## Hyperbolic-hyperbolic singular limits.(English)Zbl 0629.35079

Roughly speaking, the limit $$\epsilon \to 0^+$$ of the differential equation $$\epsilon M(u)+L(u)=0$$ is called a singular limit if, M has a higher derivative in some variable than does L. Such a singular limit is said to be uniformly well-posed if the solution is bounded independently of $$\epsilon$$ and converges as $$\epsilon$$ tends to 0 to a solution of the reduced equation $$L(u)=0$$. Singular limits of partial differential equations are categorized by the types of the full and reduced equations. For example, $$\epsilon (u_{tt}-c^ 2u_{xx})+Bu_ t+Du_ x-h=0$$ is a hyperbolic-hyperbolic singular limit. In this paper the author shows that the conditions for the well-posedness of the hyperbolic-hyperbolic singular limit $$\epsilon A(u_{tt}-c^ 2\Delta u)+Bu_ t+D^ ju_{xj}-h=0,$$ $$u(x,0)=f(x)$$, $$\epsilon u_ t(0,x)=\epsilon g(x)$$, where c is a scalar, u a vector, and A, B, and $$D^ j$$ are matrices with A and B positive definite and B and $$D^ j$$ symmetric, are $$B>0$$ and $$| D| B^{-1}<| c|$$. Further, the author explores the generalization of these conditions for systems of the form $\epsilon (Au_{tt}-Cu_{xx})+Bu_ t+Du_ x=0$ where A, B, C, and D are constant symmetric matrices, and A, B, and C are positive definite.
Reviewer: E.C.Young

### MSC:

 35L45 Initial value problems for first-order hyperbolic systems 35L15 Initial value problems for second-order hyperbolic equations 35B25 Singular perturbations in context of PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations
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### References:

 [1] Cole J.D., Perturbation Methods in Applied Mathematics (1968) · Zbl 0162.12602 [2] DOI: 10.1007/BF01211598 · Zbl 0655.76041 [3] DOI: 10.1002/cpa.3160350402 · Zbl 0478.76011 [4] DOI: 10.1016/0022-247X(83)90211-1 · Zbl 0493.34005 [5] DOI: 10.1137/0516075 · Zbl 0577.34050 [6] Freidman A., Partial Differential Equations (1976) [7] Geel R., Singular Perturbations of Hyperbolic Type, Mathematical Centre Tracts #98 (1978) · Zbl 0498.35001 [8] Genet J., Lecture Notes in Match 594 pp 201– (1977) [9] DOI: 10.1137/0516090 · Zbl 0612.35007 [10] DOI: 10.1137/0514091 · Zbl 0534.35068 [11] De Jager E.M., Nieuw Arch. Wisk 25 pp 145– (1975) [12] DOI: 10.1007/978-1-4612-5700-4 [13] DOI: 10.1002/cpa.3160340405 · Zbl 0476.76068 [14] Kreiss H.–O., Math. Scand 7 pp 71– (1959) · Zbl 0090.09801 [15] DOI: 10.1002/cpa.3160330310 · Zbl 0439.35043 [16] Lions J. L., Lecture notes in Math 323 (1973) [17] DOI: 10.1007/978-1-4612-1116-7 [18] DOI: 10.1016/0022-0396(85)90107-X · Zbl 0512.76067 [19] DOI: 10.1007/BF01229377 · Zbl 0651.73004 [20] DOI: 10.1007/BF01210792 · Zbl 0612.76082 [21] Schochet S., to appear in Commun. PDE. [22] Schochet S., to appear in J. Diff. Eq. [23] DOI: 10.1007/BF01463396 · Zbl 0639.76054 [24] Teman R., Navier Stokes Equations (1977) [25] L. R. Volevich, Differenital Equations 19 pp 1516– (1983) [26] DOI: 10.1016/0022-0396(71)90003-9 [27] Fattorini H. O., preprint [28] DOI: 10.1002/cpa.3160350503 · Zbl 0478.76091
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