The small dispersion limit of the Korteweg-de Vries equation. III

In Parts I and II we have derived explicit formulas for the distribution limit u of the solution of the KdV equation as the coefficient of u_xxx tends to zero. This formula contains n parameters β_1, …, β_n whose values, as well as whose number, depends on x and t. In Section 4 we have shown that for t<tb, n=1, and the value of β, was determined. In Section 5 we have shown that the parameters β_i satisfy a nonlinear system of partial differential equations. In Part III, Section 6 we show that for t large, n=3, and we determine the asymptotic behavior of β_1, β_2, β_3, and of u and u ², for t large. The explicit formulas show that u and u ² are O(t^?1) and O(t^-2) respectively (see formulas (6.2) and (6.24)). In Section 7 we study initial data whose value tends to zero as x→+∞, and to -1 as x→?∞. If one accepts some plausible guesses about the behavior of solutions with such initial data, we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution: . The function s(ζ) is expressible in terms of complete elliptic integrals; a similar formula is derived for U ². In Section 8 we indicate how to extend the treatment of this series of papers to multihumped (but still negative) initial data.     

On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ℝ⁺ (where * denotes convolution over (−∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^−α, b(t)=t^−β, and g(ξ)∼sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ−2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

A Threshold Result for a Non-local Parabolic Equation

In this paper, we consider the Cauchy problem: (ECP) u_t−Δu+p(x)u=u(x,t)∫u²(y,t)/∣x−y∣dy; x∈ℝ³, t>0, u(x, 0)=u₀(x)⩾0 x∈ℝ³, (0.2) The stationary problem for (ECP) is the famous Choquard–Pekar problem, and it has a unique positive solution ū(x) as long as p(x) is radial, continuous in ℝ³, p(x)⩾ā>0, and lim_{∣x∣→∞}p(x)=p¯>0. In this paper, we prove that if the initial data 0⩽u₀(x)⩽(≢)ū(x), then the corresponding solution u(x, t) exists globally and it tends to the zero steady-state solution as t→∞, if u₀(x)⩾(≢)ū(x), then the solution u(x,t) blows up in finite time. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded. This project was supported by the Program for New Century Excellent Talents of Ministry of Education of China and the National Natural Science Foundation of China (Grant No. 10671020 and 10301006).     

Periodic solutions for some second order differential equations with singularity

In this paper, we will prove the existence of infinitely many harmonic and subharmonic solutions for the second order differential equation ẍ + g(x) = f(t, x) using the phase plane analysis methods and Poincaré–Birkhoff Theorem, where the nonlinear restoring field g exhibits superlinear conditions near the infinity and strong singularity at the origin, and f(t, x) = a(t)x ^γ + b(t, x) where 0 ≤ γ ≤ 1 and b(t, x) is bounded.     

Evolution Semigroups and Product Formulas for Nonautonomous Cauchy Problems

In this paper, we study nonautonomous Cauchy problems (NCP) {^{u̇(t) = A(t)u(t)}_{u(s) = x ∈ X} for a family of linear operators (A(t))_t∈I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of (NCP). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals ∫^t_s A(τ) dτ always define generators. This yields another product formula.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

On the partially microlocal hypoellipticity for a class of totally characteristic operators with characteristics of constant multiplicity

In this paper, I study the microlocal hypoellipticity for a class of totally characteristic operators (1.1). My main result is as follows: Under the conditions (I), (II), if the indicial operator of (1.1) is microlocally hypoelliptic in the complement ofWF _x(Pu(t,·)) for anyu(t,x)∈C _b ^∞ ([0,T], ℰ),t∈[0,T], λ∈ℤ, then the operator (1.1) is microlocally hypoelliptic in the variablex. Supported by the Natural Science Foundation and Young Men's Science Foundation of Academia Sinica     

Представление и приб лижение периодических функц ий многих переменных с заданным смешанным модулем непрепывнос ти

In this paper the author studies classesH _q of periodic functions of several variables whose mixed moduli of continuity do not exceed a given modulus of continuity (t ₁ ...,t _d ). Necessary and sufficient conditions of belonging of a functionf(x ₁, ...,x _d ) to the classH _q are considered (Theorem 1). These necessary and sufficient conditions are proved under some additional assumptions on (t ₁, ...,t _d ). It is shown that additional assumptions cannot be omitted (Theorem 3). Besides, the estimates of best approximations of classesH _q with some special (t ₁, ...,t _d ) are given (Theorems 4 and 5).     

