共查询到20条相似文献,搜索用时 31 毫秒
1.
Zdeněk Skalák 《Journal of Mathematical Fluid Mechanics》2010,12(4):503-535
In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show,
for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω
⊆ R3 and β ∈[1/2, 1) , then there exist C0 > 1 and δ0 ∈ (0, 1) such that
\frac |||w(t)|||b|||w(t + d)|||b £ C0{\frac {|||w(t)|||_\beta}{|||w(t + \delta)|||_{\beta}}} \leq C_0 相似文献
2.
We study a two-dimensional nonconvex and nonlocal energy in micromagnetics defined over S
2-valued vector fields. This energy depends on two small parameters, β and e{\varepsilon} , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the
asymptotic regime b << e << 1{\beta \ll \varepsilon \ll 1} through the method of Γ-convergence. Finite energy configurations tend to become divergence-free and in-plane in the magnetic
sample except in some small regions of typical width e{\varepsilon} (called Bloch walls) where the magnetization connects two directions on S
2. We are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions.
For one-dimensional transition layers, we show by Γ-convergence analysis that the exact line density of the energy is quadratic
in the jump size. We expect the same behaviour for the two-dimensional model. In order to prove that, we investigate the concept
of entropies. In the prototype case of a periodic strip, we establish a quadratic lower bound for the energy with a non-optimal
constant. Then we introduce and study a special class of Lipschitz entropies and obtain lower bounds coinciding with the one-dimensional
Γ-limit in some particular cases. Finally, we show that entropies are not appropriate in general for proving the expected
sharp lower bound. 相似文献
3.
The streamwise evolution of an inclined circular cylinder wake was investigated by measuring all three velocity and vorticity
components using an eight-hotwire vorticity probe in a wind tunnel at a Reynolds number Red of 7,200 based on free stream velocity (U
∞) and cylinder diameter (d). The measurements were conducted at four different inclination angles (α), namely 0°, 15°, 30°, and 45° and at three downstream
locations, i.e., x/d = 10, 20, and 40 from the cylinder. At x/d = 10, the effects of α on the three coherent vorticity components are negligibly small for α ≤ 15°. When α increases further
to 45°, the maximum of coherent spanwise vorticity reduces by about 50%, while that of the streamwise vorticity increases
by about 70%. Similar results are found at x/d = 20, indicating the impaired spanwise vortices and the enhancement of the three-dimensionality of the wake with increasing
α. The streamwise decay rate of the coherent spanwise vorticity is smaller for a larger α. This is because the streamwise
spacing between the spanwise vortices is bigger for a larger α, resulting in a weak interaction between the vortices and hence
slower decaying rate in the streamwise direction. For all tested α, the coherent contribution to [`(v2)] \overline{{v^{2}}} is remarkable at x/d = 10 and 20 and significantly larger than that to [`(u2)] \overline{{u^{2}}} and [`(w2)]. \overline{{w^{2}}}. This contribution to all three Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent contribution to [`(u2)] \overline{{u^{2}}} and [`(v2)] \overline{{v^{2}}} decays slower as moving downstream for a larger α, consistent with the slow decay of the coherent spanwise vorticity for
a larger α. 相似文献
4.
Fractional calculus has gained a lot of importance during the last decades, mainly because it has become a powerful tool in
modeling several complex phenomena from various areas of science and engineering. This paper gives a new kind of perturbation
of the order of the fractional derivative with a study of the existence and uniqueness of the perturbed fractional-order evolution
equation for CDa-e0+u(t)=A CDd0+u(t)+f(t),^{C}D^{\alpha-\epsilon}_{0+}u(t)=A~^{C}D^{\delta}_{0+}u(t)+f(t),
u(0)=u
o
, α∈(0,1), and 0≤ε, δ<α under the assumption that A is the generator of a bounded C
o
-semigroup. The continuation of our solution in some different cases for α, ε and δ is discussed, as well as the importance of the obtained results is specified. 相似文献
5.
In this paper, we consider v(t) = u(t) − e
tΔ
u
0, where u(t) is the mild solution of the Navier–Stokes equations with the initial data
u0 ? L2(\mathbb Rn)?Ln(\mathbb Rn){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L
2 norm of D
β
v(t) decays like
t-\frac |b|-1 2-\frac n4{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u
1(t) such that the L
2 norm of D
β
(v(t) − u
1(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions. 相似文献
6.
Yoshihiro Ueda Tohru Nakamura Shuichi Kawashima 《Archive for Rational Mechanics and Analysis》2010,198(3):735-762
This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space.
We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection.
