共查询到20条相似文献,搜索用时 31 毫秒
1.
The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and
is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally
well posed in H
−7/4, using the ideas of an [`(F)] s{\overline F ^s}-type space [8]. Then we show that the equation is globally well posed in H
s
for s ≥ −7/4, using the ideas of the “I-method” [7]. 相似文献
2.
The Kawahara and modified Kawahara equations are fifth-order KdV type equations that have been derived to model many physical
phenomena such as gravity-capillary waves and magneto-sound propagation in plasmas. This paper establishes the local well-posedness
of the initial-value problem for the Kawahara equation in H
s
(R) with s ≥ − 7/4 and the local well-posedness for the modified Kawahara equation in H
s
(R) with s ≥ − 1/4. To prove these results, we derive a fundamental estimate on dyadic blocks for the Kawahara equation through the
[k; Z]_multiplier norm method of Tao [14] and use this to obtain new bilinear and trilinear estimates in suitable Bourgain spaces. 相似文献
3.
We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time
variable. We prove that if the coefficients of the equation belong to the Gevrey class gs0\gamma^{s_{0}} and the Cauchy data belong to gs1\gamma^{s_{1}}, then the Cauchy problem has a solution in
gs0([0,T*];gs1(\mathbbR))\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R})) for some T
*>0, provided 1≤s
1≤2−1/s
0. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s
1≤s
0. 相似文献
4.
Bin Han 《Advances in Computational Mathematics》2006,24(1-4):375-403
In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(ℝs) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented.
Rate of convergence of vector cascade algorithms in a Sobolev space Wpk(ℝs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function
vector. As a consequence, we show that if a compactly supported function vector φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz
space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167)
in the univariate setting to the multivariate setting.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 42C20, 41A25, 39B12.
Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant
G121210654. 相似文献
5.
Yu. B. Yanushanets 《Journal of Mathematical Sciences》2000,102(4):4339-4347
We consider the fundamental solution E (t,x,s;s
0) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t+c(s,grad
x)+γ)E(t,x,s;s
0)=γν∫ f((s, s′))E(t,x,s′; s0)ds′+Ωδ(t)δ(x)δ (s−s
0) that is assumed to be valid for any (t,x)∈Rn+1; morevoer, for t<0 the condition E(t,x,s; s0)=0 holds. By using the Fourier-laplace transform in space-time arguments, the problem reduces to the study of an integral
equation in the variables. For 0<ν≤1, the uniqueness and existence of the solution of the original problem are proved for any fixeds in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic
type (f(.)=1), the solution of the integral equation is given in explicit form. In the approximation of “small mean-free paths,”
various weak limits of the solution are obtained with the help of a Tauberian-type theorem, for distributions. Bibliography:
4 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 319–332.
Translated by Yu. B. Yanushanets. 相似文献
6.
Given a∈L
1(ℝ) and A the generator of an L
1-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫
−∞
t
a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L
1(ℝ) positive, nonincreasing and log-convex is already sufficient. 相似文献
7.
Xin Li 《Advances in Computational Mathematics》2009,30(3):201-230
For a Helmholtz equation Δu(x) + κ
2
u(x) = f(x) in a region of R
s
, s ≥ 2, where Δ is the Laplace operator and κ = a + ib is a complex number with b ≥ 0, a particular solution is given by a potential integral. In this paper the potential integral is approximated by using
radial bases with the order of approximation derived.
相似文献
8.
Stephen C. Preston 《Annals of Global Analysis and Geometry》2012,41(3):281-305
In this article, we study geometric aspects of the space of arcs parameterized by unit speed in the L
2 metric. Physically, this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic
equation is the nonlinear, nonlocal wave equation η
tt
= ∂
s
(σ η
s
), with
\lvert hs\rvert o 1{\lvert \eta_{s}\rvert\equiv 1} and σ given by
sss- \lvert hss\rvert2 s = -\lvert hst\rvert2{\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2}, with boundary conditions σ(t, 1) = σ(t, −1) = 0 and η(t, 0) = 0. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists
but is not smooth, and as a consequence we get a Riemannian exponential map that is continuous and even differentiable but
not C
1. This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily
short times along any geodesic. 相似文献
9.
LI Song & ZHOU Xinlong Department of Mathematics Zhejiang University Hangzhou China Instititute of Mathematics University of Duisburg-Essen D- Duisburg Germany 《中国科学A辑(英文版)》2006,49(4):439-450
This paper is concerned with multivariate refinement equations of the type where (?) is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported nonnegative sequence on Zs, and M is an s×s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation. 相似文献
10.
We study the local existence of strong solutions for the cubic nonlinear wave equation with data in H
s
(M), s<1/2, where M is a three dimensional compact Riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed
(in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large
set of initial data in H
s
(M), where s≥1/4 in the case of a boundary less manifold and s≥8/21 in the case of a manifold with boundary.
