# Asymptotic Behavior of Global Entropy Solutions for Nonstrictly Hyperbolic Systems with Linear Damping

Authors: Richard  Alexander De la Cruz Guerrero; Juan Carlos Juajibioy Otero; Leonardo Rendon
Journal: International Journal of Differential Equations
 (2014)

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## Abstract

We investigate the large time behavior of the global weak entropy solutions to the symmetric Keyfitz-Kranzer system with linear damping. It is proved that ast→∞ the entropy solutions tend to zero in the Lp norm.


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## Body

## 1. Introduction

In this paper, we consider the Cauchy problem to the symmetric system of Keyfitz-Kranzer type with linear damping(1)ut+uϕrx+au=0,vt+vϕrx+bv=0,
with initial data
(2)ux,0=u0x,vx,0=u0x.
This system models the propagation models of propagation of forward longitudinal and transverse waves of elastic string which moves in a plane; see [1, 2]. General source term for the system (1) was considered in [3]. The damping term in the system (1) represents external forces proportional to velocity, and this term can produce loss of total energy of system. Consider the scalar case; for example,
(3)ut+aux+bu=0,u(x,0)=u0(x).
From the integral representation of (3), it is easy to find the following solution:
(4)u(x,t)=u0(x-at)e-bt.
In this case, the solution of (3) tends to zero when t→∞. In Figure 1, we show the graph of solution for the advection equation with initial data
(5)u(x,0)=1-x2,ifx∈-1,1,0,otherwise.
For more general case, the behavior of solutions and its computation can be more complicated; for example, we consider Burger’s equation with a particular initial data and linear damping; this equation models the component of the velocity in one-dimensional flow
(6)ut+12u2x+u=0
with intial data
(7)ux,0=A1-x2,ifx∈-1,1,0,otherwise,
where A is a constant. By an application to the characteristics method, we have that solutions emanating from x0 are given by
(8)X(t,x0)=x0+A(1-x02)(1-e-t),u(X,t)=u0(x)e-t.
In general, in systems with source term, the characteristics are nonlinear functions and they could have asymptotic behavior; for example, the characteristics solutions for (6), (7) are asymptotic to the lines
(9)lim⁡t→∞X(t,x0)=x0+A(1-x02).
It is easy to see that the shock occurs when x0+A(1-x0)2>1. In Figures 2, 3, and 4, we show the characteristics solutions for several values of A.Figure 1
Graph of solutionut+ux+u=0, A=1.Figure 2
Characteristics foru=0, A=1.Figure 3
Linear dampingA=1/2.Figure 4
Linear dampingA=2.We are looking for conditions under which the termsa, b have a dissipative effect in the solutions of (1).Letr(x,t)=u(x,t)2+v(x,t)2, and we are going to show the following main theorem.Theorem 1.
If the initial data(u0(x),v0(x))∈L∞(R)∩L2(R), then the Cauchy problems (1) and (2) have a weak entropy solutions satisfying
(10)uL∞(Ω)+vL∞(Ω)<M.
Moreover, r(u,v) converges to zero in Lp with exponential time decay; that is,
(11)rx,tLp(R)≤Ke-Mtrx,0LpR.


## 2. Preliminaries

We start with some preliminaries about the general systems of conservation laws; see [4] chapter 5. Let f:Ω→Rn be a smooth vector field. Consider Cauchy problem for the system
(12)ut+fux=g(u),u(x,0)=u0(x).
When g(u)=0, the system (12) is called homogeneous system of conservation laws, if g(u)≠0, the system (12) is called inhomogeneous system or balance system of conservation laws. We will work also with the parabolic perturbation to the system (12); namely,
(13)ut+fux=ϵuxx+g(u),u(x,0)=u0(x).
Denote by A(u)=Df(u) the Jacobian matrix of partial derivatives of f.Definition 2.
The system (12) is strictly hyperbolic if, for every u∈Ω, the matrix A(u) has n real distinct eigenvalues λ1(u)<⋯<λn(u).Letri(u) be the corresponding eigenvector to λi(u). Then, one can see the following.Definition 3.
One says that theith characteristic field is genuinely nonlinear if
(14)∇λi(u)·ri(0)≠0.
If instead
(15)∇λi(u)·ri(0)=0,
we say that the ith characteristic field is linearly degenerate.For the following definitions, see [5, 6].Definition 4.
Ak-Riemann invariant is a smooth function wk:Rn→R, such that
(16)∇wk(u)·rk(u)=0.Definition 5.
A pair of functionη,q:Rn→R is called a entropy-entropy flux pair if it satisfies
(17)∇ηuAu=∇qu,
if η(u) is a convex function, then the pair (η,q) is called convex entropy-entropy flux pair.Definition 6.
A bounded measurable functionu(x,t) is an entropy (or admissible) solution for the Cauchy problem (12) if it satisfies the following inequality:
(18)ηut+qux+∇η(u)g(u)≤0,
in the distributional sense, where (η,q) is any convex entropy-entropy flux pair.We consider the general system of Keyfitz-Kranzer type(19)ut+uϕu,vx=0,vt+vϕu,ux=0,
to get some general observations about this type of system. Let F(u,v)=(uϕu,v,vϕu,v) in (19), and we have that the eigenvalues and eigenvector of the Jacobian matrix Df are given by
(20)λ1u,v=ϕu,v,r1=1,-ϕuϕv,(21)λ2u,v=ϕu,v+u,v·∇ϕu,v,r2=1,vu.
From (20) and (21), we have that ∇ϕ·r1=0 and ∇Z(u,v)·r2=0, where Z(u,v)=u/v, and then the Riemann invariants are given by
(22)W(u,v)=ϕ(r),Z(u,v)=uv.Lemma 7.
The system (1) is always linear degenerate in the first characteristic field. If
(23)(u,v)∇ϕ(u,v)≠0,
then the system (1) is strictly hyperbolic and nonlinear degenerate in the second characteristic field. Moreover,
(24)∇λ2u,v·r2=2u,v∇ϕu,v+u,vHϕu,vTu,
where H represents the Hessian matrix.Lemma 8.
Letη(u,v)∈C1(R+) be a Lipschitz function in a neighborhood of the origin, and let q(u,v)=ψ(u,v)+η(u,v)ϕ(u,v) be a function, such that ψ satisfies
(25)∇ψu,v=u,v·∇ηu,v-ηu,v∇ϕ(u,v).
Then, the pair
(26)nu,v,qu,v
is a entropy-entropy flux pair for the system (1). Moreover, if η(u,v) is a convex function, then the pair (26) is a convex entropy-entropy flux pair.


