共查询到20条相似文献,搜索用时 187 毫秒
1.
Consider inviscid fluids in a channel
{-1\leqq y\leqq1}{\{-1\leqq y\leqq1\}} . For the Couette flow u
0 = (y, 0), the vertical velocity of solutions to the linearized Euler equation at u
0 decays in time. Whether the same happens at the non-linear level is an open question. Here we study issues related to this
problem. First, we show that in any (vorticity)
Hs(s < \frac32){H^{s}\left(s<\frac{3}{2}\right)} neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal periods. This implies
that nonlinear inviscid damping is not true in any (vorticity)
Hs(s < \frac32){H^{s}\left(s<\frac{3}{2}\right)} neighborhood of Couette flow for any horizontal period. Indeed, the long time behaviors in such neighborhoods are very rich,
including nontrivial steady flows and stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity)
${H^{s}\left(s>\frac{3}{2}\right)}${H^{s}\left(s>\frac{3}{2}\right)} neighborhoods of Couette flow, we show that there exist no non-parallel steadily travelling flows
\varvecv(x-ct,y){\varvec{v}\left(x-ct,y\right)} , and no unstable shears. This suggests that the long time dynamics in ${H^{s}\left(s>\frac{3}{2}\right)}${H^{s}\left(s>\frac{3}{2}\right)} neighborhoods of Couette flow might be much simpler. Such contrasting dynamics in H
s
spaces with the critical power
s=\frac32{s=\frac{3}{2}} is a truly nonlinear phenomena, since the linear inviscid damping near Couette flow is true for any initial vorticity in
L
2. 相似文献
2.
Craig Cowan Pierpaolo Esposito Nassif Ghoussoub Amir Moradifam 《Archive for Rational Mechanics and Analysis》2010,198(3):763-787
We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation
D2 u=\fracl(1-u)2{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball
B ì \mathbbRN{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u
λ with 0 < u
λ < 1 exists for l ? (0,l*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution ul*{u_{\lambda^*}} is regular (supB ul* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided
N \leqq 8{N \leqq 8} while ul*{u_{\lambda^*}} is singular (supB ul* = 1){({\rm sup}_B u_{\lambda^*} =1)} for
N \geqq 9{N \geqq 9}, in which case
1-C0|x|4/3 \leqq ul* (x) \leqq 1-|x|4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where
C0:=(\fracl*[`(l)])\frac13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and
[`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}. 相似文献
3.
Mahesh Nerurkar 《Journal of Dynamics and Differential Equations》2011,23(3):451-473
Consider the class of C
r
-smooth
SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class,
(i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal
are generic, if α satisfies the following Liouville type condition:
|a-\fracpnqn| £ C exp (-qr+1+kn)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C > 0 and 0 < k < 1{0 < \kappa <1 } are some constants and
\fracPnqn{\frac{P_n}{q_n}} is some sequence of irreducible fractions. 相似文献
4.
This paper deals with the rational function approximation of the irrational transfer function
G(s) = \fracX(s)E(s) = \frac1[(t0s)2m + 2z(t0s)m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation
(t0)2m\fracd2mx(t)dt2m + 2z(t0)m\fracdmx(t)dtm + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and
the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude
and the usefulness of the approximation method. 相似文献
5.
Summary The transport of particles, caused by axial and radial diffusion and axial flow of convection, will be considered in a D.C. arc and in a laminar flame. The following mathematical model will be discussed. Assuming a steady state in both cases the mass transport may be described in cylindrical coordinates (r, z) by the following partial differential equation
where C means the particle concentration, D the coefficient of diffusion, and W the axial velocity. D and W are taken to be constant; various boundary conditions, corresponding to different approximations of the physical situation, are considered. Solutions of (0.1) are obtained in an explicit form. 相似文献
6.
S. I. Maksymenko 《Nonlinear Oscillations》2010,13(2):196-227
Let
D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin
O ? \mathbbR2 O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ¥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D+ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D
2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D
2 if and only if the 1-jet of F at O is a “rotation,” i.e.,
j1F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D+ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle. 相似文献
7.
Dongho Chae 《Journal of Mathematical Fluid Mechanics》2010,12(2):171-180
Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time
domain containing z0=(x0, t0)z_{0}=(x_{0}, t_{0}), and let Qz0,r = Bx0,r ×(t0 -r2, t0)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu
\times \frac{\omega}{|\omega|} \in
L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu
\times \frac{\omega}{|\omega|} \in
L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times
\frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times
\frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with
\frac3g + \frac2a £ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where Lγ, αx,t denotes the Serrin type of class, then z0 is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions. 相似文献
8.
Vieri Benci Marco Ghimenti Anna Maria Micheletti 《Archive for Rational Mechanics and Analysis》2012,205(2):467-492
We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12We¢(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi}, 相似文献
9.
Sébastien de Valeriola Jean Van Schaftingen 《Archive for Rational Mechanics and Analysis》2013,210(2):409-450
Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and shown to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problem $$ \left\{ \begin{aligned} -{\rm div} \left(\frac{\nabla u_{\varepsilon}}{b}\right) & = \frac{1}{\varepsilon^2} b f \left(u_{\varepsilon} - \log \tfrac{1}{\varepsilon} q \right) & & \text{ in } \; \Omega, \\u_\varepsilon & = 0 & & \text{ on } \; \partial \Omega, \end{aligned}\right.$$ for small values of ${\varepsilon > 0}$ . 相似文献
10.
This work is an extensive study of the 3 different types of positive solutions of the Matukuma equation ${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }${\frac{1}{r^{2}}\left( r^{2}\phi^{\prime}\right) ^{\prime}=-{\frac{r^{\lambda-2}}{\left( 1+r^{2}\right)^{\lambda /2}}}\phi^{p},p >1 ,\lambda >0 }: the E-solutions (regular at r = 0), the M-solutions (singular at r = 0) and the F-solutions (whose existence begins away from r = 0). An essential tool is a transformation of the equation into a 2-dimensional asymptotically autonomous system, whose
limit sets (by a theorem of H. R. Thieme) are the limit sets of Emden–Fowler systems, and serve as to characterizate the different
solutions. The emphasis lies on the study of the M-solutions. The asymptotic expansions obtained make it possible to apply the results to the important question of stellar
dynamics, solutions to which lead to galactic models (stationary solutions of the Vlasov–Poisson system) of finite radius
and/or finite mass for different p, λ. 相似文献
11.
G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form
of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation,
G-equations are Hamilton–Jacobi equations with convex (L
1 type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by
small diffusion. The nonlinear eigenvalue [`(H)]{\bar H} from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s
T. An important problem in turbulent combustion theory is to study properties of s
T, in particular how s
T depends on the flow amplitude A. In this paper, we study the behavior of [`(H)]=[`(H)](A,d){\bar H=\bar H(A,d)} as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that
$\bar H(A,d)\leqq C(d) \quad \text{for all}\ d >0 ,$\bar H(A,d)\leqq C(d) \quad \text{for all}\ d >0 , 相似文献
12.
We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}
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