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1.
We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade. © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. 相似文献
2.
M.I. Isaev 《Applicable analysis》2013,92(11):2262-2274
We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceding stability results on inverse problems of such a type. 相似文献
3.
This paper is concerned with the mathematical analysis of a time-dependent fluid-solid interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above a local rough surface. We reformulate the unbounded scattering problem into an equivalent initial-boundary value problem defined in a bounded domain by proposing a transparent boundary condition(TBC) on a hemisphere. Analyzing the reduced problem with the Lax-Milgram lemma and the abstract inversion theorem of the Laplace transform,we prove the well-posedness and stability for the reduced problem. Moreover, an a priori estimate is established directly in the time domain for the acoustic wave and elastic displacement by using the energy method. 相似文献
4.
We revisit the old problem of finding the stability and instability intervals of a second-order elliptic equation on the real line with periodic coefficients (Hill's equation). It is well known that the stability intervals correspond to the spectrum of the Bloch family of operators defined on a single period. Here we propose a characterization of the instability intervals. We introduce a new family of non-self-adjoint operators, formally equivalent to the Bloch ones but with an imaginary Bloch parameter, that we call exponential. We prove that they admit a countable infinite number of eigenvalues which, when they are real, completely characterize the intervals of instability of Hill's equation. 相似文献
5.
We analyse the problem of finding instability thresholds and global non‐linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non‐uniform temperature field in steady state. This problem was previously analysed by Georgescu and Mansutti (Int. J. Non‐Linear Mech. 1999; 34 :603–613). In our work we determine the linear instability threshold to be well above the global stability boundary found by an energy method. Since the perturbed system is not symmetric we expect this to be the case, and our analysis yields a parameter region where possible sub‐critical instabilities may be found. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
6.
Yu Chen 《纯数学与应用数学通讯》2008,61(5):627-638
Hadamard‐type instability has been known for over a century as a cause of ill‐posedness of the Cauchy problem for elliptic PDEs. This ill‐posedness manifests itself as evanescent modes growing exponentially when propagated in the reverse direction. Since every oscillating mode of the Laplace equation is evanescent, the ill‐posedness of its Cauchy problem is solely due to Hadamard‐type instability. The presence of the propagating modes and beams for the Helmholtz equation gives rise to an entirely different type of ill‐posedness, hitherto unknown to the practice, and untreated by the theory, of inverse scattering. We will present this fundamental phenomenon of ill‐posedness for the Helmholtz equation. © 2007 Wiley Periodicals, Inc. 相似文献
7.
Yue Zhao 《Applicable analysis》2013,92(4):682-692
This paper concerns the stability on the inverse source scattering problem for the one-dimensional Helmholtz equation in a two-layered medium. We show that the increasing stability can be achieved using multi-frequency wave field at the two end points of the interval which contains the compact support of the source function. 相似文献
8.
P.N. Svirkunov 《Journal of Applied Mathematics and Mechanics》1998,62(6):921-925
The problem of the symmetric instability of the steady-state motions of an incompressible ideal liquid which is stratified with respect to its density is investigated in the case of two types of motion, axially symmetric and with translational symmetry. It is shown that the sufficient condition for stability obtained in [1] using a variational method (the direct Lyapunov method) for the motions under consideration is closely related to the extremal nature of their energy; stable motions are characterized by a conditional minimum of the energy. A minimum of the energy holds in the class of states for which a potential vortex, expressed in terms of the Lagrangian invariants, angular momentum and density, is represented by the same function of these invariants. Conditions for instability are formulated and estimates of the increase in the kinetic energy of perturbations are given. 相似文献
9.
Stability of the determination of the surface impedance of an obstacle from the scattering amplitude
Mourad Bellassoued Mourad Choulli Aymen Jbalia 《Mathematical Methods in the Applied Sciences》2013,36(18):2429-2448
We prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof, which is essentially based on an elliptic Carleman inequality. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
10.
We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. 相似文献
11.