In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t
−α/4 as t → ∞, provided that the initial perturbation is in the weighted space
L2a=L2(\mathbb R+; (1+x)a dx){L^2_\alpha=L^2({\mathbb R}_+;\,(1+x)^\alpha dx)} . This convergence rate t
−α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α*(q), where α*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative
in L2a{L^2_\alpha} for α > α*(q) with another critical value α*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function
of the degenerate stationary wave. 相似文献
7.
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
8.
G. H. Keetels W. Kramer H. J. H. Clercx G. J. F. van Heijst 《Theoretical and Computational Fluid Dynamics》2011,25(5):293-300
Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re0.8{Z\propto{\rm Re}^{0.8}} and P μ Re2.25{P\propto {\rm Re}^{2.25}} for 5 × 102 ≤ Re ≤ 2 × 104 and Z μ Re0.5{Z\propto{\rm Re}^{0.5}} and P μ Re1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re
c
(here, Rec ? 2×104{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity
ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following
scaling relations are obtained: Z μ Re3/4, P μ Re9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re
c
the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate
boundary-layer theory, this yields: Z μ Re1/2{Z\propto{\rm Re}^{1/2}} and P μ Re3/2{P\propto {\rm Re}^{3/2}}. 相似文献
9.
Understanding turbulent wall-bounded flows remains an elusive goal. Most turbulent phenomena are non-linear, complex and have
broad range of scales that are difficult to completely resolve. Progress is made only in minute steps and enlightening models
are rare. Herein, we undertake the effort to bundle several experimental and numerical databases to overcome some of these
difficulties and to learn more about the kinematics of turbulent wall-bounded flows. The general scope of the present work
is to quantify the characteristics of wall-normal and spanwise Reynolds stresses, which might be different for confined (e.g.,
pipe) and semi-confined (e.g., boundary layer) flows. In particular, the peak position of wall-normal stress and a shoulder
in spanwise stress never described in detail before are investigated using select experimental and direct numerical simulation
databases available in the open literature. It is found that the positions of the
á v¢2
ñ + \left\langle {v'{^2} } \right\rangle^{ + } -peak in confined and semi-confined flow differ significantly above δ
+ ≈ 600. A similar behavior is found for the position of the
á u¢v¢
ñ + \left\langle {u'v'} \right\rangle^{ + } -peak. The upper end of the logarithmic region seems to be closely related to the position of the
á v¢2
ñ + \left\langle {v'{^2} } \right\rangle^{ + } -peak. The
á w¢2
ñ + \left\langle {w'{^2} } \right\rangle^{ + } -shoulder is found to be twice as far from the wall than the
á v¢2
ñ + \left\langle {v'{^2} } \right\rangle^{ + } -peak. It covers a significantly large portion of the typical zero-pressure-gradient turbulent boundary layer. 相似文献
10.
In a bounded domain of R n+1, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}
11.
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}}
12.
The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing
either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution
of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical
and planar geometries in the absence of solvent, b o [(h)\tilde]s/([(h)\tilde]s +[(h)\tilde]p)=0\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0, where [(h)\tilde]p\widetilde{\eta}_p and [(h)\tilde]s\widetilde{\eta}_s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and x 1 0\xi \ne 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, [(R)\tilde]\tilde{R}, and flow rate on the velocity profile, the stress components, and the film thickness, [(H)\tilde]\tilde{H}, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De=[(l)\tilde][(U)\tilde]/[(H)\tilde]De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}, and Stokes, St=[(r)\tilde][(g)\tilde][(H)\tilde]2/([(h)\tilde]p +[(h)\tilde]s )[(U)\tilde]St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}, numbers, depend on [(H)\tilde]\tilde{H} and the average film velocity, [(U)\tilde]\widetilde{U}. This makes necessary a trial and error procedure to obtain [(H)\tilde]\tilde{H}
a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to unity for a film on the inner surface. When x 1 0\xi \ne 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ. 相似文献
13.
We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}
14.
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)}
15.
We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L
∞(Ω),
W ì \mathbb Rd{\Omega \subset \mathbb R^d}) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization
of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most
general case of arbitrary bounded coefficients. For such problems, we introduce explicit and optimal finite dimensional approximations
of solutions that can be viewed as a theoretical Galerkin method with controlled error estimates, analogous to classical homogenization
approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical
concept of an e{\epsilon} family of media as in classical homogenization. We define the flux norm as the L
2 norm of the potential part of the fluxes of solutions, which is equivalent to the usual H
1-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space,
the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating
the set of solutions of the same type of PDEs with smooth coefficients in a standard space (for example, piecewise polynomial).
We refer to this property as the transfer property. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent
of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property
also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that
can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities.
These inequalities play the same role in our approach as the div-curl lemma in classical homogenization. 相似文献
16.
In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2, x ? W,-Dv +l2 v = m2 v3+bvu2, x ? W,u\geqq 0, v\geqq 0 in W, u=v=0 on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right. 相似文献
17.
Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}
|