80 research outputs found
Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD
Extending our previous work in the strictly parabolic case, we show that a
linearly unstable Lax-type viscous shock solution of a general quasilinear
hyperbolic--parabolic system of conservation laws possesses a
translation-invariant center stable manifold within which it is nonlinearly
orbitally stable with respect to small perturbations, converging
time-asymptotically to a translate of the unperturbed wave. That is, for a
shock with unstable eigenvalues, we establish conditional stability on a
codimension- manifold of initial data, with sharp rates of decay in all
. For , we recover the result of unconditional stability obtained by
Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case
is to construct an invariant manifold in the absence of parabolic smoothing.Comment: 32p
Convergence of Hill's method for nonselfadjoint operators
By the introduction of a generalized Evans function defined by an appropriate 2- modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrice
The Erpenbeck high frequency instability theorem for ZND detonations
The rigorous study of spectral stability for strong detonations was begun by
J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-D\"oring (ZND)
model, which assumes a finite reaction rate but ignores effects like viscosity
corresponding to second order derivatives, he used a normal mode analysis to
define a stability function V(\tau,\eps) whose zeros in
correspond to multidimensional perturbations of a steady detonation profile
that grow exponentially in time. Later in a remarkable paper [Er3] he provided
strong evidence, by a combination of formal and rigorous arguments, that for
certain classes of steady ZND profiles, unstable zeros of exist for
perturbations of sufficiently large transverse wavenumber \eps, even when the
von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in
the sense defined (nearly twenty years later) by Majda. In spite of a great
deal of later numerical work devoted to computing the zeros of V(\tau,\eps),
the paper \cite{Er3} remains the only work we know of that presents a detailed
and convincing theoretical argument for detecting them.
The analysis in [Er3] points the way toward, but does not constitute, a
mathematical proof that such unstable zeros exist. In this paper we identify
the mathematical issues left unresolved in [Er3] and provide proofs, together
with certain simplifications and extensions, of the main conclusions about
stability and instability of detonations contained in that paper.
The main mathematical problem, and our principal focus here, is to determine
the precise asymptotic behavior as \eps\to \infty of solutions to a linear
system of ODEs in , depending on \eps and a complex frequency as
parameters, with turning points on the half-line
Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions
Under natural spectral stability assumptions motivated by previous
investigations of the associated spectral stability problem, we determine sharp
estimates on the linearized solution operator about a multidimensional
planar periodic wave of a system of conservation laws with viscosity, yielding
linearized stability for all and dimensions and nonlinear stability and
-asymptotic behavior for and . The behavior can in
general be rather complicated, involving both convective (i.e., wave-like) and
diffusive effects
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
In a companion paper, we established nonlinear stability with detailed
diffusive rates of decay of spectrally stable periodic traveling-wave solutions
of reaction diffusion systems under small perturbations consisting of a
nonlocalized modulation plus a localized perturbation. Here, we determine
time-asymptotic behavior under such perturbations, showing that solutions
consist to leading order of a modulation whose parameter evolution is governed
by an associated Whitham averaged equation
Existence and stability of viscoelastic shock profiles
We investigate existence and stability of viscoelastic shock profiles for a
class of planar models including the incompressible shear case studied by
Antman and Malek-Madani. We establish that the resulting equations fall into
the class of symmetrizable hyperbolic--parabolic systems, hence spectral
stability implies linearized and nonlinear stability with sharp rates of decay.
The new contributions are treatment of the compressible case, formulation of a
rigorous nonlinear stability theory, including verification of stability of
small-amplitude Lax shocks, and the systematic incorporation in our
investigations of numerical Evans function computations determining stability
of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure
Shiga Toxin Binding to Glycolipids and Glycans
Background: Immunologically distinct forms of Shiga toxin (Stx1 and Stx2) display different potencies and disease outcomes, likely due to differences in host cell binding. The glycolipid globotriaosylceramide (Gb3) has been reported to be the receptor for both toxins. While there is considerable data to suggest that Gb3 can bind Stx1, binding of Stx2 to Gb3 is variable. Methodology: We used isothermal titration calorimetry (ITC) and enzyme-linked immunosorbent assay (ELISA) to examine binding of Stx1 and Stx2 to various glycans, glycosphingolipids, and glycosphingolipid mixtures in the presence or absence of membrane components, phosphatidylcholine, and cholesterol. We have also assessed the ability of glycolipids mixtures to neutralize Stx-mediated inhibition of protein synthesis in Vero kidney cells. Results: By ITC, Stx1 bound both Pk (the trisaccharide on Gb3) and P (the tetrasaccharide on globotetraosylceramide, Gb4), while Stx2 did not bind to either glycan. Binding to neutral glycolipids individually and in combination was assessed by ELISA. Stx1 bound to glycolipids Gb3 and Gb4, and Gb3 mixed with other neural glycolipids, while Stx2 only bound to Gb3 mixtures. In the presence of phosphatidylcholine and cholesterol, both Stx1 and Stx2 bound well to Gb3 or Gb4 alone or mixed with other neutral glycolipids. Pre-incubation with Gb3 in the presence of phosphatidylcholine and cholesterol neutralized Stx1, but not Stx2 toxicity to Vero cells
Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations (survey)
Analysis and Stochastic
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