## 3. Global Existence of Weak Entropy Solutions and Asymptotic Behavior

We consider the parabolic regularization of the system (1). Namely,
(27)ut+uϕrx+au=ϵuxx,vt+vϕrx+bv=ϵvxx,
with initial data
(28)uϵ(x,0)=u0ϵ*jϵ,vϵ(x,0)=v0ϵ*jϵ,
where jϵ is a mollifier. In this case, ϕ(u,v)=ϕ(r), with r=u2+v2. By (20), the eigenvectors and eigenvalues are given by
(29)λ1u,v=ϕr,r1=1,-uv,λ2u,v=ϕr+rϕ′r,r2=1,vu.
The following conditions will be necessary in our next discussion:(C1)
rϕ(r)→0, as r→0, rϕ′(r)≠0;(C2)
a>b. The condition (C1) guarantees the strict hyperbolicity to the system (27) according to Lemma 7, while condition (C2) ensures the existence of a positive invariant region. Now, we consider the following subset of R:
(30)Σ=u,v:ϕr≤C0,0<C1≤uv≤C2.
We affirm that Σ is an invariant region. Let h(u,v)=(au,bv), if (u¯,v¯)∈γ1, where γ1 is the level curve of W=ϕ(r). We have that
(31)∇W·hu¯,v¯=a+brϕ′r>0,
and if u¯,v¯∈γ2, where γ2 is the level curve of Z=u/v, we have that
(32)∇Z·hu¯,v¯=a-buv>0.
By Theorem 14.7 of [5], Σ is an invariant region for the system (27). It is easy to verify that (au,bv) satisfies the condition H1⋯H5 in [3]; thus, we have the following Lemma.Proposition 9.
If(u0,v0)∈Σ and the C condition holds, then the Cauchy problems (27), (28) have a global weak entropy solution.Now, for the global behavior of solutions, using ideas of the author in [7], we construct the following entropy-entropy flux pairs as follows:
(33)n(r)=rm,m≥2.
From (25), we have
(34)qr=m-1∫0r‍smϕ′sds+rmϕr.
Integrating by parts, we have that
(35)q(r)=mϕ(r)-mm-1∫0r‍sm-1ϕsds.
Let M=sup⁡(u,v)∈[0,∥u∥L∞]×[0,∥v∥L∞]{ϕ(r)}. Then, we have that
(36)|q(r)|≤2mMrm.
Multiplying (1) by ∇η, we have that
(37)ηrt+qrx≤-3mMrm.
Now, we choose h(x)∈C2(R) as a function, such as h′x≤1, h′′x≤1, and h(x)=x for x≥1 and set k(x)=e-h(x). Then, k′(x)≤k(x). Multiplying by k(x) in (37) and integrating over x, we have
(38)ddt∫-∞∞‍ηrgx≤∫-∞∞‍qrk′x+-3mM∫-∞∞‍rmdx.
By the inequality (36), we have
(39)ddt∫-∞∞‍η(r)k(x)dx≤-mM∫-∞∞‍rmk(x)dx.
If ψ(t)=∫-∞∞‍η(r)k(x)dx, we have
(40)ddtψt≤-mMψt.
By Gronwall’s inequality, we have
(41)ψ(t)≤e-mMtψ(0).
Thus, we have
(42)∫-∞∞‍rmtkxdx1/m≤e-Mt∫-∞∞‍rm0kxdx1/m.
Passing to limit m→∞, in (42), we have the inequality (10).



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