Hyperbolic-hyperbolic singular limits. (English) Zbl 0629.35079

Schochet, Steven

Commun. Partial Differ. Equations 12, 589-632 (1987).

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

Keywords:

singular limit; uniformly well-posed; reduced equation; hyperbolic- hyperbolic singular limit

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