A simple factorization of the finite-dimensional Galerkin operators motivates a study of the numerical stability of a Galerkin
procedure on the basis of its “potential stability” and the “conditioning” of its coordinate functions. Conditions sufficient
for stability and conditions leading to instability are thereby identified. Numerical examples of stability and instability
occurring in the application of the Galerkin method to boundary-integral equations arising in simple scattering problems are
provided and discussed within this framework. Numerical instabilities reported by other authors are examined and explained
from the same point of view.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
12.
Rupert L. Frank María del Mar González Dario D. Monticelli Jinggang Tan 《Advances in Mathematics》2015
We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre. We also prove an energy identity that yields a sharp trace Sobolev embedding. 相似文献
13.
In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow. 相似文献
14.
Summary Equilibrium equations and stability conditions for the simple deformable elastic body are derived by means of considering
a minimum of the static energy principle. The energy is supposed to be sum of the volume (elastic) and the surface terms.
The ability to change relative positions of different material particles is taken into account, and appropriate natural definitions
of the first and second variations of the energy are introduced and calculated explicitly. Considering the case of negligible
magnitude of the surface tension, we establish that an equilibrium state of a nonhydrostatically stressed simple elastic body
(of any physically reasonable elastic energy potential and of any symmetry) possessing any small smooth part of free surface
is always unstable with respect to relative transfer of the material particles along the surface. Surface tension suppresses
the mentioned instability with respect to sufficiently short disturbances of the boundary surface and thus can probably provide
local smoothness of the equilibrium shape of the crystal. We derive explicit formulas for critical wavelength for the simplest
models of the internal and surface energies and for the simplest equilibrium configurations. We also formulate the simplest
problem of mathematical physics, revealing peculiarities and difficulties of the problem of equilibrium shape of elastic crystals,
and discuss possible manifestations of the above-mentioned instability in the problems of crystal growth, materials science,
fracture, physical chemistry, and low-temperature physics. 相似文献
15.
An FDTD system associated with uniaxial perfectly matched layer(UPML) for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic(TM) mode to Maxwell’s equations is obtained by Yee’s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis. 相似文献
16.
Plamen Stefanov Gunther Uhlmann 《Proceedings of the American Mathematical Society》2004,132(5):1351-1354
We prove local uniqueness for the inverse problem in obstacle scattering at a fixed energy and fixed incident angle.
17.
Takeyuki Nagasawa Izumi Takagi 《Calculus of Variations and Partial Differential Equations》2003,16(1):63-111
Considered is a variational problem for the bending energy of closed surfaces under the prescribed area and surrounding volume.
Minimizers of this problem are interpreted as surfaces modeling the shape of red blood cells. We give a rigorous proof of
the existence of a one-parameter family of critical points bifurcating from the sphere and study their stability/instability.
In particular, for a few branches of critical points, we compute the exact values of the index and the nullity of critical
points.
Received: 8 September 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002
Partly supported by Grant-in-Aid for Exploratory Research (Nos.09874026, 11874033) and for Scientific Research (No.12640200),
Ministry of Education, Science, Sports, and Culture, Japan; and also by Sumitomo Foundation
Dedicated to Professor Takaaki Nishida on his sixtieth birthday 相似文献
18.
V. R. Khalilov 《Theoretical and Mathematical Physics》2006,149(3):1726-1740
We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of
the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and
discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm
and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state
can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm
potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the
magnetic field is taken into account in the form of the two-dimensional Dirac delta function.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006.
An erratum to this article is available at . 相似文献
19.
A. F. Kurin 《Computational Mathematics and Mathematical Physics》2009,49(3):482-495
An analytical solution to the Cauchy problem for the Hill equation is constructed by the second-order averaging method for three instability domains, stability domains near the boundaries with the instability domains, and on the boundaries themselves. An unstable exponentially decaying solution is found in the instability domains. A simple (convenient for applications) stability criterion for the trivial solution is formulated in the form of an inequality expressed in terms of the constant component, the amplitudes, and the frequencies of harmonics in the spectrum of the periodic coefficient of the Hill equation. 相似文献
20.
We study the orbital stability and instability of single-spike bound states of semi-classical nonlinear Schrödinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develop mathematical theorems which are most general on the orbital stability problem. 相似文献