共查询到20条相似文献,搜索用时 46 毫秒
1.
Long-time Asymptotic Behavior for the Derivative Schr¨odinger Equation with Finite Density Type Initial Data* 下载免费PDF全文
In this paper, the authors apply ? steepest descent method to study the Cauchy problem for the derivative nonlinear Schr¨odinger equation with finite density type initial data iqt + qxx + i(|q|2q)x = 0,q(x, 0) = q0(x),where lim/x→±∞ q0(x) = q± and |q±| = 1. Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schr¨odinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region ξ =x/t with |ξ + 2| < 1, the long time asymptotic is given by q(x, t) = T (∞)?2qrΛ(x, t) + O(t?3/4 ),in which the leading term is N(I) solitons, the second term is a residual error from a ? equation. For the region |ξ + 2| > 1, the long time asymptotic is given by q(x, t) = T (∞)?2qrΛ(x, t) ? t?1/2 if11 + O(t?3/4 ),in which the leading term is N(I) solitons, the second t?1/2 order term is soliton-radiation interactions and the third term is a residual error from a ? equation. These results are verification of the soliton resolution conjecture for the derivative Schr¨odinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively. 相似文献
2.
本文考虑当初值u0 ∈ Lp 时,四阶非线性Schr¨odinger方程的柯西问题的整体适定性. 在p /= 2的情形下,关于该柯西问题的解的存在性结论较少. 受已有二阶Schr¨odinger方程启发,借助一个关于Lp初值的分解引理和Strichartz估计,在 p > 2 且非线性项指数满足一定条件下, 本文得到了该四阶非线性Schr¨odinger方程的柯西问题的解的整体存在性. 相似文献
3.
On the Radius of Analyticity of Solutions to 3D Navier-Stokes System with Initial Data in Lp 下载免费PDF全文
Given initial data u0 ∈ Lp(R3) for some p in[3, 18/5[, the auhtors ?rst prove that 3D incompressible Navier-Stokes system has a unique solution u = uL+v with uL def = et?u0 and v ∈ e L∞([0, T]; ˙ H5/2 ? 6/p ) ∩ L1(]0, T[; ˙H9/2 ? 6/p ) for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631– 1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in ˙Hs(R3) for s∈[1/2,3/2[. 相似文献
4.
In this paper it is proved that local fundamental solution exists in some space Wm(Hn) (m∈Z), if the left invariant differential operator on the Heisenberg group Hn satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on Hn. If there exist R≥0, r,s∈R and operators {Bλ|λ∈ΓR} ∈Vs(ΓR, Mr) such that, for almost all λ∈ΓR, Bλ is the right inverse of Ⅱλ(L), then there exists E∈Wm(Hn) (when m≥0 or m even) or E∈Wm-1(Hn) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and Cλ≥ C|λ|x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2). 相似文献
5.
In this paper, the authors discuss a generalization of Lappan’s theorem to higher dimensional complex projective space and get the following result: Let f be a holomorphic mapping of ? into Pn(C), and let H1, · · · , Hq be hyperplanes in general position in Pn(C).Assume that sup {(1 ? |z|2)f?(z) : z ∈ q[ j=1 f?1(Hj )o < ∞,if q ≥ 2n2 + 3, then f is normal. 相似文献
6.
一类具不变性质的变系数偏微分方程的特解 总被引:1,自引:0,他引:1
In this paper, a few properties of general patial differential operator P beinginvariant with respect to the form F(|x|pp+|y|qq-|z|rr) are studied, where x∈Rn,y∈ Rm, z∈Rl.and explicit formulas are given for certain solution of the equation Pu=Aδ with P being a differential operator with power function coefficientswhich preserves the form (|x|pp+|y|qq-|z|rr)1/v for arbitrary even integers p, q, r,and odd integers v. 相似文献
7.
