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Slow Motion of Internal Shock Layers for the Jin–Xin System in One Space Dimension

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Abstract

This paper considers the slow motion of the shock layer exhibited by the solution to the initial-boundary value problem for a scalar hyperbolic system with relaxation. Such behavior, known as metastable dynamics, is related to the presence of a first small eigenvalue for the linearized operator around an equilibrium state; as a consequence, the time-dependent solution approaches its steady state in an asymptotically exponentially long time interval. In this contest, both rigorous and asymptotic approaches are used to analyze the slow motion of solutions to the Jin–Xin system. To describe this dynamics, we derive an ODE for the position of the internal transition layer, proving that it drifts towards the equilibrium location with a speed rate that is exponentially slow. These analytical results are also validated by numerical computations.

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Acknowledgments

I wish to thank C. Mascia for having introduced me to the problem and for guidance throughout writing the paper.

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Correspondence to Marta Strani.

Appendix

Appendix

In this section, we briefly review some results on the theory of evolution systems by A. Pazy [22, Chapter 5]. For more details and for the proofs of the Theorems, see [22, Theorem 2.3, Theorem 3.1, Theorem4.2].

Let \(X\) be a Banach space. For every \(0 \le t \le T\), let \(A(t): D(A(t)) \subset X \rightarrow X\) be a linear operator in \(X\) and let \(f(t)\) be an \(X\) valued function. Let us consider the initial value problem

$$\begin{aligned} \partial _t u = A(t) u+ f(t), \quad u(s)=u_0, \qquad 0 \le s \le t \le T. \end{aligned}$$
(6.1)

In the special case where \(A(t)=A\) is independent of \(t\), the solution to (6.1) can be represented via the formula of variations of constants

$$\begin{aligned} u(t)= T(t)u_0+ \int _0^t T(t-s)f(s) \ ds \end{aligned}$$

where \(T(t)\) is the \(C_0\) semigroup generated by \(A\). In [22] it is shown that a similar representation formula is true also when \(A(t)\) depends on time.

Definition 6.1

Let \(X\) a Banach space. A family \(\{ A(t)\}_{t \in [0,T]}\) of infinitesimal generators of \(C_0\) semigroups on \(X\) is called stable if there are constants \(M \ge 1\) and \(\omega \) (called the stability constants) such that

$$\begin{aligned} (\omega ,+\infty ) \subset \rho (A(t)) \quad \hbox {for} \quad t \in [0,T] \end{aligned}$$

and

$$\begin{aligned} \left\| \Pi _{j=1}^k R(\lambda : A(t_j))\right\| \le M(\lambda -\omega )^{-k}, \end{aligned}$$

for \(\lambda >\omega \) and for every finite sequence \(0 \le t_1 \le t_2,\ldots , t_k \le T\), \(k=1,2,\ldots \).

Remark 6.2

If, for \(t \in [0,T]\), \(A(t)\) is the infinitesimal generator of a \(C_0\) semigroup \(S_t(s)\), \(s \ge 0\), satisfying \(\Vert S_t(s)\Vert \le e^{\omega s}\), then the family \(\{ A(t)\}_{t \in [0,T]}\) is clearly stable with constants \(M=1\) and \(\omega \).

The previous remark states that, if for every fixed \(t \in [0,T]\) the operator \(A(t)\) generates a \(C_0\) semigroup \(S_t(s)\), and we can find an estimate for \(\Vert S_t(s)\Vert \) that is independent of \(t\), then the whole family \(\{ A(t)\}_{t \in [0,T]}\) is stable in the sense of Definition 6.1.

Theorem 6.3

Let \(\{ A(t)\}_{t \in [0,T]}\) be a stable family of infinitesimal generators with stability constants \(M\) and \(\omega \). Let \(B(t)\), \(0 \le t \le T\) be a bounded linear operators on \(X\). If \(\Vert B(t)\Vert \le K\) for all \(t \le T\), then \(\{ A(t)+ B(t)\}_{t \in [0,T]}\) is a stable family of infinitesimal generators with stability constants \(M\) and \(\omega + MK\).

In order to prove the existence of the so called evolution system \(U(t,s)\) for the initial value problem (6.1), let us introduce two Banach spaces \(X\) and \(Y\), with norms \(\Vert \ \Vert _X\), \(\Vert \ \Vert _Y\) respectively. Moreover, let us assume that \(Y\) is a dense subspace of \(X\) and that there exists a constant \(C\) such that \(\Vert w \Vert _X \le C \Vert w\Vert _Y\) for all \(w \in Y\).

