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On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications
Cho, Chu-Hee,Koh, Youngwoo,Seo, Ihyeok AIMS PRESS 2016 Discrete and continuous dynamical systems Vol.36 No.4
<P>In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrodinger equation in the radial case. Then we apply them to the well-posedness theory for the equation i partial derivative(t)u + vertical bar V vertical bar(alpha)u = V(x, t)u, 1 < alpha < 2, with radial (H) over dot(gamma) initial data below L-2 and radial potentials V is an element of (LtLxw)-L-r under the scaling-critical range alpha/r + n/w = alpha.</P>
Kim, Doyoon,Dong, Hongjie,Zhang, Hong AIMS PRESS 2016 Discrete and continuous dynamical systems Vol.36 No.9
<P>We prove the unique solvability in weighted Sobolev spaces of non divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for stochastic partial differential equations with BMOx coefficients.</P>
Calderón-Zygmund estimate for homogenization of parabolic systems
Byun, Sun-Sig,Jang, Yunsoo AIMS PRESS 2016 Discrete and continuous dynamical systems Vol.36 No.12
<P>We establish a global Calderon-Zygmund estimate for homogenization of a parabolic system in divergence form with discontinuous coefficients in a bounded nonsmooth domain under the assumptions that the coefficients have small BMO seminorms and the boundary of the domain is delta-flat for some delta>0 depending on the given data.</P>
Wang, Qin,Song, Kyungwoo AIMS PRESS 2016 Discrete and continuous dynamical systems Vol.36 No.3
<P>We study the regularity of sonic curves from a two-dimensional Riemann problem for the nonlinear wave system of Chaplygin gas, which is an essential step for the global existence of solutions to the two-dimensional Riemann problems. As a result, we establish the global existence of uniformly smooth solutions in the semi-hyperbolic patches up to the sonic boundary, where the degeneracy of hyperbolicity occurs. Furthermore, we show the C-1-regularity of sonic curves.</P>
Zero-one law of Hausdorff dimensions of the recurrent sets
Kim, Dong Han,Li, Bing AIMS PRESS 2016 Discrete and continuous dynamical systems Vol.36 No.10
<P>Let (Sigma, sigma) be the one-sided shift space with m symbols and R-n(x) be the first return time of x is an element of Sigma to the n-th cylinder containing x. Denote E-alpha, beta(phi) - {x is an element of Sigma : lim(n ->infinity) inf log R-n(x)/phi(n) - alpha, lim(n ->infinity) sup log R-n(x)/phi(n) - beta}, where phi : N -> R+ is a monotonically increasing function and 0 <= alpha <= beta <=+infinity. We show that the Hausdorff dimension of the set E-alpha, beta(phi) admits a dichotomy: it is either zero or one depending on phi, alpha and beta.</P>