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1.
Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T
0 > 0, ν
0 > 0 and a unique continuous family of strong solutions u
ν
(0 ≤ ν < ν
0) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T
0). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary
conditions imposed on curl
u
ν
and curl
2
u
ν
.
相似文献
2.
The transport equations for the second-order velocity structure functions 〈(δu)2〉 and 〈(δq)2〉 are used as a scale-by-scale budget to quantify the effect of initial conditions at low Reynolds numbers, typical of grid
turbulence. The validity of these equations is first investigated via hot-wire measurements of velocity and transverse vorticity
fluctuations. The transport equation for 〈(δq)2〉 is shown to be balanced at all scales, while anisotropy of the large scales leads to a significant imbalance in the equation
for 〈(δu)2〉. The effect of using similarity to evaluate the transport equation is rigorously tested. This approach has the desirable
benefit of requiring less extensive measurements to calculate the inhomogeneous term of the transport equation. The similarity
form of the 〈(δq)2〉 equation produces nearly identical results as those obtained without the similarity assumption. In the case of the 〈(δu)2〉 equation, the similarity method forces a balance at large separation, although the imbalance due to large scale anisotropy
remains. The initial conditions of the turbulence at constant R
M
≃ 10,400 (28≤ R
λ≤ 55) are changed by using three grids of different geometries. Initial conditions affect the shape and magnitude of the second-
and third-order structure functions, as well as the anisotropy of the large scales. The effect of initial conditions on the
scale-by-scale budget is restricted to the inhomogeneous term of the transport equations, while the dissipation term remains
unaffected despite the low R
λ. Scales as small as λ are affected by the changes in initial conditions. 相似文献
3.
Hermano Frid 《Archive for Rational Mechanics and Analysis》2006,181(1):177-199
We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems
of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems
whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which
are hyperplanes. In particular, we obtain the uniqueness of the self-similar L∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is
in the sense of the convergence as t→∞ in Lloc1 of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in Lloc1(ℝn), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem. 相似文献
4.
José A. Carrillo Marco Di Francesco Giuseppe Toscani 《Archive for Rational Mechanics and Analysis》2006,180(1):127-149
We investigate the long time asymptotics in L1+(R) for solutions of general nonlinear diffusion equations ut = Δϕ(u). We describe, for the first time, the intermediate asymptotics for a very large class of non-homogeneous nonlinearities
ϕ for which long time asymptotics cannot be characterized by self-similar solutions. Scaling the solutions by their own second
moment (temperature in the kinetic theory language) we obtain a universal asymptotic profile characterized by fixed points
of certain maps in probability measures spaces endowed with the Euclidean Wasserstein distance d2. In the particular case of ϕ(u) ~ um at first order when u ~ 0, we also obtain an optimal rate of convergence in L1 towards the asymptotic profile identified, in this case, as the Barenblatt self-similar solution corresponding to the exponent
m. This second result holds for a larger class of nonlinearities compared to results in the existing literature and is achieved
by a variation of the entropy dissipation method in which the nonlinear filtration equation is considered as a perturbation
of the porous medium equation. 相似文献
5.
H. Beirão da Veiga 《Journal of Mathematical Fluid Mechanics》2007,9(4):506-516
In reference [7] it is proved that the solution of the evolution Navier–Stokes equations in the whole of R
3 must be smooth if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference
[5] the authors improve the above result by showing that Lipschitz continuity may be replaced by 1/2-H?lder continuity. A
central point in the proofs is to estimate the integral of the term (ω · ∇)u · ω, where u is the velocity and ω = ∇ × u is the vorticity. In reference [4] we extend the main estimates on the above integral term to solutions under the slip boundary
condition in the half-space R
+3. This allows an immediate extension to this problem of the 1/2-H?lder sufficient condition.