Existence of quantum diffusions

Summary A quantum diffusion (A, A, j) comprises of unital *-algebras A and A and a family of identity preserving *-homomorphisms j=(j _t : t0) from A into A. Also j satisfies a system of quantum stochastic differential equations dj _t (x ₀=j _t( _j ⁱ (x ₀))dM _i ⁱ , j ₀(x ₀)=x ₀I for all x ₀A where _j ⁱ , 1i, jN are maps from A to itself and are known as the structure maps. In this paper an existence proof is given for a class of quantum diffusions, for which the structure maps are bounded in the operator norm sense. A solution to the system of quantum stochastic differential equations is first produced using a variation of the Picard iteration method. Another application of this method shows that the solution is a quantum diffusion.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems

The general problem studied has as a prototype the full non-linear Navier-Stokes equations for a slightly viscous compressible fluid including the heat transfer. The boundaries are of inflow-outflow type, i.e. non-characteristic, and the boundary conditions are the most general ones with any order of derivatives. It is assumed that the uniform Lopatinsky condition is satisfied. The goal is to prove uniform existence and boundedness of solution as the viscosity tends to zero and to justify the boundary layer asymptotics. The paper consists of two parts. In Part I the linear problem is studied. Here, uniform lower and higher order tangential estimates are derived and the existence of a solution is proved. The higher order estimates depend on the smoothness of coefficients; however this smoothness does not exceed the smoothness of the solution. In Part II the quasilinear problem is studied. It is assumed that for zero viscosity the overall initial-boundary value problem has a smooth solutionu ₀ in a time interval 0≦t≦T ₀. As a result the boundary laye, is weak and is uniformlyC ¹ bounded. This makes the linear theory applicable. an iteration scheme is set and proved to converge to the viscous solution. The convergence takes place for small viscosity and over the original time interval 0≦t≦T ₀.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

On a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration

We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than e ^–t , > 0, as t –.     

On the Blow-up Criterion of Smooth Solutions to the MHD System in BMO Space

In this paper we study the blow-up criterion of smooth solutions to the incompressible magnetohydrodynamics system in BMO space. Let （u（x,t）,b（x,t）） be smooth solutions in （0, T）. It is shown that the solution （u（x, t）, b（x, t）） can be extended beyond t = T if （u（x,t）, b（x, t）） ∈ L^1 （0, T; BMO） or the vorticity （rot u（x, t）, rot b（x, t）） ∈ L^1 （0, T; BMO） or the deformation （Def u（x, t）, Def b（x, t）） ∈ L^1 （0, T; BMO）.     

Smooth solutions to some differential-difference equations of neutral type

The paper is devoted to the scalar linear differential-difference equation of neutral type

Some reproducing identities for families of mean values

Gini, Lehmer, Beckenbach, and others studied the meanG _s (a, b) = (a ^s +b ^s )/(a ^{s 1} +b ^s-1 ) We proveTheorem 1 The identity (ina, b)G _s (G _t ,G _u ) =G _v holds if and only if (s, t, u, v) is (s, t, t, t) (the trivial solution) or one of (1, 1 –k, 1 +k, 1), (1/2, 1 –k, k, 1/2), or (0,–k, k, 0) (the exotic solutions,k is any real number)Theorem 2 IfP _s (a, b) is the power mean [(a ^s +b ^s )/2]^1/s , thenP _s (P _t ,P _u ) =P _v has only the trivial solution (s, t, u, v) = (s, t, t, t) and the exotic solution (0,t, –t, 0) The family of meansG _s (respP _s ) includes the classical arithmetic, geometric, and harmonic means     

Regularization of a non-characteristic Cauchy problem for the heat equation

The non-characteristic Cauchy problem for the heat equation u_xx(x,t) = u₁(x,t), 0 ? x ? 1, ? ∞ < t < ∞, u(0,t) = φ(t), u_x(0, t) = ψ(t), ? ∞ < t < ∞ is regularizèd when approximate expressions for φ and ψ are given. Properties of the exact solution are used to obtain an explicit stability estimate.