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1.
We study the stability and pointwise behavior of perturbed viscous shock waves for a general scalar conservation law with
constant diffusion and dispersion. Along with the usual Lax shocks, such equations are known to admit undercompressive shocks.
We unify the treatment of these two cases by introducing a new wave-tracking method based on “instantaneous projection”, giving
improved estimates even in the Lax case. Another important feature connected with the introduction of dispersion is the treatment
of a non-sectorial operator. An immediate consequence of our pointwise estimates is a simple spectral criterion for stability
in all L
p
norms, p≥ 1 for the Lax case and p > 1 for the undercompressive case.
Our approach extends immediately to the case of certain scalar equations of higher order, and would also appear suitable for
extension to systems.
Accepted May 29, 2000?Published online November 16, 2000 相似文献
2.
Daniel Matthes Ansgar Jüngel Giuseppe Toscani 《Archive for Rational Mechanics and Analysis》2011,199(2):563-596
We study families of convex Sobolev inequalities, which arise as entropy–dissipation relations for certain linear Fokker–Planck
equations. Extending the ideas recently developed by the first two authors, a refinement of the Bakry–émery method is established,
which allows us to prove non-trivial inequalities even in situations where the classical Bakry–émery criterion fails. The
main application of our theory concerns the linearized fast diffusion equation in dimensions d ≧ 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating
convex Sobolev inequalities, and prove that these bounds are sharp on a specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and
Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained
by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model
for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production. 相似文献
3.
Heinrich Freistühler Peter Szmolyan 《Archive for Rational Mechanics and Analysis》2010,195(2):353-373
A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate
frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already
in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural
idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans
function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues
correspond to zeros of an essentially analytic function E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay
on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun
and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions
hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of
the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude
shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable
spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished
that relate the small strength e{\epsilon} of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρ
ω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of
transversality. 相似文献
4.
In this article, we solve in closed form a system of nonlinear differential equations modelling the elastica in space of a thin, flexible, straight rod, loaded by a constant thrust at its free end. Common linearizations of strength of materials are of course not applicable any way, because we analyze great deformations, even if not so large to go off the linear elasticity range. By passing to cylindrical coordinates ρ, θ, z, we earn a more tractable differential system evaluating ρ as elliptic function of polar anomaly θ and also providing z through elliptic integrals of I and III kind. Deformed rod’s centerline is then completely described under both tensile or compressive load. Finally, the planar case comes out as a degeneracy, where the Bernoulli lemniscatic integral appears. 相似文献
5.
Jeffrey Humpherys Gregory Lyng Kevin Zumbrun 《Archive for Rational Mechanics and Analysis》2009,194(3):1029-1079
Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function
computations to examine the spectral stability of shock-wave solutions of the compressible Navier–Stokes equations with ideal
gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with
the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock
profile of the same equations, for which internal energy vanishes at one end state; and (ii) has no unstable (positive real
part) zeros outside a uniform ball |λ| ≦ Λ. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with
the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically.
An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant
γ in [1.2, 3] and ratio ν/μ of heat conduction to viscosity coefficient within [0.2, 5] (γ ≈ 1.4, ν/μ ≈ 1.47 for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the
multi-dimensional case, a direction that we shall pursue in a future work. 相似文献
6.
Peter Hornung 《Archive for Rational Mechanics and Analysis》2011,199(2):353-368
The nonlinear elastic energy of a thin film of thickness h is given by a functional E
h
. Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h
−α
E
h
for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally
of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary
layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the
diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible. 相似文献
7.
A Global Existence Result for the Compressible Navier–Stokes Equations in the Critical L
p
Framework
The present paper is dedicated to the global well-posedness issue for the barotropic compressible Navier–Stokes system in
the whole space
\mathbbRd{\mathbb{R}^d} with d ≧ 2. We aim at extending the work by Danchin (Inventiones Mathematicae 141(3):579–614, 2000) to a critical framework which
is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L
p
-type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class
of large highly oscillating initial velocity fields for which global existence and uniqueness holds true. In passing, we obtain new
estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows. 相似文献
8.
