigenvector associated with the eigenvalue of , behaves as the solution to the reduced system

 (1.7) ∂tω+L0AR0⋅∇xω=0,(x,t)∈Rd×R+,

where is the right eigenvector associated with the eigenvalue of , and thus, it is known in [4] that (1.6) is not true in general. The estimate (1.6) in fact depends strongly on a uniform parabolic operator. Nonetheless, this obstacle can be defeated if as in [13] or if is a simple eigenvalue of since the system (1.7) then becomes scalar and it allows us to obtain (1.6) as we will see in this paper. Another difficulty arises in the high-frequency analysis due to the loss of integrability and the fact that one cannot perform a uniform expansion of the fundamental solution as the dimension increases. Hence, the corrector as in (1.5) cannot be obtained trivially.

The aim of this paper is to study the - decay estimate for the conservative part of the solution to the system (1.1) in several dimensions for general and in in order to generalize (1.2), (1.3) and (1.4), where is not required to be symmetric but has one single eigenvalue zero. The - estimate as in (1.5) for the multi-dimensional case is still a challenge for the author.

For , consider the operators

 (1.8) E(x):=B+A(x),A(x):=A⋅x=d∑j=1Ajxj,

where and . We start with the following reasonable assumptions. Condition . [Hyperbolicity] for is uniformly diagonalizable with real linear eigenvalues i.e. there is an invertible matrix for satisfying

 supw∈Sd−1|R(w)||R−1(w)|<+∞,

for any matrix norm, such that is a diagonal matrix whose nonzero entries are real linear in .

Condition . [Diagonalizing matrix] There is a matrix uniformly diagonalizing such that is a constant matrix.

Condition . [Partial dissipation] The spectrum of is decomposed into where is simple and .

Moreover, the requisite condition for the decay of the solution to (1.1), strictly related to the Shizuta–Kawashima condition: the eigenvectors of do not belong to the kernel of for any (see [10, 17, 19] and therein), is given by Condition . [Uniform dissipation] There is a constant such that for any eigenvalue of in (1.8) for , one has

###### Remark 1.1 (Relaxing the conditions A and R).

The requirement of the linearity of the eigenvalues of the matrix satisfying the condition and the existence of the matrix satisfying the condition can be omitted by considering the dissipative structures proposed in [18, 3]. Nonetheless, the structures in [18, 3] require that the system (1.1) is Friedrich symmetrizable while in our case, the matrix is only uniformly diagonalizable. The advantage of the linearity of the eigenvalues of the matrix and the existence of the matrix is that one can construct the high-frequency asymptotic expansion of in (1.8) after subtracting a suitable Lebesgue measure zero set.

We now construct the asymptotic parabolic-limit of the solution to (1.1). Let be an oriented closed curve in the resolvent set of such that it encloses zero except for the other eigenvalues of . One sets

 (1.9) P(0)0:=−12πi∫Γ(B−zI)−1dz,Q(0)0:=12πi∫Γz−1(B−zI)−1dz,

which are the eigenprojection and the reduced resolvent coefficient associated with the eigenvalue zero of . We consider the Cauchy problem

 (1.10) {∂tU+c⋅∇xU−div(D∇xU)=0,(x,t)∈Rd×R+,U(x,0)=P(0)0u0(x),

where and is positive definite with scalar entries

 (1.11) ch:=tr(AhP(0)0),Dhℓ:=12tr(AhP(0)0AℓQ(0)0+AhQ(0)0AℓP(0)0).
###### Theorem 1.2 (Lp-Lq decay estimates).

Let be the solution to the Cauchy problem (1.1) with the initial data for . Under the assumptions , , and , the solution is decomposed into

 (1.12) u(x,t)=u(1)(x,t)+u(2)(x,t),

where

 u(1)(x,t):=F−1(e−E(ik)tP0(ik)χ(k))∗u0(x)

and is the remainder, where is the eigenprojection associated with the eigenvalue of in (1.8) converging to as and is a cut-off function with support contained in the ball , valued in , for small . Moreover, for any and , one has

 (1.13) ∥u(1)−U∥Lp≤Ct−d2(1q−1p)−12∥u0∥Lq,

where is the solution to (1.10) with the initial data , and one has

 (1.14) ∥u(2)∥L2≤Ce−ct∥u0∥L2

for some constant and for all .

