Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

ITERATIVE PROCESS TO -HEMICONTRACTIVE OPERATOR AND-STRONGLY ACCRETIVE OPERATOR EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＥｂｅａｎａｒｂｉｔｒａｒｙＢａｎａｃｈｓｐａｃｅ ,Ｅ ｂｅｉｔｓｄｕａｌｓｐａｃｅａｎｄ〈ｘ ,ｆ 〉ｂｅｔｈｅｇｅｎｅｒａｌｉｚｅｄｄｕａｌｉｔｙｐａｉｒｉｎｇｂｅｔｗｅｅｎｘ∈Ｅａｎｄｆ ∈Ｅ .ＴｈｅｍａｐｐｉｎｇＪ :Ｅ→ 2 Ｅ ｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ ∈Ｅ :〈ｘ ,ｆ 〉 =‖ｆ ‖‖ｘ‖ ,‖ｆ ‖ =‖ｘ‖ｉｓｃａｌｌｅｄｔｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇ .ＩｆＥ ｉｓｓｔｒｉｃｔｌｙｃｏｎｖｅｘ ,ｔｈｅｎＪｉｓｓｉｎｇｌｅ＿ｖａｌｕｅｄ .Ｗｅｓｈ…     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｌｌｉｐｔｉｃｓｙｓｔｅｍ(1λ)　-Δｕ=ｆ(λ,ｘ,ｕ)-ｖ　　(ｉｎΩ),-Δｖ=δｕ-γｖ(ｉｎΩ),ｕ=ｖ=0(ｏｎΩ),ｗｈｅｒｅΩｉｓａｓｍｏｏｔｈｂｏｕｎｄｅｄｄｏｍａｉｎｉｎＲＮ(Ｎ≥2)ａｎｄλｉｓａｒｅａｌｐａｒａｍｅｔｅｒ.Ｔｈｅｓｏｌｕｔｉｏｎｓ(ｕ,ｖ)ｏｆｔｈｉｓｓｙｓｔｅｍｒｅｐｒｅｓｅｎｔｓｔｅａｄｙｓｔａｔｅｓｏｌｕｔｉｏｎｓｏｆｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｓｙｓｔｅｍｓｄｅｒｉｖｅｄｆｒｏｍｓｅｖｅｒａｌａｐ…     

A class of three-level explicit schemes with higher stability properties for a dispersive equation ut=auxxx

In this paper, a class of three level explicit schemes for a dispersive equation u_t=au_xxx with stability condition |r|=|α|Δt/(Δx)³≤2.382484, are considered. The stability condition for this class of schemes is much better than |r|≤0.3849 in [1], [2] and |r|≤0.701659 in [3], and |r|≤1.1851 in [4].     

On the perturbational global attractivity of nonautomous delay differential equations

Consider the perturbed nonautonomous linear delay differential equation x(t) = - a(t)x(t-τ) + F(t, x₁, t ⩾ 0 where x₁(s)=x(t+s) for −δ≤s≤0. Suppose that a(t) ∈ C([0,∞), (0,∞)), τ≥0,F:[0, ∞) x C[−δ,0] → R is a continuous functions and F(t,0) ≡ 0. Here C[−δ,0] is the space of continuous functions Φ: [−δ,0] → R with ∥Φ∥<H for the norm | Φ |, where |·| is any norm in R and 0<H≤+∞. Most of the known papers [1–5,7] have been concerned with the local or global asymptotic behavior of the zero solution of Eq. (*) when a(t) is independent of t i. e., a(t) is autonomous. The aim in this paper is to derive the sufficient conditions for the global attractivity of the zero solution of of Eq. (*) When a(t) is nonautomous. Our results, which extend and improve the known results, are even “sharp”. At the same time, the method used in this paper can be applicable to the perturbed nonlinear equation. Project supported by the Natural Science Foundation of Hunan     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

