Asymptotic stability of viscous shock wave profiles for viscous conservation laws

This paper is concerned with the asymptotic stability of travelling wave solution to the two-dimensional steady isentropic irrotational flow with artificial viscosity. We prove that there exists a unique travelling wave solution up to a shift to the system if the end states satisfy both the Rankine–Hugoniot condition and Lax's shock condition, and that the travelling wave solution is stable if the initial disturbance is small.     

Nonlinear instability of Hele-Shaw flows with smooth viscous profiles

We rigorously derive nonlinear instability of Hele-Shaw flows moving with a constant velocity in the presence of smooth viscosity profiles where the viscosity upstream is lower than the viscosity downstream. This is a single-layer problem without any material interface. The instability of the basic flow is driven by a viscosity gradient as opposed to conventional interfacial Saffman-Taylor instability where the instability is driven by a viscosity jump across the interface. Existing analytical techniques are used in this paper to establish nonlinear instability.     

Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems

Extending to systems of hyperbolic-parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in Lp, 1?p?∞.     

The geometric structures and instability of entropic dynamical models

In this paper, we characterize two entropic dynamical (ED) models from the viewpoint of information geometry and give the geometric structures of the associated statistical manifolds of the models. The scalar curvatures and the geodesics are obtained. Also the instability of entropic dynamical models is studied from the behavior of the geodesics lengths, statistical volume elements and Jacobi vector fields.     

Nonlinear instability for nonhomogeneous incompressible viscous fluids

We investigate the nonlinear instability of a smooth steady density profile solution to the three-dimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions to the linearized problem that grow in time in the Sobolev space H ^k , thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, which is different from the previous known results.     

Stopping criteria for the Ando-Li-Mathias and Bini-Meini-Poloni geometric means

We provide an upper bound for the number of iterations necessary to achieve a desired level of accuracy for the Ando-Li-Mathias [Linear Algebra Appl. 385 (2004) 305-334] and Bini-Meini-Poloni [Math. Comput. 79 (2010) 437-452] symmetrization procedures for computing the geometric mean of n positive definite matrices, where accuracy is measured by the spectral norm and the Thompson metric on the convex cone of positive definite matrices. It is shown that the upper bound for the number of iterations depends only on the diameter of the set of n matrices and the desired convergence tolerance. A striking result is that the upper bound decreases as n increases on any bounded region of positive definite matrices.     

Asymptotic stability and instability criteria for nonautonomous systems

The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if V^{(m + 1)} \mathop {V}\limits^{(m + {1})} (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that [(V)\dot] \dot{V} (t, x) ≤ −c(∥x∥) can be weakened and replaced by that [(V)\dot] \dot{V} (t, x) ≤ 0 and − (−[(V)\dot] \dot{V} (tk, x)| + − [(V)\ddot] \ddot{V} (tk, x)| + ⋯ + − V^(m) \mathop {V}\limits^{(m)} (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]     

The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = (?1, 1), $$\frac{{\Lambda _U (z)}} {{\cos \tfrac{{U(z)\pi }} {2}}} \leqslant \frac{4} {\pi }\frac{1} {{1 - \left| z \right|^2 }}$$ holds for z ∈ D, where Λ _U (z) is the maximum dilation of U at z. The inequality is sharp for any z ∈ D and any value of U(z), and the equality occurs for some point in D if and only if $U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}$ , z ∈ D, with a Möbius transformation φ of D onto itself.     

Local tracking and stability for degenerate viscous shock waves

We study the pointwise behavior of perturbed degenerate (sonic) shock waves for scalar conservation laws with nonconstant diffusion. Building on the pointwise Green's function approach of Zumbrun and Howard (Indiana U. Math. J. 47(3) (1998) 741) we extend the linear analysis to an equation with nonintegrable coefficients. In lieu of working with the integrated equation, we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems (Indiana U. Math. J. 47(3) (1998) 741).     

Pointwise decaying rate of large perturbation around viscous shock for scalar viscous conservation law

In this paper, we consider the large perturbation around the viscous shock of the scalar conservation law with viscosity in one dimension case. We divide the time region into t ? T ₀ and t > T ₀ for a fixed constant T ₀ when applying energy method. Since T ₀ is fixed, the case t ? T ₀ is easy to deal with and when t > T ₀, from the decaying property of the solution, there is a priori estimate for the solution. Thus we can succeed to control the nonlinear term and get the pointwise estimate for the perturbation by the weighted energy method.     

Stability and instability criteria for Kaplan–Yorke solutions

We derive sufficient conditions for the stability and instability of periodic solutions p:\mathbbR ? \mathbbRp:\mathbb{R} \to \mathbb{R} of Kaplan–Yorke type to the equation $\ifmmode\expandafter\dot\else\expandafter\.\fi{x}(t) = \alpha f(x(t),x(t - 1)),$\ifmmode\expandafter\dot\else\expandafter\.\fi{x}(t) = \alpha f(x(t),x(t - 1)), where f is even in the first and odd in the second argument. The criteria are based on the monotonicity of the coefficient in a transformed version of the variational equation. For the special case of cubic f, we show that this monotonicity property is satisfied if and only if the set { (p(t),p(t - 1))|t ? \mathbbR} ì \mathbbR² \{ (p(t),p(t - 1))|t \in \mathbb{R}\} \subset \mathbb{R}^{2} is contained in a region E defined by a quadratic form (bounded by an an ellipse or a hyperbola). The coefficients of this quadratic form are expressible in terms of the Taylor coefficients of f. Further, the parameter α in the equation and the amplitude z of the periodic solution are related by an elliptic integral. Using the relation between this integral and the arithmeticgeometric mean, we obtain upper and lower estimates on this relation, and on the inverse function. Combining these estimates with the inequality that defines the region E, we obtain stability criteria explicit in terms of the Taylor coefficients of f. These criteria go well beyond local stability analysis, as examples show.     

Pointwise convergence to shock waves for viscous conservation laws

We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal L_p convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞. The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere. © 1997 John Wiley & Sons, Inc.