Multiplicity of Solutions of the Weighted p-Laplacian with Isolated Singularity and Diffusion Suppressed by Convection 下载免费PDF全文
In this paper, the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and di?usion suppressed by convection ?div(|x|α|?u|p?2?u) + λ 1/|x|β |?u|p?2?u · x = |x|γg(|x|) in B \ {0} subject to nonlinear Robin boundary value condition |x|α|?u|p?2?u · ?n = A ? ρu on ?B,where λ > 0, B ? RN(N ≥ 2) is the unit ball centered at the origin, α > 0, p > 1, β ∈ R, γ > ?N, g ∈ C([0,1]) with g(0) > 0, A ∈ R, ρ > 0 and ?n is the unit outward normal. The same problem with di?usion promoted by convection, namely λ ≤ 0, has already been discussed by the last two authors (Song-Yin (2012)), where the existence, nonexistence and classi?cation of singularities for solutions are presented. Completely di?erent from [Song, H. J. and Yin, J. X., Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection, Math. Meth. Appl. Sci., 35, 2012, 1089–1100], in the present case λ > 0, namely the di?usion is suppressed by the convection, non-singular solutions are not only existent but also may be in?nite which vary according only to the values of solutions at the isolated singular point. At the same time, the singular solutions may exist only if the di?usion dominates the convection. 相似文献
8.
Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws 下载免费PDF全文
Yi ZHOU 《数学年刊B辑(英文版)》2022,43(4):499-508
Let u(t, x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solutions belong to, for example, H1 because by Sobolev embedding theorem H1 functions are H¨older continuous. However, the author notes that from any point (t, x), he can draw a generalized characteristic downward which meets the initial axis at y = α(t, x). If he regards u as a function of (t, y), it indeed belongs to H1 as a function of y if the initial data belongs to H1 . He may call this generalized persistence (of high regularity) of the entropy weak solutions.The main purpose of this paper is to prove some kinds of generalized persistence (of high regularity) for the scalar and 2 × 2 Temple system of hyperbolic conservation laws in one space dimension. 相似文献
9.
该文分析了四阶椭圆方程△2u=|x|aup-1,x∈Ω; u=\Delta u=0 , x ∈аΩ, (Ω表示Rn中以原点为中心的球)基态解的集中性态,并证明了当p趋近于 2*=\frac{2n}{n-4} (n>4)时基态解up集中在Ω的边界附近. 相似文献
10.
Almansi-Type Decomposition Theorem for Bi-k-regular Functions in the Clifford Algebra Cl2n+2,0(R)* 下载免费PDF全文
Lixia LIU 《数学年刊B辑(英文版)》2022,43(2):253-264
Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domain Ω Rn+1 × Rn+1 centered at the origin with values in the Clifford algebra Cl2n+2,0(R) is proved. As a corollary, Almansi-type decomposition theorem for biharmonic functions of degree k is given. 相似文献
11.
The authors show that if Θ = (θjk) is a 3 × 3 totally irrational real skewsymmetric matrix, where θjk ∈ [0, 1) for j, k = 1, 2, 3, then for any ε > 0, there exists δ > 0 satisfying the following: For any unital C*-algebra A with the cancellation property,strict comparison and nonempty tracial state space, any four unitaries u1, u2, u3, w ∈ A such that (1) ukuj - e2πiθjk ujukk < δ, wujw-1 = u-1j, w2 = 1A for j, k = 1, 2, 3; (2)τ (aw) = 0 and τ ((ukujuk*uj* )n ) = e2πinθjk for all n ∈ N, all a ∈ C*(u1, u2, u3), j, k = 1, 2, 3 and all tracial states τ on A, where C*(u1, u2, u3) is the C*-subalgebra generated by u1, u2 and u3, there exists a 4-tuple of unitaries u1, u2, u3, w in A such that ukuj = e2πinθjk ukuj, w uj w-1 = u-1 j, w2 = 1A and k uj - ujk < ε, k w - wk < ε for j, k = 1, 2, 3. The above conclusion is also called that the rotation relations of three unitaries with the flip action is stable under the above conditions. 相似文献
12.