Periodicity and strict oscillation for generalized lyness equations

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎｘｎ 1=ｘｎ(ａ ｂｘｎ)ｘｎ- 1,　　ｎ=0 ,1 ,2 ,… ,(1 )ｗｈｅｒｅａ ,ｂ∈ [0 ,∞ )ｗｉｔｈａ ｂ>0 (2 )ａｎｄｗｈｅｒｅｔｈｅｉｎｉｔｉａｌｖａｌｕｅｓｘ- 1ａｎｄｘ0 ａｒｅａｒｂｉｔｒａｒｙｐｏｓｉｔｉｖｅｎｕｍｂｅｒｓ.Ｅｑ .(1 )ｉｓｒｅｇａｒｄｅｄａｓａｇｅｎｅｒａｌｉｚｅｄＬｙｎｅｓｓｅｑｕａｔｉｏｎｂｙＧ .Ｌａｄａｓｉｎ [1 ] .Ｏｂｖｉｏｕｓｌｙ ,ｕｎｄｅｒｃｏｎｄｉｔｉｏｎ (2 ) ,…     

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      11. Stability of flexural-gravity waves and quadratic interactions      A. V. Marchenko 《Fluid Dynamics》1999,34(1):78-86 The processes leading to strong distortion of the amplitude of a periodic flexural-gravity wave propagating in a fluid layer of finite depth beneath an infinite thin elastic plate are investigated. It is shown that three-wave resonance with noise harmonics of the flexural-gravity waves may lead to instability of a wave with the wave vector k w , where |k w |<k min. Depending on |k w |, either two copropagating noise harmonics or two noise harmonics whose wave vectors make a nonzero angle with the vector k w may be most strongly amplified during the initial instants of time. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–101, January–February, 1999. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-010-1746) and the International Association for Assistance and Cooperation with Scientists from the Independent States of the former Soviet Union and the Russian Foundation for Basic Research (INTAS-RFBR 95-0435).      12. Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations      李仁贵  刘立山 《应用数学和力学(英文版)》2001,22(4):495-500 ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…      13. MAXIMAL ELEMENTS FOR G-B-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅱ)      Ding Xie-ping 《应用数学和力学(英文版)》2003,24(9):1017-1024 By applying existence theorems of maximal elements for a family of GB-majorized mappings in a product space of G-convex spaces, some coincidence theorem, Fan-Browder type fixed point theorem and some existence theorems of solutions for a system of minimax inequalities are proved under noncompact setting of G-convex spaces. These theorems improve and generalize many important known results in literature. Contributed by Duping Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Duping Xie-ping (1938∼), Professor (E-mail: dingxip@sichu.edu.cn)      14. Isolated Singularity for the Stationary Navier—Stokes System      H. J. Choe 《Journal of Mathematical Fluid Mechanics》2000,2(2):151-184 In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in BR \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|-(n - 1)/2) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L2n/(n - 1)(BR) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? Lb(BR) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.      