an:04092007
Zbl 0667.35047
Roytburd, V.; Slemrod, M.
An application of the method of compensated compactness to a problem in phase transitions
EN
Material instabilities in continuous mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 427-463 (1988).
1988
a
35L65 35B40 82B26 35B65
system of evolution equations; asymptotic behaviour of the solution; nonlinearity; Lipschitz continuous function; weak-star limits; compensated compactness; conservation laws
[For the entire collection see Zbl 0627.00023.]
The authors consider the system of evolution equations
\[
u_ t+(f(v))_ x=\epsilon u_{xx};\quad v_ t-u_ x=\epsilon v_{xx}
\]
with initial conditions \(u(x,0)=u_ 0(x)\), \(v(x,0)=v_ 0(x)\) and \(- \infty <x<+\infty,\) \(t>0\). They study the asymptotic behaviour of the solution-pair \(\{u(\epsilon),v(\epsilon)\}\) of the above system as \(\epsilon \to 0^+\). A typical nonlinearity they consider is a Lipschitz continuous function f which is non-increasing and constant in some interval (a,b), consequently, the limiting problem \(\epsilon =0\) is not strictly hyperbolic. The authors show that under suitable assumptions on \(u_ 0,v_ 0\) and f, the weak-star limits satisfy the ``inviscid'' \(\epsilon =0\) problem. The main ingredients in their proof are the use of the method of compensated compactness [due to \textit{L. Tartar}, Res. Notes Math. 39, 136-212 (1979; Zbl 0437.35004) and NATO ASI Ser. Ser., C 111, 263-285 (1983; Zbl 0536.35003); and \textit{F. Murat}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 489-507 (1978; Zbl 0399.46022), and ibid. 8, 69-102 (1981; Zbl 0464.46034)] and fundamental ideas of \textit{R. DiPerna} [Commun. Math. Phys. 91, 1-30 (1983; Zbl 0533.76071) and Arch. Ration. Mech. Anal. 82, 27-70 (1983; Zbl 0519.35054)] for the analysis of \(2\times 2\) systems of conservation laws.
G.A.Perla-Menzala
0627.00023; 0437.35004; 0536.35003; 0399.46022; 0464.46034; 0533.76071; 0519.35054