Asymptotic behaviour of solutions of hyperbolic conservation laws ut + (um)x = 0 as m → ∞ with inconsistent initial values

SynopsisWe study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

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A singular perturbation problem for the Lavrent'ev-Bitsadze equation

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028066 ◽

1983 ◽

Vol 26 (1) ◽

pp. 49-66

Author(s):

R. J. Weinacht

Keyword(s):

Singular Perturbation ◽

Mixed Type ◽

Singular Integral Equations ◽

Singular Perturbation Problem ◽

Equations Of Mixed Type ◽

Perturbation Problem ◽

Small Positive Parameter ◽

Carry Over ◽

Image Position ◽

Asymptotic Validity

In this note we consider a singular perturbation problem for the equationwhere K(y) = sgn y and. Ε is a small (positive) parameter. This equation for ε≠O is elliptic for y<0 and hyperbolic for y>0. Many of the results carry over to more difficult and interesting problems for equations of mixed type. The particularly simple model treated here permits the elimination of some complications in the analysis involving singular integral equations while preserving the main qualitative features of more general cases. For a special Tricomi-like problem for (1.1) we construct asymptotic expansions in ε, including boundary layer corrections, of the solution. A proof of uniform asymptotic validity of the lowest order terms is given.

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A singular perturbation problem for nonlinear damped hyperbolic equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500024987 ◽

1989 ◽

Vol 111 (1-2) ◽

pp. 21-31 ◽

Cited By ~ 2

Author(s):

Piotr Biler

Keyword(s):

Singular Perturbation ◽

Asymptotic Behaviour ◽

Positive Operator ◽

Hyperbolic Equations ◽

Strong Solutions ◽

Singular Perturbation Problem ◽

Nonlinear Hyperbolic Equations ◽

Convex Functional ◽

Perturbation Problem ◽

Asymptotic Behaviour Of Solutions

SynopsisWe consider damped nonlinear hyperbolic equations utt + Aut + αAu + βA2u + G(u) = 0, where A is a positive operator and G is the Gateaux derivative of a convex functional. We examine the asymptotic behaviour of solutions and the convergence of strong solutions to these equations when the parameter β tends to zero.

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The Schrödinger equation as a singular perturbation problem

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500010982 ◽

1978 ◽

Vol 82 (1-2) ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

E. M. de Jager ◽

T. Küpper

Keyword(s):

Schrödinger Equation ◽

Singular Perturbation ◽

Schrodinger Equation ◽

Eigenvalue Problems ◽

Singular Perturbation Problem ◽

Perturbation Problem ◽

First Approximation ◽

Image Position

SynopsisComparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problemsandwith φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.

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Cauchy problem for hyperbolic conservation laws with a relaxation term

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023088 ◽

1996 ◽

Vol 126 (4) ◽

pp. 821-828 ◽

Cited By ~ 6

Author(s):

Christian Klingenberg ◽

Yun-guang Lu

Keyword(s):

Cauchy Problem ◽

Conservation Laws ◽

Hyperbolic Conservation Laws ◽

Original System ◽

Young Measures ◽

Weak Continuity ◽

Hyperbolic Conservation ◽

Relaxation Term ◽

The Cauchy Problem ◽

Image Position

This paper considers the Cauchy problem for hyperbolic conservation laws arising in chromatography:with bounded measurable initial data, where the relaxation term g(δ, u, v) converges to zero as the parameter δ > 0 tends to zero. We show that a solution of the equilibrium equationis given by the limit of the solutions of the viscous approximationof the original system as the dissipation ε and the relaxation δ go to zero related by δ = O(ε). The proof of convergence is obtained by a simplified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.

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