15. Singular Limit and Homogenization for Flame Propagation in Periodic Excitable Media      Luis A.?CaffarelliEmail author  Ki-Ahm?Lee  Antoine?Mellet 《Archive for Rational Mechanics and Analysis》2004,172(2):153-190 This paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media. The model involves two parameters: the period of the medium |L| and a singular parameter related to the activation energy. The existence of pulsating travelling fronts for fixed and |L| was proved by Berestycki & Hamel in [BH]. In the present paper, we investigate the behaviour of such solutions when More precisely, we establish that pulsating travelling fronts behave like travelling waves, when the period |L| is small and . We also study the convergence of the solution, as goes to zero (and |L| is fixed), toward a solution of a free boundary problem.      16. Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues      张振宇 《应用数学和力学(英文版)》2014,35(4):437-452 An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2,, λrof the matrix satisfy |λ1| |λr| and |λs| |λs+1|(s r-1), then associated with any eigenvalue λi(i s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi/λs+1|q+1, where the approximate method only uses the eigenpairs corresponding to λ1, λ2,, λs. A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.      17. On the Homoclinic Orbits in a Class of Two-Degree-of-Freedom Systems under the Resonance Conditions      王茂南  徐振源 《应用数学和力学(英文版)》2001,22(3):340-352 ＩｎｔｒｏｄｕｃｔｉｏｎＴｗｏ＿ｄｅｇｒｅｅ＿ｏｆ＿ｆｒｅｅｄｏｍｓｙｓｔｅｍｓｈａｖｉｎｇｃｕｂｉｃｎｏｎｌｉｎｅａｒｉｔｉｅｓａｒｅｅｘｔｅｎｓｉｖｅｌｙｕｓｅｄｉｎｐｈｙｓｉｃｓ,ｍｅｃｈａｎｉｃｓ.Ｆｏｒｅｘａｍｐｌｅ :ｔｈｅｌａｒｇｅ＿ａｍｐｌｉｔｕｄｅｖｉｂｒａｔｉｏｎｓｏｆｓｔｒｉｎｇｓ,ｂｅａｍｓ,ｍｅｍｂｒａｎｅｓａｎｄｐｌａｔｅｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎ＿ｉｓｏｌａｔｉｏｎｓｙｓｔｅｍｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎａｂｓｏｒｂｅｒｓ,ｔｈｅｍｏｔｉｏｎｏｆｓｐｈｅ…      18. Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations      Bernold Fiedler  Hiroshi Matano 《Journal of Dynamics and Differential Equations》2007,19(4):867-893 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.      19. On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows      Mahesh Nerurkar 《Journal of Dynamics and Differential Equations》2011,23(3):451-473 Consider the class of C r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracpnqn| ￡ C exp (-qr+1+kn)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracPnqn{\frac{P_n}{q_n}} is some sequence of irreducible fractions.      20. The MLP method for subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems      唐驾时 《应用数学和力学(英文版)》2000,21(10):1153-1160 ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｏｆｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｓｉｓｓｔｉｌｌｔｈｅｍａｉｎｔａｓｋｉｎｔｈｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃａｌｒｅｇｉｏｎ .ＴｈｅｒｅａｒｅｍａｔｕｒｅｍｅｔｈｏｄｓｉｎｃｌｕｄｉｎｇｔｈｅｄｅｖｅｌｏｐｅｄＫＢＭｍｅｔｈｏｄ[1],ｔｈｅｍｏｄｉｆｉｅｄＬＰｍｅｔｈｏｄ[2 ]ａｎｄｔｈｅｓｔｒｏｂｏｓｃｏｐｉｃｍｅｔｈｏｄ[3]ｅｔａｌ.Ｈｏｗｅｖｅｒ,ｗｈｅｎｔｈｅｓｅｍｅｔｈｏｄｓａｒｅａ…