Adrien Blanchet Matteo Bonforte Jean Dolbeault Gabriele Grillo Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2009,191(2):347-385
We consider non-negative solutions of the fast diffusion equation u
t
= Δ
u
m
with m ∈ (0, 1) in the Euclidean space , d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ m
c
= (d − 2)/d, or as t approaches the extinction time when m < m
c
. For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for
t large enough if m ≧ m
c
, or close enough to the extinction time if m < m
c
. Such results are new in the range m ≦ m
c
where previous approaches fail. In the range m
c
< m < 1, we improve on known results. 相似文献
9.
Nicola Costanzino Jeffrey Humpherys Toan Nguyen Kevin Zumbrun 《Archive for Rational Mechanics and Analysis》2009,192(3):537-587
Building on the work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive,
or shock-like, boundary layers of the isentropic compressible Navier–Stokes equations with γ-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our analytical results
include convergence of the Evans function in the shock and large-amplitude limits and stability in the large-amplitude limit,
the first rigorous stability result for other than the nearly constant case, for all . Together with these analytical results, our numerical investigations indicate stability for γ ϵ [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit
(not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara
using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an
additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure.
Inflow boundary layers turn out to have quite delicate stability in both large-displacement (shock) and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun
in order to decide stability. 相似文献
10.
In the present paper, we endeavor to accomplish a diagram, which demarcates the validity ranges for interfacial wave theories
in a two-layer system, to meet the needs of design in ocean engineering. On the basis of the available solutions of periodic
and solitary waves, we propose a guideline as principle to identify the validity regions of the interfacial wave theories
in terms of wave period T, wave height H, upper layer thickness d
1, and lower layer thickness d
2, instead of only one parameter–water depth d as in the water surface wave circumstance. The diagram proposed here happens to be Le Méhauté’s plot for free surface waves
if water depth ratio r = d
1/d
2 approaches to infinity and the upper layer water density ρ
1 to zero. On the contrary, the diagram for water surface waves can be used for two-layer interfacial waves if gravity acceleration
g in it is replaced by the reduced gravity defined in this study under the condition of σ = (ρ
2 − ρ
1)/ρ
2 → 1.0 and r > 1.0. In the end, several figures of the validity ranges for various interfacial wave theories in the two-layer fluid are
given and compared with the results for surface waves.
The project supported by the Knowledge Innovation Project of CAS (KJCX-YW-L02), the National 863 Project of China (2006AA09A103-4),
China National Oil Corporation in Beijing (CNOOC), and the National Natural Science Foundation of China (10672056). 相似文献
11.
Jack X. Xin 《Archive for Rational Mechanics and Analysis》1992,121(3):205-233
We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in R
n, n1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem. 相似文献
12.
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem,
we determine sharp L
p
estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws
with viscosity, yielding linearized L
1 ∩ L
p
→ L
p
stability for all
p \geqq 2{p \geqq 2} and dimensions
d \geqq 1{d \geqq 1} and nonlinear L
1 ∩ H
s
→ L
p
∩ H
s
stability and L
2-asymptotic behavior for
p\geqq 2{p\geqq 2} and
d\geqq 3{d\geqq 3} . The behavior can in general be rather complicated, involving both convective (that is, wave-like) and diffusive effects. 相似文献
13.
The onset of convective instability in an initially quiescent, stably stratified fluid layer between two horizontal plates
is analyzed with linear theory. The bottom boundary is heated suddenly from below, subjected to a step change in surface temperature.
