The Problem on Evolution of a Self-Gravitating Isolated Fluid Mass That is not Subject to the Surface Tension Forces

We establish the solvability of the nonstationary free boundary problem for the Navier–Stokes equations which describes the evolution of a self-gravitating isolated fluid mass that is not subject to the surface tension forces. Bibliography: 12 titles.     

L q-Estimates for a Solution to the Problem about the Evolution of an Isolated Amount of a Fluid

We construct the L _q-theory of the nonstationary free boundary problem for the Navier–Stokes equations under the action of surface tension forces. Although the main theorem is proved for some finite time interval, it can be used for extending the solution to a larger interval since the smoothness of the solution is preserved for t < 0. Bibliography: 10 titles.     

Classical Solvability of a Model Problem in a Half-Space, Related to the Motion of an Isolated Mass of a Compressible Fluid

For an arbitrary finite time interval, the unique solvability of a linear half-space problem is obtained in Hölder classes of functions. The problem arises as the result of the linearization of a free boundary problem for the Navier--Stokes system governing the unsteady motion of a finite mass of a compressible fluid. The boundary conditions in the linear problem are noncoercive because of the surface tension acting on the free boundary. This fact presents the main difficulty in the problem, while the differential system in itself is parabolic in the sense of Petrovskii. The principal idea of the investigation is to reduce the noncoercive problem to a coercive one with zero coefficient of the surface tension. Bibliography: 6 titles.     

On an Initialization Problem

Mathematical Notes -     

On an Embedding Problem

The following theorem is proved. Let n be an odd integer; if all primes that occur in the canonical decomposition of the integer with odd multiplicities have the form , and , then the splitting field of the polynomial is embeddable in a nonsplit extension of degree 48. Bibliography: 2 titles.     

一类大气演化方程组的Cauchy问题

在分层理论的框架下讨论一类大气演化方程组的Cauchy问题,证明了:1) 惯性力对一类大气演化方程组Cauchy问题的适定性判别标准没有影响;2) 可压缩性对粘性大气方程组Cauchy问题的适定性判别标准没有影响,但对无粘大气方程组,可压缩性改变Cauchy问题适定性判别标准;3) 所论方程组在t=0超平面上的Cauchy问题均是不适定的,并不受粘性和可压缩性的影响;4) 可压无粘大气方程与运动静止初始条件构成的Cauchy问题是不适定的．     

On the Cauchy Problem of Evolution p-Laplacian Equation with Nonlinear Gradient Term

The authors study the existence of solution to p-Laplacian equation with nonlinear forcing term under optimal assumptions on the initial data, which are assumed to be measures. The existence of local solution is obtained.     

On the Cauchy Problem of Evolution p-Laplacian Equation with Nonlinear Gradient Term

The authors study the existence of solution to p-Laplacian equation with non-linear forcing term under optimal assumptions on the initial data,which are assumed to be measures.The existence of local solution is obtained.     

On Groups with an Isolated 2-Subgroup

Mathematical Notes - In the paper, sufficient conditions for a group with an isolated 2-subgroup to be a Frobenius group are found.     

On the Cauchy Problem Describing an Electron-Phonon Interaction

In this paper, a model is derived to describe a quartic anharmonic interatomic interaction with an external potential involving a pair electron-phonon. The authors study the corresponding Cauchy Problem in the semilinear and quasilinear cases.     

On an Extension Problem for Polynomials

Consider the following problem: given complex numbers a₁, ...,a_n, find an L function f of minimum norm whose Fourier coefficientsc_k(f) are equal to a_k for k between 0 and n. We show the uniquenessof this function, and we estimate its norm. The operator-valuedcase is also discussed. 2000 Mathematics Subject Classification30E05, 47A20, 47A56, 47A57.     

Note on an Open Problem of Higher Order Nonlinear Evolution Equations

In view of a new idea on initial conditions, an open problem of nonlinear evolution equations with higher order, which was given by J. L. Lions, is solved. Effect of our results is shown on an example.     

Optimal Control of an Evolution Problem Involving Time-Dependent Maximal Monotone Operators

Journal of Optimization Theory and Applications - We consider a control problem in a finite-dimensional setting, which consists in finding a minimizer for a standard functional defined by way of...     

On an Operator Theory Approach to the Corona Problem

This paper deals with an operator theory approach to the coronaconjecture for H(Dⁿ). Treil gave a counter-example to this conjecturein the case where n = 1 for operator-valued functions; thusone might hope to use this to disprove the corona conjecturefor H(Dⁿ) (for n 2). This paper shows that this natural approachtowards a negative answer fails. On the other hand, the secondresult here shows that ‘commutant lifting’ cannotbe true for more than two contractions for any constant. Thisobstructs a natural attempted proof of the corona conjecturefor H(Dⁿ) (for n 2) by our previous result.     

On an Extremal Problem About the Arc-Length of Algebraic Polynomials

In this work we investigate polynomials of maximal (minimal) arc-length in the interval [−1, 1] amongst all monic polynomials of fixed degree n with n real zeros in [−1, 1].