Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

Lin, Xiao-Biao

doi:10.1007/s10884-005-9001-2

Cited by 5 publications

(4 citation statements)

References 39 publications

(53 reference statements)

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“…We will also show that non-zero eigenvalues are the zeros of the determinant of the "SLEP" matrix as defined in [14,10].…”

Section: Eigenvalues and Resonance Valuesmentioning

confidence: 87%

“…Moreover, this condition is equivalent to that s is the root of the SLEP function (determinant of the corresponding SLEP matrix [15]) as defined in [10].…”

Section: Moreover X → H(x S) Is a Continuous Function From Rmentioning

confidence: 99%

“…In contrast to the "fast eigenvalues" of (1.6) which are of O( 1 ) and come from the dynamics of the singular layers near the shocks of the conservation laws, the critical eigenvalues are so called "slow eigenvalues" in [14,16,10] which are of O(1) and are the eigenvalues of the inviscid system (1.7).…”

Section: Introductionmentioning

confidence: 99%

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L² semigroup and linear stability for Riemann solutions of conservation laws

Lin

2005

Dynamics of Partial Differential Equations

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Abstract. Riemann solutions for the systems of conservation laws uτ +f (u) ξ = 0 are self-similar solutions of the form u = u(ξ/τ ). Using the change of variables x = ξ/τ, t = ln(τ ), Riemann solutions become stationary to the system ut + (Df (u) − xI)ux = 0. For the linear variational system around the Riemann solution with n-Lax shocks, we introduce a semigroup in the Hilbert space with weighted L 2 norm. We show that (A) the region λ > −η consists of normal points only. (B) Eigenvalues of the linear system correspond to zeros of the determinant of a transcendental matrix. They lie on vertical lines in the complex plane. There can be resonance values where the response of the system to forcing terms can be arbitrarily large, see Definition 6.2. Resonance values also lie on vertical lines in the complex plane. (C) Solutions of the linear system are O(e γt ) for any constant γ that is greater than the largest real parts of the eigenvalues and the coordinates of resonance lines. This work can be applied to the linear and nonlinear stability of Riemann solutions of conservation laws and the stability of nearby solutions of the Dafermos regularizations ut + (Df (u) − xI)ux = uxx.

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“…We will also show that non-zero eigenvalues are the zeros of the determinant of the "SLEP" matrix as defined in [14,10].…”

Section: Eigenvalues and Resonance Valuesmentioning

confidence: 87%

“…Moreover, this condition is equivalent to that s is the root of the SLEP function (determinant of the corresponding SLEP matrix [15]) as defined in [10].…”

Section: Moreover X → H(x S) Is a Continuous Function From Rmentioning

confidence: 99%

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L² semigroup and linear stability for Riemann solutions of conservation laws

Lin

2005

Dynamics of Partial Differential Equations

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“…The method of factorization can be used to convert a second order singularly perturbed equation to a first order system of which the fast and slow variables naturally split [4,16,25,37].…”

Section: Asymptotic Factorization and The Riccati Equationmentioning

confidence: 99%