The Existence of Weak Solutions to the 2D Euler Equations with Initial Vorticity in BMO

We shall prove that there exists a weak solution to the 2D Euler equations with initial vorticity in BMO and initial velocity in L^∞.     

Solvability of the Dirichlet Problem in W~（2,p ）for a Class of Elliptic Equations with Singular Data

We prove an existence and uniqueness result for the Dirichlet problem for a class of elliptic equations with singular data in weighted Sobolev spaces.     

On the United Theory of the Family of Euler—Haller Type Methods with Cubical Convergence in Banach Spaces

The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average,and a united convergence theory with its applications is presented.     

The Smoothness of Weak Solutions to the System of Second Order Differential Equations with Non—negative Characteristics

In this paper,we will discuss smoothness of weak solutions for the system of second orderdifferential equations eith non-negative characteristies.First of all,we establish boundary,and interiorestimates and then we prove that solutions of regularization problem satisfy Lipschitz condition.     

Additional Note on Partial Regularity of Weak Solutions of the Navier-Stokes Equations in the Class L ∞(0, T, L 3(Ω)3)

We present a simplified proof of a theorem proved recently concerning the number of singular points of weak solutions to the Navier-Stokes equations. If a weak solution u belongs to L (0, T, L ³()³), then the set of all possible singular points of u in is at most finite at every time t₀ (0,T).     

Existence of Weak Solutions of Two-dimensional Euler Equations with Initial Vorticity in Lorentz Space L （2,1） （ R^2）

Some estimates on 2-D Euler equations are given when initial vorticity ω0 belongs to a Lorentz space L(2, 1). Then based on these estimates, it is proved that there exist global weak solutions of two dimensional Euler equations when ωo ∈ L(2, 1).     

Navier-Stokes equations under slip boundary conditions: Lower bounds to the minimal amplitude of possible time-discontinuities of solutions with two components in L∞(L3)

Science China Mathematics - The main purpose of this paper is to extend the result obtained by Beirão da Veiga (2000) from the whole-space case to slip boundary cases. Denote by ū two...     

On a greedy algorithm in L 1(0, 1) with regard to subsystems of the Haar system and on ω-quasigreedy bases

All quasigreedy subsystems of the Haar system in L ¹(0, 1) are described. The problem of renormalizing the Haar system so that it becomes a quasigreedy basis is also studied.     

A generalization of Serre’s condition $$mathrm {(F)}$$ ( F ) with applications to the finiteness of unramified cohomology

Mathematische Zeitschrift - In this paper, we introduce a condition ( $$mathrm {F}_m'$$ ) on a field K, for a positive integer m, that generalizes Serre’s condition (F) and which still...     

Supplement to the paper “the averaging operator with respect to a countable partition on a minimal symmetric ideal of the space L1(0, 1)”

Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let tL₁(0,1), let N_t be the smallest rearrangement-invariant order ideal vector lattice in L₁(0,1), containing t. In the paper one investigates the properties of the image E(N_t¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, gL₁(0,1), such that E(N_t¦A)N_g. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(tA)QE(t*A*), where is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1Q2⁵. One formulates the problem of the smallest value of Q.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 137–141, 1986.     

Asymptotics and Blow-up for Mass Critical Nonlinear Dispersive Equations(Dedicated to Professor Ham Brezis on the occasion of his 70th birthday)

The author considers mass critical nonlinear Schrdinger and Korteweg-de Vries equations. A review on results related to the blow-up of solution of these equations is given.     

Simon’s team’s contributions to scientific progress in mathematics education: A commentary on the Learning Through Activity (LTA) research program

I discuss two ways in which the Learning Through Activity (LTA) research program contributes to scientific progress in mathematics education: (a) providing general and content-specific constructs to explain conceptual learning and instructional design that corroborate and/or elaborate on previous work and (b) raising new questions/issues. The general constructs include using instructional design as testable models of learning and using theoretical constructs to guide real-time, instructional adaptations. In this sense, the general constructs promote understanding of linkages between conceptual learning and instruction in mathematics. The concept-specific constructs consist of empirically-grounded, hypothetical learning trajectories (HLTs) for fractional and multiplicative reasoning. Each HLT consists of specific, intended conceptual changes and tasks that can bring them forth. Questions raised for me by the LTA work involve inconsistencies between the stance on learning and reported teaching-learning interactions that effectively led to students’ abstraction of the intended mathematical concepts.     

Corrigendum to “Global existence of solutions to 2-D Navier–Stokes flow with non-decaying initial data in half-plane” [J. Differential Equations 265 (10) (2018) 5352–5383]

A misuse of notations generates confusion in the statements. It does not influence the proofs that are coherently developed with the right notations.     

The Hencky strain energy ‖log U‖2 measures the geodesic distance of the deformation gradient to SO(n) in the canonical left-invariant Riemannian metric on GL(n)

The well-known isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set of rigid rotations in a canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity requires the Riemannian metric to be left-GL(n) invariant, isotropy requires the Riemannian metric to be right-O(n) invariant. The latter two conditions are satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n). Surprisingly, the final result is basically independent of the chosen parameters. In deriving the result, geodesics on GL(n) have to be parametrized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)