###### Remark 1.3 (Finite speed of propagation).

In the case where the solution to the system (1.1) has finite speed of propagation, since the fundamental solution associated with has compact support contained in the wave cone for some constant , one can decompose into , where

 u(1)(x,t):=F−1(e−E(ik)tχ(k))∗u0(x),

and is the remainder, where is a cut-off function with support contained in the ball , valued in , for any , and the estimates (1.13) and (1.14) still hold for . This fact will be proved in the subsequent sections. For instance, it is the case where the system (1.1) is Friedrich symmetrizable. Nonetheless, in one dimension , the case can be treated since the Cauchy integral theorem holds for the whole complex plane, and thus, one can use the estimates for the asymptotic expansion of the fundamental solution in the high frequency after changing paths of integrals of holomorphic functions (see [13]).

Moreover, consider the one-dimensional linear Goldstein–Kac system

It can be checked easily that satisfies the linear damped wave equation

 ⎧⎪⎨⎪⎩∂2ttw−∂2xxw+∂tw=0,(x,t)∈R×R+,w(x,0)=w0(x),∂tw(x,0)=w1(x),

where and are appropriate initial data. It then follows from [11] that

 (1.15) ∥∥w−ϕ−e−t2w0(x+t)+w0(x−t)2∥∥Lp≤Ct−12(1q−1p)−1∥(w0,w1)∥Lq,

for any and , where is the solution to the heat equation

 {∂tϕ−∂2xxϕ=0,(x,t)∈R×R+,ϕ(x,0)=w0(x)+w1(x).

Without regarding the exponentially decaying term in (1.15), there is a difference of a quantity of between the decay rates (1.15) and (1.13). The difference can be explained by a symmetry property that the one-dimensional linear Goldstein–Kac system possesses. Such kind of symmetry properties is already studied in [13] based on the existence of an invertible matrix commuting with and anti-commuting with the matrix of one-dimensional dissipative linear hyperbolic systems. More general, in several dimensions , the symmetry property is given by Condition . [Symmetry] There is an invertible matrix for such that

 SB=BS,SA=−AS,

where is given by (1.8) for .

###### Remark 1.4.

In practice, it is easy to check the condition if there is a constant invertible matrix satisfying and for all since by definition.

We will show that under the conditions , and , the decay rate in the estimate (1.13) increases. We primarily refine the asymptotic profile .

With the coefficients in (1.9) and in (1.11), we consider the Cauchy problem

 (1.16) {∂tU−div(D∇xU)=0,(x,t)∈Rd×R+,U(x,0)=P(0)0u0(x)+P(1)0⋅∇xu0(x),

where with matrix entries

 (1.17) P(1)0h:=−P(0)0AhQ(0)0−Q(0)0AhP(0)0.
###### Theorem 1.5 (Optimal decay rate).

Under the same hypotheses of Theorem 1.2, if the condition holds in addition, the solution is also decomposed into as in (1.12) such that for any and , one has

 (1.18) ∥u(1)−U∥Lp≤Ct−d2(1q−1p)−1∥u0∥Lq,

where is the solution to (1.16) with the initial data .

The paper is organized as follows. Section 2 is devoted to proofs and examples of Theorem 1.2 and Theorem 1.5, where the proofs are based on the estimates obtained in Section 5. In order to prove these estimates in Section 5, we primarily invoke some useful tools of the Fourier analysis and the perturbation analysis in Section 3. With these tools, we construct the asymptotic expansions of the operator in (1.8) in Section 4 in order to obtain the asymptotic expansions of the fundamental solution to the system (1.1) to be able to prove the estimates in Section 5.

### Notations and Definitions

We introduce here the notations and definitions which will be used frequently through out this paper. See [1, 2] for more details.

###### Definition 1.6.

Let be a function from to a Banach space equipped with norm , we define the Lebesgue spaces for consisting of functions satisfying

 ∥u∥Lp:=(∫Rd|u(x)|pdx)1/p<+∞,1≤p<∞,

and satisfying

 ∥u∥L∞:=esssupRd|u(x)|<+∞.

Let be the multi-index with . One denotes by

 ∂αf:=∂|α|f∂xα11…∂xαdd,

where , the partial derivatives of a smooth function on . Then, for smooths functions and on , we have the Leibniz rule

 ∂α(fg)=∑ν≤α(αν)∂νf∂α−νg,

where

 (αν):=α!ν!(α−ν)!=α1!…αd!ν1!…νd!(α1−ν1)!…(αd−νd)!

is the multi-index binomial coefficient, means that for all and the difference is defined by

 α−ν:=(α1−ν1,…,αd−νd).
###### Definition 1.7.

