Geodesic completeness for Sobolev H s -metrics on the diffeomorphism group of the circle

We prove that the weak Riemannian metric induced by the fractional Sobolev norm H ^s on the diffeomorphism group of the circle is geodesically complete, provided that s > 3/2.     

Locally Supported,Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes

In this paper, biorthogonal wavelets are constructed on nonuniform meshes. Both primal and dual wavelets are locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces (H^s) for (|s| < 3/2). The wavelets at the primal side span standard Lagrange finite element spaces.     

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces Hs, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the Hs norm of the solution, in terms of the Hs norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in Hs.     

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.     

Non‐uniform dependence and persistence properties for coupled Camassa–Holm equations

This paper deals with the non‐uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well‐posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces H^s with s > 3/2. On the other hand, the persistence properties in weighted L^p spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Sharp sobolev embeddings and related hardy inequalities: The critical case

The paper deals with sharp embeddings of the Sobolev spaces H^s_p(IRⁿ) and the Besov spaces B^s_p,p(IRⁿ) into rearrangement—invariant spaces and related Hardy inequalities. Here 1 < p < ∞ and s = n/p.     

A Nonhomogeneous Boundary-Value Problem for the Korteweg–de Vries Equation Posed on a Finite Domain

Abstract

Studied here is an initial- and boundary-value problem for the Korteweg–de Vries equation posed on a bounded interval with nonhomogeneous boundary conditions. This particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed. It is shown here that this initial-boundary-value problem is globally well-posed in the L ₂-based Sobolev space H  ^s (0, 1) for any s ≥ 0. In addition, the mapping that associates to appropriate initial- and boundary-data the corresponding solution is shown to be analytic as a function between appropriate Banach spaces.     

Blowup in a three‐dimensional vector model for the Euler equations

We present a three‐dimensional vector model given in terms of an infinite system of nonlinearly coupled ordinary differential equations. This model has structural similarities with the Euler equations for incompressible, inviscid fluid flows. It mimics certain important properties of the Euler equations, namely, conservation of energy and divergence‐free velocity. It is proven for certain families of initial data that the model system permits local existence in time for initial conditions in Sobolev spaces H^s, s > ; and blowup occurs in the sense that the H^{3/2 + ?} norm becomes unbounded in finite time. © 2004 Wiley Periodicals, Inc.     

Further regularity results for almost cubic nonlinear Schrödinger equation

We present three results related with the regularity of solutions of the almost cubic NLS. In the first one, following Ozawa’s idea, we establish mass and energy conservation for the solutions without regularizing the initial datum. Our second result is the H^s well-posedness for the Cauchy problem for 0<s<1. Finally, we show that the same solutions are also in some Bourgain spaces for possibly a smaller time interval. In all of our results, the non-local nonlinear term in the equation is shown to act like a cubic nonlinearity on the appropriate Sobolev and Besov spaces.     

Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity

We consider the global attractor for the weakly damped forced KdV equation in Sobolev spaces [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}for s < 0. Under the assumption that the external forcing term belongs to [(L)\dot]²(T),{\dot{L}^2({\mathbf T}),} we prove the existence of the global attractor in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})} for −1/2 ≤ s < 0, which is identical to the one in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and thus is compact in H ³(T). The argument is a combination of the I-method and decomposing the solution into two parts, one of which is uniformly bounded in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and the other decays exponentially in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}.     

The regularity of local solutions for a generalized Camassa-Holm type equation

The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space Hs （R） with 1 〈 s ≤ 3/2 is derived.     

Elliptic Operators in a Refined Scale of Functional Spaces

We study the theory of elliptic boundary-value problems in a refined two-sided scale of the Hormander spaces H ^s , ^ϕ, where s ∈ R and ϕ is a functional parameter slowly varying at +∞. For the Sobolev spaces H ^s , the function ϕ(|ξ|) ≡ 1. We establish that the considered operators possess the Fredholm property, and solutions are globally and locally regular. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 689–696, May, 2005.     

Multiwavelet sampling theorem in Sobolev spaces

This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (H s (R) , H-s (R)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ H s (R) with s 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L 2 (R), whose Sobolev exponents are greater than 1 /2.     

Rawlins' problem for half-plane diffraction: its generalized eigenfunctions with real wave numbers

This paper deals with a famous diffraction problem for a single half-plane Σ: x>0, y=0 as an obstacle and for some time-harmonic plane incident wave field. Rawlins in 1975 was the first to solve the mixed (Dirichlet/Neumann) boundary value problem for the scalar Helmholtz equation. He also was the first to solve the equivalent pair of coupled Wiener–Hopf equations explicitly by factoring their discontinuous 2×2 Fourier matrix symbol in 1980. Although for real wave numbers k the usual factorization procedure fails it will serve as the basis: Following the lines given by Ali Mehmeti in his habilitation thesis [1] for the (Dirichlet/Dirichlet) boundary value problem we combine the idea of integral path deforming along the branch cuts of the characteristic square root √(ξ²−k²) given in Meister's book [13] with the modern Wiener–Hopf method solution derived by Speck [24] explicitly in a H^1+ε, ε⩾0, Sobolev space setting. The symmetry of the intermediate spaces H^s, H^-s, ∣s∣<1 2, which is due to generalized factorization, plays a key role in deforming the Fourier integral paths in order to get Laplace transform representations of the generalized eigenfunctions of the problem. As a remarkable fact 0<ε<¼ must hold here. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Résolution des équations de Navier-Stokes dans des espaces anisotropes

In this paper we prove the global existence and uniqueness of the solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which H^δ in the i-th direction, δ₁+δ₂+δ₃=½,½< δ_i <½ and in a space which is L² in the first two directions and B_2,1^½ in the third direction, where H and B (respectively) denote the usual homogeneous Sobolev and Besov (respectively) spaces.     

The Cauchy Problem for the Fifth Order Shallow Water Equation

The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu＋αδx^5u＋βδx^3u＋rδxu＋μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s（s≥-3/8） by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s（s〉0）, the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data.     

The Cauchy problem for the Novikov equation

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces ${H^s(\mathfrak{R})}$ with ${s < \frac{3}{2}}$ in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in ${H^{s}(\mathfrak{R})}$ with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in ${H^{s}(\mathfrak{R})}$ with s > 3/2.     

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ? _z  + a(z,x,D _x ) with Re (a) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L(H ^(s),H ^(s)) of these operators is provided, which yields a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞.