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1.
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. 相似文献
2.
Yoshiyuki Kagei Michael Růžička Gudrun Thäter 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(4):447-467
We discuss a model limit problem which arises as a first step in the mathematical justification of our Boussinesq-type approximation
[4], which takes into account dissipative heating in natural convection. We treat a simplified highly non linear system depending
on a (perturbation) parameter ε. The main difficulty is that for ε ≠ 0 the velocity is not solenoidal. First we prove that
our system has weak solutions for each fixed ε. Moreover, while the chosen perturbation parameter ε tends to zero we show,
that we arrive at the usual incompressible case and the standard Boussinesq approximation. 相似文献
3.
In this article we consider the following generalized quasi-geostrophic equation
4.
Thermosolutal convection at infinite Prandtl number: initial layer and infinite Prandtl number limit
Jungho Park 《Applicable analysis》2013,92(9):1829-1847
We examine the initial layer problem and the infinite Prandtl number limit of the thermosolutal convection, which is applicable to magma chambers. We derive the effective approximating system of the Boussinesq system at large Prandtl number using two time scale approach [M. Holmes, Introduction to Perturbation Methods, Springer, New York, 1995, A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York, American Mathematical Society, Providence, RI, 2003]. We show that the effective approximating system is nothing but the infinite Prandtl number system with initial layer terms. We also show that the solutions of the Boussinesq system converge to solutions of the effective approximating system with the convergence rate of O(?). 相似文献
5.
In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces Lp, with p∈[1,∞]. Local results for arbitrary initial data are also given. 相似文献
6.
In this paper we prove the local-in-time well-posedness for the 2D non-dissipative quasi-geostrophic equation, and study the blow-up criterion in the critical Besov spaces. These results improve the previous one by Constantin et al. [P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity 7 (1994) 1495–1533]. 相似文献
7.
The steady laminar natural convection of a fluid having chemical reaction of order n past a semi-infinite vertical plate is considered. The solution of the problem by means of one-parameter group method reduces the number of independent variables by one leading to a system of nonlinear ordinary differential equations. Two different similarity transformations are found. In each case the set of differential equations are solved numerically using Runge–Kutta and the shooting method. For each transformation different Schmidt numbers and chemical reaction orders are tested. 相似文献
8.
Isom Herron Jeremy Goodman 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(4):615-622
Axisymmetric stability of viscous resistive magnetized Couette flow is re-examined, with emphasis on flows that would be hydrodynamically
stable according Rayleigh’s criterion: opposing gradients of angular velocity and specific angular momentum. In this regime,
mag- netorotational instability (MRI) may occur. The governing system in cylindrical coordinates is of tenth order. It is
proved, by methods based on those of Synge and Chandrasekhar, that by dropping one term from the system, MRI is suppressed,
in fact no instability at all occurs, with insulating boundary conditions. This term is often neglected because it has the
magnetic Prandtl number, which is very small, as a factor; nevertheless it is crucially important.
(Received: August 11, 2005; revised: January 3, 2006) 相似文献
9.
Dynamics of systems on infinite lattices 总被引:1,自引:0,他引:1
Bixiang Wang 《Journal of Differential Equations》2006,221(1):224-245
The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l2 space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems. 相似文献
10.
The method of bifurcation of planar dynamical systems and method of numerical simulation of differential equations are employed to investigate the modified dispersive water wave equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. In different regions, different types of travelling solutions including solitary wave solutions, shock wave solutions and periodic wave solutions are simulated. Furthermore, with a generalized projective Riccati equation method, several new explicit exact solutions are obtained. 相似文献
11.
We study steady linear gravity waves of small amplitude travelling on a current of constant vorticity. For positive vorticity the situation resembles that of Stokes waves, but if the vorticity is large enough the particle trajectories are affected. For negative vorticity we show that there may appear internal waves and vortices, wherein the particle trajectories are not ellipses. 相似文献
12.
We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense. 相似文献
13.
Songsong Lu 《Journal of Differential Equations》2006,230(1):196-212
First, the existence and structure of uniform attractors in H is proved for nonautonomous 2D Navier-Stokes equations on bounded domain with a new class of distribution forces, termed normal in (see Definition 3.1), which are translation bounded but not translation compact in . Then, the properties of the kernel section are investigated. Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained. 相似文献
14.
Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive ‘extreme’ waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories. 相似文献
15.
In this paper, we investigate the existence and uniqueness of the solution to the Cauchy problem for a class of nonlinear wave equations of higher order and prove the existence and nonexistence of global solutions to this problem by a potential well method. 相似文献
16.
17.
B. Scarpellini 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(3):412-438
In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form
It is shown that this equilibrium is Ljapounov stable under small perturbations in
where
provided that the parameters a, b, , are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.Received: February 5, 2003; revised: September 29, 2003Dedicated to Prof. H. Amann on the occasion of his 65. birthday 相似文献
18.
We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. ?ukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context. 相似文献
19.
Peter H. Lesky 《Journal of Differential Equations》2008,244(4):945-971
We consider initial-boundary value problems for the equations of isotropic elasticity for several mixed boundary conditions in infinite wave guides, as well as Maxwell equations. With the help of decompositions of the displacement field into divergence- and curl-free parts, respectively, which are compatible with the boundary conditions, we obtain sharp decay rates for the solutions. The decomposed systems correspond to the second-order Maxwell equations for the electric and the magnetic field with electric and magnetic boundary conditions, respectively. 相似文献
20.
In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions. 相似文献