Stability of flexural-gravity waves and quadratic interactions

The processes leading to strong distortion of the amplitude of a periodic flexural-gravity wave propagating in a fluid layer of finite depth beneath an infinite thin elastic plate are investigated. It is shown that three-wave resonance with noise harmonics of the flexural-gravity waves may lead to instability of a wave with the wave vector k _w , where |k _w |<k _min. Depending on |k _w |, either two copropagating noise harmonics or two noise harmonics whose wave vectors make a nonzero angle with the vector k _w may be most strongly amplified during the initial instants of time. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–101, January–February, 1999. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-010-1746) and the International Association for Assistance and Cooperation with Scientists from the Independent States of the former Soviet Union and the Russian Foundation for Basic Research (INTAS-RFBR 95-0435).     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

MAXIMAL ELEMENTS FOR G-B-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅱ)

By applying existence theorems of maximal elements for a family of G_B-majorized mappings in a product space of G-convex spaces, some coincidence theorem, Fan-Browder type fixed point theorem and some existence theorems of solutions for a system of minimax inequalities are proved under noncompact setting of G-convex spaces. These theorems improve and generalize many important known results in literature. Contributed by Duping Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Duping Xie-ping (1938∼), Professor (E-mail: dingxip@sichu.edu.cn)     

Isolated Singularity for the Stationary Navier—Stokes System

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in B_R \{0} B_R \setminus \{0\} , n 3 3 n \geq 3 and |u(x)| = o (|x|^{-(n - 1)/2}) |u(x)| = o\,(|x|^{-(n - 1)/2}) as |x| ? 0 |x| \to 0 or u ? L^{2n/(n - 1)}(B_R) u \in L^{2n/(n - 1)}(B_R) , then (u, p) is a distribution solution and if in addition, u ? L^b(B_R) u \in L^{\beta}(B_R) for some b > n \beta > n then ( u, p) is smooth in BR.     

Singular Limit and Homogenization for Flame Propagation in Periodic Excitable Media

This paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media. The model involves two parameters: the period of the medium |L| and a singular parameter related to the activation energy. The existence of pulsating travelling fronts for fixed and |L| was proved by Berestycki & Hamel in [BH]. In the present paper, we investigate the behaviour of such solutions when More precisely, we establish that pulsating travelling fronts behave like travelling waves, when the period |L| is small and . We also study the convergence of the solution, as goes to zero (and |L| is fixed), toward a solution of a free boundary problem.     

Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ1, λ2,, λrof the matrix satisfy |λ1| |λr| and |λs| |λs+1|(s r-1), then associated with any eigenvalue λi(i s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λi/λs+1|q+1, where the approximate method only uses the eigenpairs corresponding to λ1, λ2,, λs. A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.     

On the Homoclinic Orbits in a Class of Two-Degree-of-Freedom Systems under the Resonance Conditions

ＩｎｔｒｏｄｕｃｔｉｏｎＴｗｏ＿ｄｅｇｒｅｅ＿ｏｆ＿ｆｒｅｅｄｏｍｓｙｓｔｅｍｓｈａｖｉｎｇｃｕｂｉｃｎｏｎｌｉｎｅａｒｉｔｉｅｓａｒｅｅｘｔｅｎｓｉｖｅｌｙｕｓｅｄｉｎｐｈｙｓｉｃｓ,ｍｅｃｈａｎｉｃｓ.Ｆｏｒｅｘａｍｐｌｅ :ｔｈｅｌａｒｇｅ＿ａｍｐｌｉｔｕｄｅｖｉｂｒａｔｉｏｎｓｏｆｓｔｒｉｎｇｓ,ｂｅａｍｓ,ｍｅｍｂｒａｎｅｓａｎｄｐｌａｔｅｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎ＿ｉｓｏｌａｔｉｏｎｓｙｓｔｅｍｓ ,ｄｙｎａｍｉｃｖｉｂｒａｔｉｏｎａｂｓｏｒｂｅｒｓ,ｔｈｅｍｏｔｉｏｎｏｆｓｐｈｅ…     

Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations

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.     

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

The MLP method for subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｏｆｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｓｉｓｓｔｉｌｌｔｈｅｍａｉｎｔａｓｋｉｎｔｈｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃａｌｒｅｇｉｏｎ .ＴｈｅｒｅａｒｅｍａｔｕｒｅｍｅｔｈｏｄｓｉｎｃｌｕｄｉｎｇｔｈｅｄｅｖｅｌｏｐｅｄＫＢＭｍｅｔｈｏｄ[1],ｔｈｅｍｏｄｉｆｉｅｄＬＰｍｅｔｈｏｄ[2 ]ａｎｄｔｈｅｓｔｒｏｂｏｓｃｏｐｉｃｍｅｔｈｏｄ[3]ｅｔａｌ.Ｈｏｗｅｖｅｒ,ｗｈｅｎｔｈｅｓｅｍｅｔｈｏｄｓａｒｅａ…