The regularity of bidimensional solutions of the third grade fluids equations

It was recently proved in [1,2] that the third grade fluids equations have a unique global bidimensional solution provided that the initial velocity belongs to the Sobolev space H². Here, we complete this result by proving that this solution preserves the Sobolev regularity of the initial data, i.e., if the initial velocity belongs to H^m, m ≥ 2, then the evolved velocity v(t, ·) also belongs to H^m for every time t.     

Null controllability in function space of nonlinear retarded systems with limited control

The question of controlling both linear and nonlinear retarded functional differential equations from an initial function in a Sobolev space W₂⁽¹⁾ to zero in W₂⁽¹⁾ is considered when the controls are square integrable with values in a compact subset U of m-dimensional Euclidean space with zero in its interior. Sufficient computable criteria for null controllability are given.     

Spectral element methods for parabolic problems

A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k   is proportional to h²$h^2$. At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method.     

Analytic semigroups of pseudodifferential operators on vector-valued Sobolev spaces

In this paper we study continuity and invertibility of pseudodifferential operators with non-regular Banach space valued symbols. The corresponding pseudodifferential operators generate analytic semigroups on the Sobolev spaces W _p ^k (? ⁿ , E) with k ∈ ?₀, 1 ≤ p ≤ ∞. Here E is an arbitrary Banach space. We also apply the theory to solve non-autonomous parabolic pseudodifferential equations in Sobolev spaces.     

To the Sobolev embedding theorem for the limiting exponent

We establish embeddings of the Sobolev space W _p ^s and the space B _pq ^s (with the limiting exponent) in certain spaces of locally integrable functions of zero smoothness. This refines the embedding of the Sobolev space in the Lorentz and Lorentz-Zygmund spaces. Similar problems are considered for the case of irregular domains and for the potential space.     

One Initial Boundary-Value Problem for Integro-Differential Equation of the Second Order With Power Nonlinearity

In the Sobolev space W _∞ ² (?⁺) we investigate one initial boundary-value problem for integro-differential equation of the second order with power nonlinearity on a semi-axis. Assuming that summary-difference even kernel serves for the considered kernel as minorant in the sense of M. A. Krasnosel’skii, we prove the existence of a nonnegative (nontrivial) solution in the Sobolev spaceW _∞ ² (?⁺). We also calculate the limits of constructed solution at infinity.     

Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ? ⁿ to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2n-dimensional torus $\mathbb{T}^{2n} $ . This reduction enables us to identify the Schwartz space on ? ⁿ with the space of smooth sections Γ^∞(T ²ⁿ , ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ? ⁿ , we construct an index formula in accordance with the classical Atya-Singer formula.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Another note on the embedding of the Sobolev space for the limiting exponent

The embedding of the Sobolev spaces W _p ^s (? ⁿ ) in a Lizorkin-type space of locally summable functions of zero smoothness is established. This result is extended to the case of the embedding of Sobolev spaces on nonregular domains of n-dimensional Euclidean space. The formulation of the theorem depends on the geometric parameters of the domain of the functions.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

The sharp Strichartz and Sobolev–Strichartz inequalities for the fourth‐order Schrödinger equation

In this paper, we will obtain that there exists a maximizer for the non‐endpoint Strichartz inequalities for the fourth‐order Schrödinger equation with initial data in the L²( R ^d) space in all dimensions, and then we obtain a maximizer also for the non‐endpoint Sobolev–Strichartz inequality for the fourth‐order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

On the global existence and wave-breaking criteria for the two-component Camassa-Holm system

Considered herein is a two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. A wave-breaking criterion for strong solutions is determined in the lowest Sobolev space Hs, by using the localization analysis in the transport equation theory. Moreover, an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this Sobolev space Hs.     

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.     

Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds

In this work, we aim to prove algebra properties for generalized Sobolev spaces W ^s,p ?? L ^?? on a Riemannian manifold (or more general homogeneous type space as graphs), where W ^s,p is of Bessel-type W ^s,p := (1+L)^?s/m (L ^p ) with an operator L generating a heat semigroup satisfying off-diagonal decays. We do not require any assumption on the gradient of the semigroup. Instead, we propose two different approaches (one by paraproducts associated to the heat semigroup and another one using functionals). We also study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non-bounded geometry) and even in euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.     

A theorem of global existence of solutions to nonlinear wave equations in four space dimensions

In this paper, the following nonlinear wave equation is considered; □u=F(u, D u, D _x D u),x∈R ⁿ ,t>0; u=u ⁰ (x), u _t =u ¹ (x), t=0. We prove that if the space dimensionn ≥ 4 and the nonlinearityF is smooth and satisfies a mild condition in a small neighborhood of the origin, then the above problem admits a unique and smooth global solution (in time) whenever the initial data are small and smooth. The strategy in proof is to use and improve Global Sobolev Inequalities in Minkowski space (see [8]), and to develop a generalized energy estimate for solutions.     

The boundedness for commutators of maximal hypersingular integrals with rough kernels

In this paper, we prove that the commutators of maximal hypersingular integrals with rough kernels are bounded from the Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ), which is a substantial improvement and an extension of some known results.     

Embedding of Sobolev spaces with limit exponent revisited

An embedding of the Sobolev spaces W _p ^s (? ⁿ ) in Lizorkin-type spaces of locally integrable functions of smoothness zero is obtained; a similar assertion for Riesz and Bessel potentials is presented. The embedding theorem is extended to Sobolev spaces on irregular domains in n-dimensional Euclidean space. The statement of the theorem depends on geometric parameters of the domain of functions.     

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ε⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.