The Schwartz space is the set of smooth functions on such that for any , we have

 ∥u∥k,S:=sup|α|≤k,x∈Rd(1+|x|)k|∂αu(x)|<+∞.

One denotes by the dual space of and is called a tempered distribution. For , the Fourier transform is defined by

 ^u(k):=∫Rde−ix⋅ku(x)dx,

where is the usual scalar product on , and the inverse Fourier transform of also denoted by is given by

 u(x):=1(2π)d∫Rdeix⋅k^u(k)dk.

On the other hand, we can define the Fourier transform of tempered distributions by the inner product on , namely

 ⟨^u(k),ϕ(k)⟩L2=⟨u(x),^ϕ(x)⟩L2,∀ϕ∈S(Rd).
###### Definition 1.8.

Let , the Sobolev space consists of tempered distributions such that and

 ∥u∥Hs:=(∫Rd(1+|k|2)s|^u(k)|2dk)1/2<+∞.
###### Definition 1.9.

Let , is called a Fourier multiplier on for if the convolution for all and if

 ∥ρ∥Mp:=sup∥ϕ∥Lp=1∥F−1(ρ(k))∗ϕ∥Lp<+∞.

The linear space of all such is denoted by equipped with norm .

## 2. Proofs and Examples of Theorem 1.2 and Theorem 1.5

For , let be in (1.8). Let and be in (1.11). Let be in (1.9) and be in (1.17).

For , consider , the kernel associated with the system (1.1), and , the kernel associated with the system (1.10). Note that

 (2.1) ~Φt∗U0(x)=Φt∗u0(x),

where

 (2.2) Φt(x):=Φ(x,t)=F−1(e−c⋅ikt−k⋅DktP(0)0)∈Rn×n.

Consider also the kernel associated with the system (1.16). One has

 (2.3) ~Ψt∗U0(x)=Ψt∗u0(x),

where

 (2.4) Ψt(x):=Ψ(x,t)=F−1(e−k⋅Dkt(P(0)0+P(1)0⋅ik))∈Rn×n.

We are now able to give the proofs of Theorem 1.2 and Theorem 1.5 by using the estimates which will be proved later in Section 5.

###### Proof of Theorem 1.2.

Let be the solution to (1.1) with the initial data and be the solution to (1.10) with the initial data . One has

 u(x,t)=Γt∗u0(x),U(x,t)=~Φt∗U0(x).

Moreover, by the relation (2.1), one has

 u(x,t)−U(x,t)=(Γt−Φt)∗u0(x),

where is given by (2.2).

On the other hand, we decompose

 u(x,t)=u(1)(x,t)+u(2)(x,t),

where

 u(1)(x,t)=F−1(e−E(ik)tP0(ik)χ1(k))∗u0(x)

and is the remainder, where is the eigenprojection associated with the eigenvalue of in (1.8) converging to as and is a cut-off function with support contained in the ball , valued in , for small .

Therefore, by Proposition 5.6, Proposition 5.8 and Proposition 5.10, for , there is a constant such that we have

 ∥u(1)−U∥Lp ≤∥F−1((^Γt(k)P0(ik)−^Φt(k))χ1(k))∗u0∥Lp +∥F−1(^Φt(k)χ2(k))∗u0∥Lp+∥F−1(^Φt(k)χ3(k))∗u0∥Lp ≤Ct−d2(1q−1p)−12∥u0∥Lp,∀t≥1,

where and is a cut-off function with support contained in , valued in , for large .

Finally, by Proposition 5.6, Proposition 5.7 and Proposition 5.9, one also has

 ∥u(2)∥L2 ≤∥F−1(^Γt(k)(I−P0(k))χ1(k))∗u0∥L2 +∥F−1(^Γt(k)(1−χ1(k)))∗u0∥L2 ≤Ce−ct∥u0∥L2,∀t≥1,

for some constants and . The proof is done. ∎

###### Example 2.1.

Consider the three-dimensional linear Goldstein–Kac system (1.1) where

 Aj=⎛⎜ ⎜ ⎜⎝vj1000vj2000vj3⎞⎟ ⎟ ⎟⎠,B=⎛⎜⎝b+c−c−b−ca+c−a−b−aa+b⎞⎟⎠,

where for and , and the initial data is . Let for , we also consider the system (1.10) where and if , the matrix is given by

Moreover, the initial data is chosen as

 U0=13(u10+u20+u30,u10+u20+u30,u10+u20+u30)T∈R3.