Mathematics Subject Classification (2000) 35Q55, 35BXX, 37K05, 37L50, 81Q20 相似文献
11.
In the paper, the equation
is considered in the scale of the weighted spaces H
β
s
(ℝ
n
) (q > 1, a
kα
∈ ℂ). We prove that if the expression
does not vanish on the set {ξ ∈ ℝ
n
∖ 0, |z| ≤ q
β−s+n
/2−2m}, then this equation has a unique solution u ∈ H
β
s+2m
(ℝ
n
) for every function f ∈ H
β
s
(ℝ
n
) provided that β, s ≠ ∈ ℝ, β − s ≠ n/2 + p, and β − s − 2m ≠ − n/2 − p (p = 0, 1, ...).
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 37–55, 2007. 相似文献
12.
B. de Malafosse 《Acta Mathematica Hungarica》2009,122(3):217-230
We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between
s
h,l
(λ, μ) = s
α
0((Δ − λI)
h
) + s
β
(c)((Δ − μI)
l
) and s
γ
. Among other things the aim of this paper is to reduce the set (s
h,l
(λ, μ), s
γ
to a set of the form S
τ,γ
.
相似文献
13.
Andreas de Vries 《manuscripta mathematica》1995,88(1):233-246
Analogously to the recently published treatise on the massless spin-s wave fields fors=1/2 ands=1, cf. [32], the covariant Dirac equation in certain coordinate charts is rewritten as an evolution equation. As a result
it is proved that the Diract operatorD in the whole outer space of a Kerr-Newman black hole is symmetric. This is different from the behavior of the Maxwell operator
which admits superradiance in case of a rotating black hole, cf. [32]. An interpretation of this symmetry may be that there
is no particle creation by black holes, cf. [24, 16, 10, 15]. Moreover, the operatorA=−ih
−1
D in expanding Robertson-Walker universes is shown to be dissipative, whereas in the contracting case −A is dissipative.
This article was processed by the author using the LaTEX style filecljour1 from Springer-Verlag 相似文献
14.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
15.
Summary We introduce the concepts of γ-semi-open set, γs-set, γs-set, generalized γs-set, generalized γs-set, semi-T1γspace and semi-R0γspace by using γ-open-sets. 相似文献
16.
Ying Lungan 《数学学报(英文版)》1988,4(3):210-226
Three dimensional initial boundary value problem of the Navier-Stokes equation is considered. The equation is split in an
Euler equation and a non-stationary Stokes equation within each time step. Unlike the conventional approach, we apply a non-homogeneous
Stokes equation instead of homogeneous one. Under the hypothesis that the original problem possesses a smooth solution, the
estimate of theH
s+1 norm, 0≦s<3/2, of the approximate solutions and the order of theL
2 norm of the errors is obtained.
This work was supported by the Science Foundation of Academia Sinica under grant (84)-103. 相似文献
17.
Local and Global Existence of Solutions to
Initial Value Problems of Modified Nonlinear
Kawahara Equations 总被引:3,自引:0,他引:3
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(5):1035-1044
This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the second the third +β the third partial dervative of u to x ,the second the thire +γ the fifth partial dervative of u to x = 0,(x,t)∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo(x) ∈ H^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2. 相似文献
18.
Mario Petrich 《Semigroup Forum》2011,83(3):412-446
Let S be a semigroup and s,t∈S. We say that t is an associate of s if s=sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s
∗, we say that G is an associate subgroup of S and consider the mapping s→s
∗ as a unary operation on S. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms.
Among them, only those satisfying the identity (st)∗=t
∗
s
∗, called medial, have a structure theorem, due to Blyth and Martins. 相似文献
19.
Xingyu Yang 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):273-285
We consider the global attractor for the weakly damped forced KdV equation in Sobolev spaces [(H)\dot]s(T){\dot{H}^s({\mathbf T})}for s < 0. Under the assumption that the external forcing term belongs to [(L)\dot]2(T),{\dot{L}^2({\mathbf T}),} we prove the existence of the global attractor in [(H)\dot]s(T){\dot{H}^s({\mathbf T})} for −1/2 ≤ s < 0, which is identical to the one in [(L)\dot]2(T){\dot{L}^2({\mathbf T})} and thus is compact in H
3(T). The argument is a combination of the I-method and decomposing the solution into two parts, one of which is uniformly bounded
in [(L)\dot]2(T){\dot{L}^2({\mathbf T})} and the other decays exponentially in [(H)\dot]s(T){\dot{H}^s({\mathbf T})}. 相似文献
20.
Let R
k,s
(n) denote the number of solutions of the equation n = x2 + y1k + y2k + ?+ ysk{n= x^2 + y_1^k + y_2^k + \cdots + y_s^k} in natural numbers x, y
1, . . . , y
s
. By a straightforward application of the circle method, an asymptotic formula for R
k,s
(n) is obtained when k ≥ 3 and s ≥ 2
k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2
k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R
5,17(n) by this strategy. 相似文献