The aim of these notes is to show that under the non-slip boundary condition the above integral term may be estimated as well
in a similar, even simpler, way. Nevertheless, without further hypotheses, we are not able now to extend to the non slip (or
adherence) boundary condition the 1/2-H?lder sufficient condition. This is not due to the “nonlinear" term (ω · ∇)u · ω but to a boundary integral which is due to the combination of viscosity and adherence to the boundary. On the other hand,
by appealing to the properties of Green functions, we are able to consider here a regular, arbitrary open set Ω.
相似文献
6.
Asymptotic Variational Wave Equations 总被引:1,自引:0,他引:1
Alberto Bressan Ping Zhang Yuxi Zheng 《Archive for Rational Mechanics and Analysis》2007,183(1):163-185
We investigate the equation (u
t
+(f(u))
x
)
x
=f
′ ′(u) (u
x
)2/2 where f(u) is a given smooth function. Typically f(u)=u
2/2 or u
3/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u
tt
− c(u) (c(u)u
x
)
x
=0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data. 相似文献
7.
Chien-Cheng Chang Jeng-Yu Su Sheng-Yuan Lei 《Theoretical and Computational Fluid Dynamics》1998,10(1-4):71-90
The paper is aimed at reviewing and adding some new results to our recent work on a force theory for viscous compressible
flows around a finite body. It has been proposed to analyze aerodynamic forces directly in terms of fluid elements of nonzero
vorticity and density gradient. Let ρ denote the density, u the velocity, and ω the vorticity. It is demonstrated that for largely separated flows about bluff bodies, there are two
major source elements:
R
e(x) =−?u
2∇ρ·∇ϕ and V
e(x) =−u×ω·∇ϕ,
where ϕ is an acyclic potential, generated by the solid body moving with unit velocity in the negative direction of the force
considered. In particular, under mild conditions, the (unique) choice of ϕ enforces that the elements R
e(x) and V
e(x) decay rapidly away from the body. Four kinds of finite body are considered: a circular cylinder, a sphere, a hemi sphere-cylinder,
and a delta wing of elliptic section—all in transonic-to-supersonic regimes. From an extensive numerical study carried out
for these bodies, it is found that these two elements contribute to 95% or more of the total drag or lift for all the cases
under consideration. Moreover, R
e(x) due to density gradient becomes progressively important relative to V
e(x) due to vorticity as the Mach number increases. The present method of force analysis enables effective analysis and assessment
of relative importance of aerodynamics forces, contributed from individual flow structures. The analysis could therefore be
very much useful in view of the rapid growth in numerical fluid dynamics; detailed (either local or global) flow information
is often available. The paper is dedicated to Sir James Lighthill in honor of his great contributions to aeronautics on the
occasion of the publication of his collected works.
Received 3 January 1997 and accepted 11 April 1997 相似文献
8.
We consider reaction diffusion equations of the prototype form u
t
= u
xx
+ λ u + |u|
p-1
u on the interval 0 < x < π, with p > 1 and λ > m
2. We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up
profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u
1, . . . ,u
m
. Since u
k
= ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x
1, . . . ,x
m
of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree
argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively.
Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work
by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all
negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano
and Fiedler (2007) (in preparation).
Dedicated to Palo Brunovsky on the occasion of his birthday. 相似文献
9.
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
10.
We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u
1(x),...,u
N
(x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be
, a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14]. 相似文献
11.
A new subgrid scale (SGS) modelling concept for large-eddy simulation (LES) of incompressible flow is proposed based on the
three-dimensional spatial velocity increment δ u
i
. The new model is inspired by the structure function formulation developed by Métais and Lesieur [39] and applied in the context of the scale similarity type formulation. First, the similarity between the SGS stress tensor τ
ij
and the velocity increment tensor Q
ij
= δ u
i
δ u
j
is analyzed analytically and numerically using a priori tests of fully developed pipe flow at Re
τ = 180. Both forward and backward energy transfers between resolved and unresolved scales of the flow are well predicted with
a SGS model based on Q
ij
. Secondly, a posteriori tests are performed for two families of turbulent shear flows. LES of fully developed pipe flow up to Re
τ = 520 and LES of round turbulent jet at Re
D
= 25000 carried out with a dynamic version of the model provide promising results that confirm the power of this approach
for wall-bounded and free shear flows. 相似文献
12.