The critical time t
c to mark the onset of Rayleigh-Bénard convection is predicted by propagation theory. This theory uses the length scaled by
, where α denotes thermal diffusivity. Under the normal mode analysis the dimensionless disturbance equations are obtained
as a function of τ(=αt/d
2) and ζ(=Z/), where d is the fluid layer depth and Z is the vertical distance. The resulting equations are transformed to self-similar ones by using scaling and finally fixing
τ as τc under the frame of coordinates τ and ζ. For a given γ, Pr and τc, the minimum value of Ra is obtained from the marginal stability curve. Here γ denotes the temperature ratio to represent the degree of stabilizing
effect, Pr is the Prandtl number and Ra is the Rayleigh number. With γ=0, the minimum Ra value approaches the well-known value of 1708 as τc increases. However, it is inversely proportional to τc
3/2 as τc decreases. With increasing γ, the system becomes more stable. It is interesting that in the present system, propagation theory
produces the stability criteria to bound the available experimental data over the whole domain of time.
Received 5 November 2001 and accepted 29 March 2002 Published online: 2 October 2002
RID="*"
ID="*" This work has been supported by both SK Chemicals Co. Ltd. and LG Chemical Ltd., Seoul under the Brain Korea 21 Project
of the Ministry of Education.
Communicated by H.J.S. Fernando 相似文献
14.
This paper presents an experimental investigation of one-dimensional moving shock waves in vertical soap films. The shock
waves were generated by bursting the films with a perforating spark. Images of propagating shock waves and small disturbances
were recorded using a fast line scan CCD camera. An aureole and a shock wave preceding the rim of the expanding hole were
clearly observed. These images are similar to the x-t diagrams in gas dynamics and give the velocities of shock and sound waves. The moving shock waves cause jumps in thickness.
The variations of the induced Mach number, M2 and the ratio of film thickness across the shock wave, δ
2/δ
1, are plotted versus the shock Mach number, M
s. Both results suggest that soap films are analogous to compressible gases with a specific heat ratio of γ≅1.0.
Published online: 15 October 2002 相似文献
15.
Matteo Bonforte Gabriele Grillo Juan Luis Vázquez 《Archive for Rational Mechanics and Analysis》2010,196(2):631-680
We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ut=D(um/m) = div (um-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for
x ? \mathbb Rd{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular
to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted
here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold
(\mathbb Rd,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard
\mathbb Rd{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated
with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics,
which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear
Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known
exponential decay of such representation for all other values of m. 相似文献
16.
We continue work by the second author and co-workers onsolitary wave solutions of nonlinear beam equations and their stabilityand interaction properties. The equations are partial differentialequations that are fourth-order in space and second-order in time.First, we highlight similarities between the intricate structure ofsolitary wave solutions for two different nonlinearities; apiecewise-linear term versus an exponential approximation to thisnonlinearity which was shown in earlier work to possess remarkablystable solitary waves. Second, we compare two different numericalmethods for solving the time dependent problem. One uses a fixed griddiscretization and the other a moving mesh method. We use these methodsto shed light on the nonlinear dynamics of the solitary waves. Earlywork has reported how even quite complex solitary waves appear stable,and that stable waves appear to interact like solitons. Here we show twofurther effects. The first effect is that large complex waves can, as aresult of roundoff error, spontaneously decompose into two simplerwaves, a process we call fission. The second is the fusion of twostable waves into another plus a small amount of radiation. 相似文献
17.
Doka Yamigno Serge Mibaile Justin Gambo Betchewe Kofane Timoleon Crepin 《Nonlinear dynamics》2017,87(1):13-23
We present the iterative classical point symmetry analysis of a shallow water wave equation in \(2+1\) dimensions and that of its corresponding nonisospectral, two-component Lax pair. A few reductions arise and are identified with celebrate equations in the Physics and Mathematics literature of nonlinear waves. We pay particular attention to the isospectral or nonisospectral nature of the reduced spectral problems. 相似文献
18.
S. Benzoni-Gavage 《Archive for Rational Mechanics and Analysis》1999,150(1):23-55
We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional
van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic,
and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity
criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar
phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy
a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity
on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions.
The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal
structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition
between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions.
As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform
Lopatinsky condition.
Accepted December 14, 1998 相似文献
19.
For elliptic equations ε2Δu − V(x) u + f(u) = 0, x ∈ R
N
, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component
of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal.
An erratum to this article can be found at 相似文献
20.
We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws. 相似文献