在这篇文章中, 作者研究涉及凹凸非线性项的Kirchhoff型问题-(a + b ∫R3|▽u|2dx) Δu + λV (x)u = μf(x)|u|q?2u + |u|p?2u, x ∈ R3,u ∈ H1(R3),其中a,b > 0 是常数, λ, μ > 0 是参数, 1 < q < 2, 4 < p < 6 且 V 是一个非负连续位势. 在f(x) 和 V 的合适条件下,此问题正解的存在性和集中性能够通过Nehari 流形和Ekeland 变分原理得到. 相似文献
13.
许兴业 《数学物理学报(A辑)》2009,29(1):87-93
该文以Schauder-Tychonoff不动点定理为工具,建立了一类Rn上带奇异性的非线性双调和方程 Δ2u=f(|x|, u,| u|)u-β (x ∈ Rn, n ≥ 3, β > 0) 正的径向对称整体解的存在性定理,并给出了解的有关性质,所得的结果丰富和发展了文献[1--5]的结果. 相似文献
14.
通过建立Heisenberg群上无穷远处的集中列紧原理, 研究了如下$p$ -次Laplace方程
-ΔH, pu=λg(ξ)|u|q-2u+f (ξ)|u|p*-2u,在Hn上,
u∈ D1, p(Hn),
其中ξ∈Hn,λ∈R,1
j, 且m, j为整数. 相似文献
15.
In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??Bu + λu = a(z)f(u), u ≥ 0 in RN+,where N = n + 1 ≥ 3, λ > 0, z = t,x1,· · · ,xn, and ?B = (t?t)2 + ?2x1 + · · · + ?2xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f. 相似文献
16.
It is proved that for each 2n×2n symplectic matrix S, there exists con-tinuous linear map Fs: S′(Rn)→S′(Rn), unique up to a constant factor, such that(?) Fs is called the generalized Fourier transformation. Some properties and applica-tions of Fs are obtained. In especial , the lower and upper boundedness of Fs in Hm(Rn) is proved and a new proof of L. Hormander Theorem is given. 相似文献
17.
Let D be a domain in the n-dimensional Euclidean space Rn, n ≥ 2, and let E be a compact in D. The paper presents conditions on the compact E under which any homeomorphic mapping
f = D ∖ E → Rn can be extended to a continuous mapping f = D → Rˉn = Rn ⋃ {∞}. These conditions define the class of NCS-compacts, which, for n = 2, coincides with the class of topologically removable
compacts for conformal and quasiconformal mappings. Bibliography: 11 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 213–220. 相似文献
18.
该文研究椭圆型方程
{Δpu+m|u|p-2u-Δqu+n|u|q-2u=g(x, u), x∈RN,
u∈ W1, p(RN)∩W1, q(RN)
弱解在全空间RN上的衰减性, 其中m, n ≥ 0, N≥3, 1 < q < p < N, g(x, u)关于u满足类渐近线性. 证明了该方程的
弱解在无穷远处关于|x|呈指数衰减性. 相似文献
19.
Mengjie ZHANG 《数学年刊B辑(英文版)》2022,43(3):425-442
In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extremal functions on a compact Riemann surface (Σ, g) with smooth boundary ?Σ. Explicitly, let λ1(?Σ) = inf u∈W1,2 (Σ,g),R ?Σ udsg=0,u6≡0 R Σ(|?gu|2 + u2 )dvg R ?Σ u2 dsg and H = n u ∈ W1,2 (Σ, g) : Z Σ(|?gu|2 + u2 )dvg ? α Z ?Σ u2dsg ≤ 1 and Z ?Σ u dsg = 0o ,where W1,2 (Σ, g) denotes the usual Sobolev space and ?g stands for the gradient operator.By the method of blow-up analysis, we obtain sup u∈H Z ?Σ e πu2 dsg ( < +∞, 0 ≤ α < λ1(?Σ),= +∞, α ≥ λ1(?Σ).Moreover, the author proves the above supremum is attained by a function uα ∈ H∩C∞(Σ)for any 0 ≤ α < λ1(?Σ). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363–386], [Yang,Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary,Pacific J. Math., 227, 2006, 177–200] and [Yang, Y., Extremal functions for TrudingerMoser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015,3161–3193] 相似文献
20.
Shi Yang Zheng 《数学学报(英文版)》2014,30(10):1735-1747
In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents. 相似文献