Definition 6.4

Let \(A\) be the infinitesimal generator of a \(C_0\) semigroup \(S(s)\), \(s \ge 0\), on \(X\). \(Y\) is called \(A\)-admissible if it is an invariant subspace of \(S(s)\), and the restriction \(\tilde{S}(s)\) of \(S(s)\) to \(Y\) is a \(C_0\) semigroup on \(Y\). Moreover, the infinitesimal generator of the semigroup \(\tilde{S}(s)\) on \(Y\), denoted here with \(\tilde{A}\), is called the part of \(A\) in \(Y\).

Next, let us fix \(t \in [0,T]\), and let \(A(t)\) be the infinitesimal generator of a \(C_0\) semigroup \(S_t(s)\) on \(X\). The following assumptions are made

(H1):

\(\{ A(t)\}_{t \in [0,T]}\) is a stable family with stability constants \(M\) and \(\omega \).

(H2):

Y is \(A(t)\)-admissible for \(t\in [0,T]\) and the family \(\{ \tilde{A}(t)\}_{t \in [0,T]}\) is a stable family in \(Y\) with stability constants \(\tilde{M}\), \(\tilde{\omega }\).

(H3):

For \(t \in [0,T]\), \(Y \subset D(A(t))\), \(A(t)\) is a bounded operator from \(Y\) into \(X\) and \(t \rightarrow A(t)\) in continuous in the \(B(X,Y)\) norm.

Remark 6.5

The assumption that the family \(\{ A(t)\}_{t \in [0,T]}\) satisfies (H2) is not always easy to check. A sufficient condition for (H2) which can be effectively checked in many applications states that (H2) holds if there is a family \(\{ Q(t)\}\) of isomorphisms of \(Y\) onto \(X\) such that \(\Vert Q(t) \Vert _{Y \rightarrow X}\) and \(\Vert Q(t)^{-1} \Vert _{Y \rightarrow X}\) are uniformly bounded and \(t \rightarrow Q(t)\) is of bounded variation in the \(B(Y,X)\) norm (for more details, see [22, Chapter 5]).

Remark 6.6

Condition (H3) can be replaced by the weaker condition

(H3)\(^{\prime }\) :

For \(t\in [0,T]\), \(Y \subset D(A(t))\) and \(A(t) \in L^1([0,T];B(Y,X))\).

Theorem 6.7

Let \(A(t)\), \(0 \le t \le T\) be the infinitesimal generator of a \(C_0\) semigroup \(S_t(s)\), \(s \ge 0\) on \(X\). If the family \(\{ A(t)\}_{t \in [0,T]}\) satisfies the conditions (H1)–(H3), then there exists a unique evolution system \(U(t,s)\), \(0 \le s \le t \le T\), in \(X\) satisfying

$$\begin{aligned} \Vert U(t,s) \Vert \le Me^{\omega (t-s)}, \quad \mathrm{for } \quad 0 \le s \le t \le T. \end{aligned}$$
(6.2)

Moreover, if \(f \in C([s,T],X)\), the solution to (6.1) can be written as

$$\begin{aligned} u(t)= U(t,s)u_0 + \int _s^t U(t,r) f(r) \ dr, \end{aligned}$$
(6.3)

for all \(0 \le s \le t \le T\).

One special case where the conditions of Theorem 6.7 can be easily checked is the case of an operator whose domain is independent on \(t\), i.e. \(D(A(t)) \equiv D\). In this case we can take \(D\) as the Banach space denoted by \(Y\), and the following Theorem holds.

Theorem 6.8

Let \(\{ A(t)\}_{t \in [0,T]}\) be a stable family of infinitesimal generators of \(C_0\) semigroups on \(X\). If \(D(A(t))=D\) is independent on \(t\) and for \(u_0 \in D\), \(A(t)u_0\) is continuously differentiable in \(X\), then there exists a unique evolution system \(U(t,s)\), \(0 \le s \le t \le T\), satisfying (6.2). Morevoer, if \(f \in C([s,T],X)\), then, for every \(u_0 \in D\), the initial value problem (6.1) has a unique solution given by (6.3).

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Strani, M. Slow Motion of Internal Shock Layers for the Jin–Xin System in One Space Dimension. J Dyn Diff Equat 27, 1–27 (2015). https://doi.org/10.1007/s10884-014-9418-6

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