A variational approach to transonic potential flow problems

This paper continues considerations of transonic potential flow problems by variational methods. A functional which is associated with a boundary value problem for the (full) potential equation and which possesses a real physical meaning is minimized over a class of admissible functions. These functions have to satisfy a non‐linear local entropy condition and a certain boundness constraint. Though this class is not a compact set of the underlying Hilbert space and though the functional need not be convex, the existence of a solution to the established variational problem can be proved by direct methods of the calculus of variations. Furthermore, some properties of minimizers concerning uniqueness, relation to the potential equation, and behaviour on supersonic regions are derived. Copyright © 2000 John Wiley & Sons, Ltd.     

The solution of transonic flow problems by the method of stabilization

The paper deals with the study of the perturbated Full Potential Equation. The proofs of existence, unicity and regularity of the solution are given. The main result consists in the fact that the resulting strong solution is the generic solution for the Transonic Flow Problem     

Remarks on the solvability of a convolution integral equation on a finite interval

We present some results on the solvability of an integral equation of the second kind with a difference kernel on a finite interval, construct a counterexample to an assertion, earlier believed to have been proved, on the solvability of this equation, and pose an open problem.     

关于Q-曲率流的几个注记

标准球面上的预定纯量曲率问题已经得到很好的研究,为此,几种不同的方法发展起来了.它的高维对应的问题即预定Q-曲率问题也吸引了众多人的注意.负梯度流的方法在处理此类问题上似乎十分有效,至少在曲率备选函数是正的情形下是如此的.本文的目的在于指出,对于指数非线性增长的预定Q-曲率问题,Q-曲率备选函数的正性假设是不必要的.     

Two-dimensional choked transonic flow

Summary Choked transonic flow about a thin two-dimensional airfoil in a wide wind tunnel is analyzed. Asymptotic expansions are constructed in terms of the transonic similarity variableK _T as K_T0 (and the tunnel height), in an inner region, near the body, and in an outer region, near the walls, and are matched. The inner expansion consists of sonic free flow plus a correction. The outer expansion accounts for the meeting of the sonic line and the limit characteristic at the wall. The magnitude of the correction to sonic free flight at the body as well as the dependence of the choking Mach number on the wind tunnel height are determined. A boundary value problem is formulated to determine the correction to the free flight sonic flow.

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Zusammenfassung Es wird die schallnahe Sperrströmung um einen zweidimensionalen Flügel in einem weiten Windkanal analysiert. Asymptotische Reihenentwicklungen werden in der transsonischen Ähnlich-KeitsvariablenK _T im inneren Bereich in der Nähe des Körpers und im äußeren Bereich nahe der Wandung konstruiert und verglichen. Die innere Reihenentwicklung besteht aus freier Schallströmung und einem Korrektionsglied. Die äußere Reihenentwicklung steht für das Zusammentreffen der Schallinie und der Grenzcharacteristik an der Wandung. Es werden die Größe des Korrektionsgliedes zur freien Schallströmung am Körper, als auch die Abhängigkeit der Machschen Zahl der freien Strömung von der Höhe des Windkanals bestimmt. Ein Randwertproblem wird formuliert um die Korrektur zur freien Schallströmung zu bestimmen.

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Research partly sponsored by AFOSR under Grant No. 82-0155 and by NSF under Grant DMS-8401738. The U.S. Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation thereon.     

On solvability of a space-periodic problem for the nonstationary VT-equation of transonic gas dynamics

We consider the problem of existence and uniqueness of a solution to the nonstationary VT-equation of transonic gas dynamics which is periodic in an appropriate space variable and study the question of smoothness of this solution in time.     

Nonlinear singular sturm-liouville problems and an application to transonic flow through a nozzle

We consider a class of singular Sturm-Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction-diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.     

Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let f_r(P, d) denote the smallest positive integer such that given any sequence of f_r(p, d) elements in (Z/pZ(^d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(^d. That f₁(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f₁(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of f_r(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of f_r(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f₁(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.     

On the solvability of some nonlinear Elliptic problems

The solvability of second-order nonlinear elliptic equations in weighted Sobolev spaces is analyzed. An additional condition ensuring the solvability of such equations is that the average of the desired solution over some circle of fixed radius is zero. Examples are equations containing a weighted p-Laplacian and the Euler equations.     

On the solvability of physically nonlinear problems of Thermoelasticity

We study the problem of existence and uniqueness of generalized solutions of nonlinear vector boundary-value problems arising in the physically nonlinear theory of thermoelasticity. We prove the convergence of iteration processes in the space W ¹ ₂.     

A critical case for the solvability of stefan-like problems

We consider a one-phase one-dimensional Stefan problem with general data with the aim to investigate some open questions on existence of classical solutions. We show how existence and nonexistence are discriminated by the behavior of the initial datum in the neighborhood of the starting point of the free boundary.     

Viscosity method in a transonic flow

We prove that the solution of a nonviscous compressible transonic flow can be obtained as a limit of viscous solutions, if the viscosity and heat conductivity tend to zero. To obtain an isentropic irrotational flow it is necessary to control the entropy and temperature on the boundary in a convenient way     

Remarks on elliptic singular perturbation problems

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton—Jacobi—Bellman equations with a small parameter.H. Ishii was supported in part by the AFOSR under Grant No. AFOSR 85-0315 and the Division of Applied Mathematics, Brown University.     

Remarks on elliptic problems involving the Caffarelli-Kohn-Nirenberg inequalities

We give some regularity results of the solutions and a Liouville type theorem to singular elliptic equations involving the Caffarelli-Kohn- Nirenberg inequalities.