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1.
Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L
2 Initial Data 总被引:6,自引:0,他引:6
Shang Bin CUI Dong Gao DENG Shuang Ping TAO 《数学学报(英文版)》2006,22(5):1457-1466
In this paper we study solvability of the Cauchy problem of the Kawahara equation 偏导dtu + au偏导dzu + β偏导d^3xu +γ偏导d^5xu = 0 with L^2 initial data. By working on the Bourgain space X^r,s(R^2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to H^r(R) and -1 〈 r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L^2(R). 相似文献
2.
In this paper we consider the Cauchy problem for a higher order modified Camassa–Holm equation. By using the Fourier restriction
norm method introduced by Bourgain, we establish the local well-posedness for the initial data in the H
s
(R) with ${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.}${s > -n+\frac{5}{4},\,n\in {\bf N}^{+}.} As a consequence of the conservation of the energy ||u||H1(R),{{||u||_{H^{1}(R)},}} we have the global well-posedness for the initial data in H
1(R). 相似文献
3.
Let G be a locally compact Abelian group with Haar measure. The authors discuss some basic properties of Lw1^r (G)∩ L(p, q, w2dμ)(G) spaces. Then the necessary conditions for compact embeddings of the spaces Lw1^r (R^d)∩ L(p, q, w2dμ)(R^d) are showed. 相似文献
4.
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
5.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
6.
Wojciech M. Zaja̧czkowski 《Mathematical Methods in the Applied Sciences》2011,34(5):544-562
We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??3 with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L2, ?µ(ΩT), initial velocity v0∈H(Ω), µ∈?+\? there exist velocity v∈H(ΩT) and the pressure p, ?p∈L2, ?µ(ΩT) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:
- 1. the slip boundary condition and
- 2. the Helmholtz–Weyl decomposition.
7.
A basic integral equation of random fields estimation theory by the criterion of minimum of variance of the estimation error is of the form Rh = f, where
and R(x, y) is a covariance function.The singular perturbation problem we study consists of finding the asymptotic behavior of the solution to the equation
as
0.$$" align="middle" border="0">
The domain D can be an interval or a domain in Rn, n > 1. The class of operators R is defined by the class of their kernels R(x,y) which solve the equation Q(x, Dx)R(x, y) = P(x, Dx)δ(x − y), where Q(x, Dx) and Px, Dx) are elliptic differential operators. 相似文献
8.
Global Existence of Solutions for the Kawahara Equation in Sobolev Spaces of Negative Indices 总被引:1,自引:0,他引:1
Hua WANG Shang Bin CUI Dong Gao DENG 《数学学报(英文版)》2007,23(8):1435-1446
We first prove that the Cauchy problem of the Kawahara equation, δtu + uδxu +βδx^3u+γδx^5u = 0, is locally solvable if the initial data belong to H^r(R) and r〉 r≥-7/5, thus improving the known local well-posedness result of this equation. Next we use this local result and the method of "almost conservation law" to prove that global solutions exist if the initial data belong to H^r(R) and r〉-1/2. 相似文献
9.
Violeta Petkova 《Archiv der Mathematik》2009,93(4):357-368
We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces
Lw2(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with
f ? Cc(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces
L2w(\mathbb Rk){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form. 相似文献
10.
Let R be a prime ring with extended centroid F and let δ be an F-algebraic continuous derivation of R with the associated inner derivation ad(b). Factorize the minimal polynomial μ(λ) of b over F into distinct irreducible factors m(l)=?ipi(l)ni{\mu(\lambda)=\prod_i\pi_i(\lambda)^{n_i}} . Set ℓ to be the maximum of n
i
. Let R(d)=def.{x ? R | d(x)=0}{R^{(\delta)}{\mathop{=}\limits^{{\rm def.}}}\{x\in R\mid \delta(x)=0\}} be the subring of constants of δ on R. Denote the prime radical of a ring A by P(A){{\mathcal{P}}(A)} . It is shown among other things that
P(R(d))2l-1=0 \textand P(R(d))=R(d)?P(CR(b)){\mathcal{P}}(R^{(\delta)})^{2^\ell-1}=0\quad\text{and}\quad{\mathcal{P}}(R^{(\delta)})=R^{(\delta)}\cap {\mathcal{P}}(C_R(b)) 相似文献
11.
Behrooz Mirzaii 《Mathematische Annalen》2008,340(1):159-184
The homology of GL
n
(R) and SL
n
(R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H
4(GL3(R), k) → H
4(GL4(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the
K-group K
4(R). 相似文献
12.
Dale Alspach 《Israel Journal of Mathematics》2000,117(1):239-259
It is shown that a
-space with separable dual constructed by Bourgain and Delbaen has small Szlenk index and thus does not have a quotient isomorphic
toC(ωω). It follows that this is a
-space which is the same size asc
0 in the sense of the Szlenk index but does not containc
0. This has some consequences in the theory of uniform homeomorphism of Banach spaces. 相似文献
13.
The wave equation, ∂
tt
u=Δu, in ℝ
n+1, considered with initial data u(x,0)=f∈H
s
(ℝ
n
) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H
s
(ℝ
n
) to L
q
(ℝ
n
). 相似文献
14.
Zhou Fujun Cui Shangbin 《高校应用数学学报(英文版)》2005,20(4):441-447
The solvability of the fifth-order nonlinear dispersive equation δtu+au (δxu)^2+βδx^3u+γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5(R) and s〉1/4,then the Cauchy problem has a unique solution in C([-T,T],H^5(R)) for some T〉0. 相似文献
15.
Yan Guo 《纯数学与应用数学通讯》2006,59(5):626-687
Given a normalized Maxwellian μ and n ≥ 1, we establish the global‐in‐time validity of a diffusive expansion for a solution Fε to the rescaled Boltzmann equation (diffusive scaling) inside a periodic box ??3. We assume that in the initial expansion (0.1) at t = 0, the fluid parts of these fm(0,x,v) have arbitrary divergence‐free velocity fields as well as temperature fields for all 1 ≤ m ≤ n while f1(0,x,v) has small amplitude in H2. For m ≥ 2, these fm(t,x,v) are determined by a sequence of linear Navier‐Stokes‐Fourier systems iteratively. More importantly, the remainder f(t,x,v) is proven to decay in time uniformly in ε via a unified nonlinear energy method. In particular, our results lead to an error estimate for f1(t,x,v), the well‐known Navier‐Stokes‐Fourier approximation, and beyond. The collision kernel Q includes hard‐sphere, the cutoff inverse‐power, as well as the Coulomb interactions. © 2005 Wiley Periodicals, Inc. 相似文献
16.
Nina Chernyavskaya 《Mathematische Nachrichten》2002,243(1):5-18
We consider a boundary value problem where f(x) ∈ Lp(R), p ∈ [1,∞] (L∞(R) ≔ C(R) and 0 ≤ q(x) ∈ Lloc1( R). Boundary value problem (0.1) is called correctly solvable in the given space Lp(R) if for any f(x) ∈ Lp(R) there is a unique solution y(x) ∞ Lp(R) and the following inequality holds with absolute constant c(p) ∈ (0,∞). We find criteria for correct solvability of the problem (0.1) in Lp(R). 相似文献
17.
Giovanni Di Lena Davide Franco Mario Martelli Basilio Messano 《Mediterranean Journal of Mathematics》2011,8(4):473-489
The main purpose of this paper is to investigate dynamical systems
F : \mathbbR2 ? \mathbbR2{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that
f : \mathbbR2 ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x0 = (x
0, y
0), such that the orbit
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