Theorem 1.2 then implies that the solution to the three-dimensional Goldstein–Kac system can be decomposed into such that the difference decays in at the rate with respect to in as for any , where is the solution to the above system (1.10). The formulas of and in fact coincide the formulas obtained by using the graph theory as in Example 3.3 p. 412 in [12].

We give the proof of Theorem 1.5.

###### Proof of Theorem 1.5.

The proof is similar to the proof of Theorem 1.2 where and are substituted by and respectively once considering to be the solution to (1.16). We finish the proof. ∎

###### Example 2.2.

Consider the two-dimensional linearized isentropic Euler equations with damping

 (2.5) {∂tρ+divv=0,∂tv+∇xρ=−v,(x,t)∈R2×R+,

which can be written in the vectorial form

 ∂tu+A1∂x1u+A2∂x2u+Bu=0,

where with the initial data and

 A1=⎛⎜⎝010100000⎞⎟⎠,A2=⎛⎜⎝001000100⎞⎟⎠,B=⎛⎜⎝000010001⎞⎟⎠.

Moreover, the matrix satisfying the condition and the matrix satisfying the condition are given by

 R(w1,w2)=12⎛⎜⎝101−w12w2w1−w2−2w1w2⎞⎟⎠,S=⎛⎜⎝−100010001⎞⎟⎠.

Then, Theorem 1.5 implies that , where has the asymptotic profile, which is the solution to the Cauchy problem

 {∂tU−ΔxU=0,(x,t)∈R2×R+,U(x,0)=U0(x),

where

 U0=(ρ0−∂x1v10−∂x2v20,−∂x1ρ0,−∂x2ρ0)T∈R3.

Moreover, decays in at the optimal rate with respect to in as for any . This result is comparable with [7] since satisfying (2.5) also satisfies the linear damped wave equation

 ⎧⎪⎨⎪⎩∂2ttρ−Δxρ+∂tρ=0,(x,t)∈R2×R+,ρ(x,0)=ρ0(x),∂tρ(x,0)=−∂x1v10(x)−∂x2v20(x).

The proofs of Theorem 2.1 in [7] then implies that decays in at the rate with respect to in as for any , where is the solution to the heat equation

 {∂tϕ−Δxϕ=0,(x,t)∈R2×R+,ϕ(x,0)=ρ0(x)−∂x1v10(x)−∂x2v20(x).
###### Remark 2.3 (Proof of the case of finite speed of propagation).

In the case where has compact support contained in the wave cone for some constant , also by Proposition 5.6 - Proposition 5.10, can be refined by

 u(1)(x,t)=F−1(e−E(ik)tχ1(k))∗u0(x),

where is a cut-off function with support contained in the ball , valued in , for any . The proof is then similar to the above proofs. Moreover, this property holds for the above two examples since they are in fact symmetric hyperbolic systems.

## 3. Useful lemmas

This section is devoted to some useful facts of the Fourier analysis in [1, 2] and the perturbation analysis in [9]. They will be used in Section 4 and Section 5.

### 3.1. Fourier analysis

We introduce here the two well-known inequalities which are the Young inequality and the complex interpolation inequality. On the other hand, we also introduce a powerful Fourier multiplier estimate which is the estimate (3.1) given by Lemma 3.3. The multiplier estimates are very helpful to study the - estimate for .

###### Lemma 3.1 (Young’s inequality).

For satisfying and any and , one has and the inequality

 ∥f∗g∥Lr≤∥f∥Lp∥g∥Lq.
###### Proof.

See the proof of Lemma 1.4 p. 5 in [1]. ∎

###### Lemma 3.2 (Complex interpolation inequality).

Consider a linear operator which continuously maps into for with . Let be such that

then continuously maps into and one has

 ∥T∥L(Lpθ;Lqθ)≤∥T∥L(Lp0;Lq0)∥T∥L(Lp1;Lq1).
###### Proof.

See the proof of Corollary 1.12 p. 12 in [1]. ∎

If