We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space
We show, among other things, that there are two positive constants
and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to Lq(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with
and u ∈ L∞(Ω) if a ∈ L∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W1-1/q,q(∂Ω), with
then ∇u, p ∈ Lq(Ω) and if a ∈ C0,μ(∂Ω), with μ ∈ [0, α), then
also, natural estimates holds. 相似文献
13.
14.
We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u
tt
− c(u)(c(u)u
x
)
x
= 0. We allow for initial data u|
t = 0 and u
t
|
t=0 that contain measures. We assume that
0 < k-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (ut2+c(u)2ux2)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby
singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times
of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples. 相似文献
15.
Michael G. Crandall 《Archive for Rational Mechanics and Analysis》2003,167(4):271-279
For 1<p<∞, the equation which characterizes minima of the functional u↦∫
U
|Du|
p
,dx subject to fixed values of u on ∂U is −Δ
p
u=0. Here −Δ
p
is the well-known ``p-Laplacian'. When p=∞ the corresponding functional is u↦|| |Du|2||
L∞(U)
. A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers'. Aronsson showed that then the appropriate equation is
−Δ∞
u=0, that is, u is ``infinity harmonic' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity
sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals
u↦||f(x,u,Du)||
L∞(U)
by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses.
(Accepted September 6, 2002)
Published online April 14, 2003
Communicated by S. Muller 相似文献
16.
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
17.
As an example of an extended, formally gradient dynamical system, we consider the damped hyperbolic equation u
tt+u
t=u+F(x, u) in R
N
, where F is a locally Lipschitz nonlinearity. Using local energy estimates, we study the semiflow defined by this equation in the uniformly local energy space H1
ul(R
N
)×L2
ul(R
N
). If N2, we show in particular that there exist no periodic orbits, except for equilibria, and we give a lower bound on the time needed for a bounded trajectory to return in a small neighborhood of the initial point. We also prove that any nonequilibrium point has a neighborhood which is never visited on average by the trajectories of the system, and we conclude that any bounded trajectory converges on average to the set of equilibria. Some counter-examples are constructed, which show that these results cannot be extended to higher space dimensions. 相似文献
18.
Yoshihisa Morita Hirokazu Ninomiya 《Journal of Dynamics and Differential Equations》2006,18(4):841-861
We deal with a reaction–diffusion equation u
t
= u
xx
+ f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c
1
t) (c
1 < 0) and ψ2(x + c
2
t) (c
2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all
. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c
1
t) and ψ2(x + c
2
t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c
1, we show the existence of an entire solution which behaves as ψ1( − x + c
1
t) in
and φ(x + ct) in
for t≈ − ∞. 相似文献
19.
Yasunori Maekawa 《Journal of Mathematical Fluid Mechanics》2008,10(1):89-105
In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations
equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities
at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the
initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the
Oseen vortex.
相似文献
20.
Low-order moments of the increments δu andδv where u and v are the axial and radial velocity fluctuations respectively, have been obtained using single and X-hot wires mainly on the
axis of a fully developed pipe flow for different values of the Taylor microscale Reynolds numberR
λ. The mean energy dissipation rate〉ε〈 was inferred from the uspectrum after the latter was corrected for the spatial resolution of the hot-wire probes. The corrected Kolmogorov-normalized
second-order structure functions show a continuous evolution withR
λ. In particular, the scaling exponentζ
v
, corresponding to the v structure function, continues to increase with R
λ in contrast to the nearly unchanged value of ζ
u
. The Kolmogorov constant for δu shows a smaller rate of increase with R
λ than that forδv. The level of agreement with local isotropy is examined in the context of the competing influences ofR
λ and the mean shear. There is close but not perfect agreement between the present results on the pipe axis and those on the
centreline of a fully developed channel flow.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献