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An application of the method of compensated compactness to a problem in phase transitions. (English) Zbl 0667.35047
Material instabilities in continuous mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 427-463 (1988).
[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 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 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 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.
Reviewer: G.A.Perla-Menzala

MSC:
35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
82B26 Phase transitions (general) in equilibrium statistical mechanics
35B65 Smoothness and regularity of solutions to PDEs