The existence and uniqueness of the local solution for a Camassa-Holm type equation

A shallow water equation of Camassa-Holm type, containing nonlinear dissipative effect, is investigated. Using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself, we establish the existence and uniqueness of its local solution in Sobolev space H^s(R) with . Meanwhile, a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order Sobolev space H^s with are presented.     

Codimension reduction and second fundamental form of CR submanifolds in complex space forms

Considering n-dimensional real submanifolds M of a complex space form which are CR submanifolds of CR dimension , we study the condition h(FX,Y)+h(X,FY)=0 on the structure tensor F naturally induced from the almost complex structure J of the ambient manifold and on the second fundamental form h of submanifolds M.     

The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations

In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L^p(Ω), and , respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L^p(Ω), and , the results in this part are new and appear to be optimal.     

Symplectic duality of symmetric spaces

Let M⊂Cn be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ωhyp. Denote by (M^∗,ωFS) the compact dual of (M,ωhyp), where ωFS is the Fubini-Study form on M^∗. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism satisfying and for the pull-back of ΨM, where ω₀ is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ΨM, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of Cn. As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin.     

On continuous choice of retractions onto nonconvex subsets

For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate can be improved to and the constant can be replaced by the root of the equation α+α²+α³=1.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

4-dimensional minimal CR submanifolds of the sphere S satisfying Chen's equality

In this paper, we classify 4-dimensional minimal CR submanifolds M of the nearly Kähler 6-sphere S⁶(1) which satisfy Chen's equality, i.e. , where δM(p)=τ(p)−infK(p) for every p∈M.     

On a decay property of solutions to the Haraux-Weissler equation

We give a sufficient condition that non-radial H¹-solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space , where ρ is the weight function exp(|x|²/4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space .     

Low-type submanifolds of real space forms via the immersions by projectors

We undertake a comprehensive study of submanifolds of low Chen-type (1, 2, or 3) in non-flat real space forms, immersed into a suitable (pseudo) Euclidean space of symmetric matrices by projection operators. Some previous results for submanifolds of the unit sphere (obtained in [A. Ros, Eigenvalue inequalities for minimal submanifolds and P-manifolds, Math. Z. 187 (1984) 393–404; M. Barros, B.Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J. 108 (1987) 77–91; I. Dimitrić, Spherical hypersurfaces with low type quadric representation, Tokyo J. Math. 13 (1990) 469–492; J.T. Lu, Hypersurfaces of a sphere with 3-type quadric representation, Kodai Math. J. 17 (1994) 290–298]) are generalized and extended to real projective and hyperbolic spaces as well as to the sphere. In particular, we give a characterization of 2-type submanifolds of these space forms with parallel mean curvature vector. We classify 2-type hypersurfaces in these spaces and give two sets of necessary conditions for a minimal hypersurface to be of 3-type and for a hypersurface with constant mean curvature to be mass-symmetric and of 3-type. These conditions are then used to classify such hypersurfaces of dimension n5. For example, the complete minimal hypersurfaces of the unit sphere Sⁿ⁺¹ which are of 3-type via the immersion by projectors are exactly the 3-dimensional Cartan minimal hypersurface and the Clifford minimal hypersurfaces M_k,n−k for n≠2k. An interesting characterization of horospheres in is also obtained.     

Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces

We study the Cauchy problem of the Ostrovsky equation , with βγ<0. By establishing a bilinear estimate on the anisotropic Bourgain space Xs,ω,b, we prove that the Cauchy problem of this equation is locally well-posed in the anisotropic Sobolev space H(s,ω)(R) for any and some . Using this result and conservation laws of this equation, we also prove that the Cauchy problem of this equation is globally well-posed in H(s,ω)(R) for s?0.     

A note on the boundedness of Calderón-Zygmund operators on Hardy spaces

It was well known that Calderón-Zygmund operators T are bounded on Hp for provided T^∗(1)=0. A new Hardy space , where b is a para-accretive function, was introduced in [Y. Han, M. Lee, C. Lin, Hardy spaces and the Tb-theorem, J. Geom. Anal. 14 (2004) 291-318] and the authors proved that Calderón-Zygmund operators T are bounded from the classical Hardy space Hp to the new Hardy space if T^∗(b)=0. In this note, we give a simple and direct proof of the boundedness of Calderón-Zygmund operators via the vector-valued singular integral operator theory.     

Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Geometry of the tetrahedron space

Let X^° be the space of all labeled tetrahedra in P³. In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285-312] we constructed a smooth symmetric compactification of X^°. In this article we show that the complement is a divisor with normal crossings, and we compute the cohomology ring .     

Estimations anisotropes dans les convexes de type fini

Let Ω be a smoothly bounded convex domain of finite type m and f be a (0,1)-form -closed in Ω. It is proved that the equation admits a solution u belonging to the space Λ1(Ω) (respectively to the anisotropic space Γ^α_(ρ) of McNeal-Stein, for all α,0<α<1/m) if the anisotropic norm - introduced by Bruna-Charpentier-Dupain - is finite (respectively if the Euclidian norm ‖f‖_∞ of the form f is finite).