共查询到20条相似文献,搜索用时 15 毫秒
1.
Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ?3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C ? r ?1+ε |t|?ε/2 for ? T 0 ≤ t < 0 and 0 < C ? < ∞ allowed to be large. We prove that v is regular at time zero. 相似文献
2.
We consider the elliptic equation ? Δu = f(u) in the whole ?2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?2m . This is a solution which changes sign in ?2m and vanishes only on the Simons cone 𝒞 = {(x 1, x 2) ∈ ? m × ? m : |x 1| = |x 2|}. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established. 相似文献
3.
4.
We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v ? v *|β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v ? v *|?1(v ? v *)·σ and σ ∈ 𝕊 d?1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 d?1, we prove that there exists a > 0 such that moments with weight exp (amin {t, 1}|v|β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients. 相似文献
5.
Ricardo Weder∗ 《偏微分方程通讯》2013,38(5):675-691
We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|?ρ), A(x) = O(|x|?ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L 2(K) where K is any connected bounded open set in ? n ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V 1, A 1) and (V 2, A 2) in ? n , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V j and the magnetic fields F j : = curl A j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field. 相似文献
6.
The distinguishing number D(G) of a graph is the least integer d such that there is a d‐labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G ≠ K2, K3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [Electron J Combin, 12 ( 1 ), #N17] on powers of prime graphs, and results of Klav?ar and Zhu [Eu J Combin, to appear]. More generally, we also prove that d(G □ H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2|H| ? |H|. Under additional conditions similar results hold for powers of graphs with respect to the strong and the direct product. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 250–260, 2006 相似文献
7.
Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G. 相似文献
8.
Rubén A. Hidalgo 《Journal of Difference Equations and Applications》2013,19(9):953-969
Let Q be a m × m real matrix and f j : ? → ?, j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x j and f j (x j ), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph 𝒢 is just any linear operator D:C 0(𝒢) → C 0(𝒢), where C 0(𝒢) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:V × ? → ?, one has associated a partial difference equation Dμ = F(v,μ), and one searches for solutions μ ∈ C 0(𝒢). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (𝒢, d), where 𝒢 is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator ?2. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91–107) was considered the nonlinear elliptic partial difference equations ?2 u = F(u), for the metric d = 1. 相似文献
9.
Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures
Bang-Yen Chen 《Annals of Global Analysis and Geometry》2010,38(2):145-160
Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature K
D
of a surface in Euclidean 4-space
\mathbb E4{\mathbb {E}^4} satisfy K + |K
D
| ≤ H
2, where H
2 is the squared mean curvature. A surface in
\mathbb E4{\mathbb {E}^4} is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in
\mathbb E4{\mathbb {E}^4} form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely
classify Wintgen ideal surfaces in
\mathbb E4{\mathbb E^4} satisfying |K| = |K
D
| identically. 相似文献
10.
Piotr Niemiec 《Integral Equations and Operator Theory》2011,71(2):151-160
Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator
T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?K|2 £ á|T|x, x?Há|T*|y, y?K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function
f: \mathbbR+ ? \mathbbR+{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined. 相似文献
11.
Suppose D is a subset of all positive integers. The distance graph G(Z, D) with distance set D is the graph with vertex set Z, and two vertices x and y are adjacent if and only if |x − y| ≡ D. This paper studies the chromatic number χ(Z, D) of G(Z, D). In particular, we prove that χ(Z, D) ≤ |D| + 1 when |D| is finite. Exact values of χ(G, D) are also determined for some D with |D| = 3. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 287–294, 1997 相似文献
12.
《偏微分方程通讯》2013,38(4):451-482
ABSTRACT We consider the Schrödinger equation in ?2, with external Yang–Mills potentials that decay exponentially as |x| → ∞. We prove that the scattering amplitude at fixed positive energy determines the potentials uniquely modulo a gauge transformation, assuming that potentials are small. 相似文献
13.
14.
Let D be a simple digraph without loops or digons. For any v ? V(D) v\in V(D) , the first out-neighborhood N+(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N++(v) of v is the set of all vertices with out-distance 2 from v. We show that every simple digraph without loops or digons contains a vertex v such that |N++(v)| 3 g|N+(v)| |N^{++}(v)|\geq\gamma|N^+(v)| , where % = 0.657298... is the unique real root of the equation 2x3 + x2 -1 = 0. 相似文献
15.
Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ? ?0, D[Γ] be the semigroup ring of Γ over D (and hence D ? D[Γ] ? D[X]), and D + X n K[X] = {a + X n g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X n K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X n K[X]. 相似文献
16.
Yanyan Li 《纯数学与应用数学通讯》1993,46(3):303-340
In this paper, we study the equation –Δu = K(x)u5 in ?3 and provide a large class of positive functions K(x) for which we obtain infinitely many positive solutions which decay at infinity at the rate of |x|?1. © 1993 John Wiley & Sons, Inc. 相似文献
17.
Mitsuhiro Miyazaki 《代数通讯》2013,41(9):2894-2900
Let K be a field, D a finite distributive lattice and P the set of all join-irreducible elements of D. We show that if {y ∈ P | y ≥ x} is pure for any x ∈ P, then the Hibi ring ? K (D) is level. Using this result and the argument of sagbi basis theory, we show that the homogeneous coordinate rings of Schubert subvarieties of Grassmannians are level. 相似文献
18.
Anders Yeo 《Journal of Graph Theory》1999,32(2):123-136
The local irregularity of a digraph D is defined as il(D) = max {|d+ (x) − d− (x)| : x ϵ V(D)}. Let T be a tournament, let Γ = {V1, V2, …, Vc} be a partition of V(T) such that |V1| ≥ |V2| ≥ … ≥ |Vc|, and let D be the multipartite tournament obtained by deleting all the arcs with both end points in the same set in Γ. We prove that, if |V(T)| ≥ max{2il(T) + 2|V1| + 2|V2| − 2, il(T) + 3|V1| − 1}, then D is Hamiltonian. Furthermore, if T is regular (i.e., il(T) = 0), then we state slightly better lower bounds for |V(T)| such that we still can guarantee that D is Hamiltonian. Finally, we show that our results are best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 123–136, 1999 相似文献
19.
Cheng He 《数学学报(英文版)》1999,15(2):153-164
By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability,
we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain
with uniformlyC
3 boundary, under the assumption that |a|
L
2(Θ) + |f|
L
1(0,∞;L
2(Θ)) or |∇a|
L
2(Θ) + |f|
L
2(0,∞;L
2(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary
conditions is also discussed.
This work is supported by foundation of Institute of Mathematics, Academia Sinica 相似文献
20.
I. Alrasasi 《代数通讯》2013,41(4):1385-1400
Let D be an integral domain with quotient field K. A Bhargava ring over D is defined to be 𝔹 x (D): = {f ∈ K[X] | ? a ∈ D, f(xX + a) ∈ D[X]}, where x ∈ D. A Bhargava ring over D is a subring of the ring of integer-valued polynomials over D. In this article, we study the prime ideal structure and calculate the Krull and valuative dimension of Bhargava rings over a general domain D. 相似文献