The authors consider the scattering phenomena of the defocusing $\dot H^s$-critical NLS. It is shown that if a solution of the defocusing NLS remains bounded in the critical homogeneous Sobolev norm on its maximal interval of existence, then the solution is global and scatters.
This article deals with an inverse problem of reconstructing two time independent coefficients in the reaction diffusion system from the final time space discretized measurement using the optimization method with the help of the smooth interpolation technique. The main objective of the article is to analyse the asymptotic behavior of the solution of the inverse problem for the linearly coupled reaction diffusion system with respect to the homogeneous Dirichlet boundary condition.
The author considers the L p boundedness for two kinds of Carleson-type maximal operators with variable kernels $\frac{{\Omega \left( {x,y'} \right)}}{{\left| y \right|^n }}$, where Ω(x, y′) ∈ L ∞(ℝ n)×W 2 s (S n−1) for some s > 0.
For relatively prime positive integers u 0 and r, and for 0 ≤ k ≤ n, define u k:= u 0 + kr. Let L n:= lcm(u 0, u 1, ..., u n) and let a, l ≥ 2 be any integers. In this paper, the authors show that, for integers α ≥ a, r ≥ max(a,l − 1) and n ≥ lαr, the following inequality holds $L_n \geqslant u_0 r^{\left( {l - 1} \right)\alpha + a - l} \left( {r + 1} \right)^n .$ Particularly, letting l = 3 yields an improvement on the best previous lower bound on L n obtained by Hong and Kominers in 2010.
The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy (BTH). As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the (1, 2)-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the (1, 2)-BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of (N, 1)-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of (N, 1)-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation (a useful ecological competition model) are given.
This paper is concerned with the pointwise estimates for the sharp function of two kinds of maximal commutators of multilinear singular integral operators T Σb * and T Πb *, which are generalized by a weighted BMO function b and a multilinear singular integral operator T, respectively. As applications, some commutator theorems are established.
The author mainly studies the difference of the weak solutions generated by a wave front tracking algorithm to the steady Euler system and the isothermal Euler system. Under the hypothesis that the initial data are of sufficiently small total variation, it is proved that the difference between the solutions of the steady Euler system and the system of isothermal supersonic flow can be bounded by the cube of the total variation of the initial perturbation.
This paper considers the steady Swift-Hohenberg equation $u'''' + \beta ^2 u'' + u^3 - u = 0.$ Using the dynamic approach, the authors prove that it has a homoclinic solution for each $\beta \in \left[ {\sqrt[4]{8} - \varepsilon _0 ,\sqrt[4]{8}} \right)$, where ε 0 is a small positive constant. This slightly complements Santra and Wei’s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038–2056], which stated that it admits a homoclinic solution for each β ∈ (0, β 0) where β 0 = 0.9342 ....
Let F be an algebraically closed field of prime characteristic p > 3, and W (n) the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W (n) are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W (n) is proved to have one block.
In this paper, the reflection phenomenon of a vapor shock front (both sides of the front are in the vapor phase) in a van der Waals fluid is considered. Both the 1-dimensional case and the multidimensional case are investigated. The authors find that under certain conditions, the reflected wave can be a single shock, or a single subsonic phase boundary, or one weak shock together with one subsonic phase boundary, which depends on the strength of the incident shock. This is different from the known result for the reflection of shock fronts in a gas dynamical system due to Chen in 1989.
Let l 1, l 2, ..., l g be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that (k + l 1)2, (k + l 2)2, ..., (k + l g)2 can not be expressed as 2 n + p α is at least c(g)x, where p is an odd prime and the constant c(g) depends only on g.