Sharp local well-posedness for a fifth-order shallow water wave equation

In this paper we prove that the already-established local well-posedness in the range s>−5/4 of the Cauchy problem with an initial Hs(R) data for a fifth-order shallow water wave equation is extendable to s=−5/4 by using the space. This is sharp in the sense that the ill-posedness in the range s<−5/4 of this initial value problem is already known.     

Well-posedness and weak rotation limit for the Ostrovsky equation

We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,a with s>−a/2−3/4 and 0?a?−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hs with s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L². To show this result, we prove a bilinear estimate which is uniform with respect to γ.     

On the well-posedness of the Schrödinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sobolev spaces L²(R)×H−3/4(R), and Hs(R)×H−3/4(R) (s>−1/16) for the resonant case. The new ingredient is that we use the -type space, introduced by the first author in Guo (2009) [10], to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares (2007) [6].     

On the low regularity of the modified Korteweg-de Vries equation with a dissipative term

We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u³)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues.     

Riesz extremal measures on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|^−s with d−2?s<d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s=d−2 yields interesting phenomena, which we investigate in detail. A weak^∗ asymptotic analysis is provided as s→⁺(d−2).     

Well posedness for a higher order modified Camassa-Holm equation

We show that the Cauchy problem for a higher order modification of the Camassa-Holm equation is locally well posed for initial data in the Sobolev space Hs(R) for s>s^′, where 1/4?s^′<1/2 and the value of s^′ depends on the order of equation. Employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions.     

Well-posedness for the Cauchy problem associated to the Hirota-Satsuma equation: Periodic case

We consider a system of Korteweg-de Vries (KdV) equations coupled through nonlinear terms, called the Hirota-Satsuma system. We study the initial value problem (IVP) associated to this system in the periodic case, for given data in Sobolev spaces Hs×Hs+1 with regularity below the one given by the conservation laws. Using the Fourier transform restriction norm method, we prove local well-posedness whenever s>−1/2. Also, with some restriction on the parameters of the system, we use the recent technique introduced by Colliander et al., called I-method and almost conserved quantities, to prove global well-posedness for s>−3/14.     

Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation

In this article we establish the bilinear estimates corresponding to the 1D and 2D NLS with a quadratic nonlinearity , which imply the local well-posedness of the Cauchy problem in Hs for s?−1 in the 1D case and for s>−1 in the 2D case. This is a continuation of our study [N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity , Commun. Pure Appl. Anal. 7 (2008) 1123-1143] on the 1D NLS with nonlinearity . Previous papers by Kenig, Ponce and Vega, and Colliander, Delort, Kenig and Staffilani established local well-posedness for s>−3/4 in 1D and in 2D, respectively, and when the nonlinearity is restricted to cu², papers by Bejenaru and Tao, and Bejenaru and De Silva improved these results to s?−1 in 1D and s>−1 in 2D. The bilinear estimate for 2D also yields an improvement on the growth rate of Sobolev norms of finite energy global-in-time solutions to the 2D cubic NLS.     

Magic labelings of triangles

The first proof is given that for every even integer s≥4, the graph consisting of s vertex disjoint copies of C₃, (denoted sC₃) is vertex-magic. Hence it is also edge-magic. It is shown that for each even integer s≥6, sC₃ has vertex-magic total labelings with at least 2s−2 different magic constants. If s≡2mod4, two extra vertex-magic total labelings with the highest possible and lowest possible magic constants are given. If s=2⋅3^k, k≥1, it is shown that sC₃ has a vertex-magic total labeling with magic constant h if and only if (1/2)(15s+4)≤h≤(1/2)(21s+2). It is also shown that 2C₃ is not vertex-magic. If s is odd, vertex-magic total labelings for sC₃ with s+1 different magic constants are provided.     

Nonresonance between the first two eigenvalues for a Steklov problem

In this paper, we study the solvability of the Steklov problem Δ_pu=|u|^p−2u in Ω, on ∂Ω, under assumptions on the asymptotic behaviour of the quotients f(x,s)/|s|^p−2s and pF(x,s)/|s|^p which extends the classical results with Dirichlet boundary conditions that for a.e. x∈∂Ω, the limits at the infinity of these quotients lie between the first two eigenvalues.     

The well-posedness of the Korteweg-de Vries-Benjamin-Ono equation

The well-posedness of the Cauchy problem to the generalized Korteweg-de Vries-Benjamin-Ono equation is considered. Local results for data in (s?−1/8) and the global well-posedness for data in are obtained if l=2. Moreover, for l=3, the problem is locally well-posed for data in H^s (s?1/4). The main idea is to use the Fourier restriction norm method.     

Cauchy problem for the multi-dimensional Boussinesq type equation

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt−Δu+Δ²u=Δσ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ∈C(R), σ(s) is of polynomial growth order, say p (>1), either , s∈R, where β>0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N=1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown.     

Eigenvalues and discrete boundary value problems for the one-dimensional p-laplacian

In this paper, we characterize the eigenvalues and show existence of positive solutions to discrete boundary value problem (here ?(s)=|s|^p−2s, p>1 and λ>0 is a parameter)

     

On a family of relative quartic Thue inequalities

We consider the relative Thue inequalities

|X⁴−t²X²Y²+s²Y⁴|?²|t|−²|s|−2,     

Proper partial geometries with Singer groups and pseudogeometric partial difference sets

A partial geometry admitting a Singer group G is equivalent to a partial difference set in G admitting a certain decomposition into cosets of line stabilizers. We develop methods for the classification of these objects, in particular, for the case of abelian Singer groups. As an application, we show that a proper partial geometry Π=pg(s+1,t+1,2) with an abelian Singer group G can only exist if t=2(s+2) and G is an elementary abelian 3-group of order ³(s+1) or Π is the Van Lint-Schrijver partial geometry. As part of the proof, we show that the Diophantine equation (m³−1)/2=(²rw−1)/(r²−1) has no solutions in integers m,r?1, w?2, settling a case of Goormaghtigh's equation.     

On global solution to the Klein-Gordon-Hartree equation below energy space

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R³. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S. Klainerman and D. Tataru, we establish the Hs (s<1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs₀×Hs₀−1 where but not , for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.     

Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R¹⁺¹. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H^s1 and wave data in H^s2 ×H^s2?1 for 3/4?α < s₁ ≤ 0 and ?1/2 < s₂ < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α ≤ 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s₁ = 0 and ?1/2 < s₂ < 1/2 by using the conservation law for the L² norm of u.     

Global rough solutions to the critical generalized KdV equation

We prove that the initial value problem (IVP) for the critical generalized KdV equation ut+uxxx+x(u⁵)=0 on the real line is globally well-posed in Hs(R) if s>3/5 with the appropriate smallness assumption on the initial data.     

Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness

We study the existence, boundary behavior and uniqueness of solutions for the singular elliptic system −Δu=u^−pv^−q,−Δv=u^−rv^−s,u>0,v>0,x∈Ω,u|_∂Ω=v|_∂Ω=0, where Ω is a bounded domain with smooth boundary in R^N, p,s≥0 and q,r>0. Our results are obtained in a range of p,q,r,s different from those in [M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295-3318].     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.