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     

Invariant Measure for the Markov Process Corresponding to a PDE System

In this paper, we consider the Markov process （X^∈（t）, Z^∈（t）） corresponding to a weakly coupled elliptic PDE system with a small parameter ∈ 〉 0. We first prove that （X^∈（t）, Z^∈（t）） has the Feller continuity by the coupling method, and then prove that （X^∈（t）, Z^∈（t）） has an invariant measure μ^∈（·） by the Foster-Lyapunov inequality. Finally, we establish a large deviations principle for μ^∈（·） as the small parameter e tends to zero.     

Remark on the Regularities of Kato’s Solutions to Navier-Stokes Equations with Initial Data in L^d(R^d)

Motivated by the results of J. Y. Chemin in ＂J. Anal. Math., 77, 1999, 27- 50＂ and G. Furioli et al in ＂Revista Mat. Iberoamer., 16, 2002, 605-667＂, the author considers further regularities of the mild solutions to Navier-Stokes equation with initial data uo ∈ L^d（R^d）. In particular, it is proved that if u C ∈（[0, T^＊）; L^d（R^d）） is a mild solution of （NSv）, then u（t,x）- e^vt△uo ∈ L^∞（（0, T）;B2/4^1,∞）~∩L^1 （（0, T）; B2/4^3 ,∞） for any T 〈 T^＊.     

Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L ¹(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L ^p (0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L ¹(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators if one of them generates a C ₀-semigroup. Günter Lumer in memoriam     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

Implicit Euler scheme for an abstract evolution inequality

For a triple {V, H, V*} of Hilbert spaces, we consider an evolution inclusion of the form u′(t)+A(t)u(t)+δϕ(t, u(t)) ∋ f(t), u(0) = u0, t ∈ (0, T ], where A(t) and ϕ(t, ·), t ∈ [0, T], are a family of nonlinear operators from V to V * and a family of convex lower semicontinuous functionals with common effective domain D(ϕ) ⊂ V. We indicate conditions on the data under which there exists a unique solution of the problem in the space H ¹(0, T; V)∩W _∞¹ (0, T;H) and the implicit Euler method has first-order accuracy in the energy norm.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu(m)bTu(n)ag}m,n∈p on local field are investigated. Accurately, that the canonical dual of frame {Eu(m)bTu(n)ag}m,n∈p also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu(m)bTu(n)ag}m,n∈p to be a frame for L2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L2(K) are established. An example is finally given.     

Exact solution of one boundary-value problem

We study the boundary-value perlodic problem u _tt −u _xx =F(x, t), u(0, t)=u(π, t)=0, u(x, t+T)=u(x, t), (x, t) ∈ R ². By using the Vejvoda-Shtedry operator, we determine a solution of this problem. Ternopol Pedagogical Institute, Temopol. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 998–1001, July, 1997.     

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.     

Higher order commutators for a class of rough operators

In this paper we study the (L ^p (u ^p ),L ^q (v ^q )) boundedness of the higher order commutatorsT _Ω,α,b ^m andM _Ω,α,b ^m formed by the fractional integral operatorT _Ω,α , the fractional maximal operatorM _Ω,α , and a functionb(x) in BMO(v), respectively. Our results imporve and extend the corresponding results obtained by Segovia and Torrea in 1993 [9]. The project was supported by NNSF of China.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

An integral strong law of large numbers for processes with independent increments

For a stochastically continuous stochastic process with independent increments overD[0,T], letN(t,ε) be the number of smaple function jumps that occur in the interval [0,t] of sizes less than −ε or greater than ε, where ε>0. LetM(t,ε)=EN(t,ε), and assumeM(t,0+)=∞ for 0<t≦T. If lim_{ε ↓0}(M(t,ε)/M(T,ε)) exists and is positive for eacht∈(0,T], then lim_{ε ↓0}(N(t,ε)/M(T,ε)) for allt∈(0,T] with probability one. The research of Howard G. Tucker was supported in part by the National Science Foundation, Grant No. MCS76-03591A01.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

Interior transition layers of solutions to the perturbed elliptic Sine-Gordon equation on an interval

We consider the perturbed elliptic Sine-Gordon equation on an interval-u″t+γsinu(t)=μf(u(t)),t ∈I := (-T, T),u(t) > 0,t ∈I,u(±T)=0 where λ, μ>0 are parameters andT>0 is a constant. By applying variational methods subject to the constraint depending on λ, we obtain eigenpairs (μ,u)=(μ(λ),u _λ) which solve this eigenvalue problem for a given λ>0. Then we study the asymptotic behavior ofu _λ and μ(λ) as λ→∞. Especially, we study the location of interior transition layers ofu _λ as λ→∞. This research has been supported by the Japan Society for the Promotion of Science.     

The nonlinear version of Pazy’s local existence theorem

LetX be a real Banach space,U ⊂X a given open set,A ⊂X×X am-dissipative set andF:C(0,a;U) →L ^∞(0,a;X) a continuous mapping. Assume thatA generates a nonlinear semigroup of contractionsS(t): {ie221-2}) → {ie221-3}), strongly continuous at the origin, withS(t) compact for allt>0. Then, for eachu ₀ ∈ {ie221-4}) ∩U there existsT ∈ ]0,a] such that the following initial value problem: (du(t))/(dt) ∈Au(t) +F(u)(t),u(0)=u ₀, has at least one integral solution on [0,T]. Some extensions and applications are also included.     

An Endpoint Estimate for the Commutator of Singular Integrals

It is well known that the commutator Tb of the singular integral operator T with a BMO function b is bounded on L^P（R^n）, 1 〈 p 〈 ∞. In this paper, we consider the endpoint estimates for a kind of commutator of singular integrals. A BMO-type estimate for Tb is obtained under the assumption b ∈ LMO.     

On a smooth solution of a boundary-value problem

We study a periodic boundary-value problem for the quasilinear equation u _tt −u _xx =F[u, u _t , u _x ], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998.