求解复杂湍流的非线性涡黏性系数模型和代数应力模型

非线性涡黏性系数模型和代数应力模型联系了线性涡黏性系数湍流模型和完整的微分 雷诺应力模型.随着它们受到日益关注,其形式也越来越多样化.本篇综述的目的是对这些模 型加以总结并比较它们之间的共同点及不同之处,指出它们与完整微分雷诺应力模型之间的 关系,以及相对于线性涡黏性系数模型而言它们在预报流场上所具有的优势.     

Continuation of blowup solutions of nonlinear heat equations in several space dimensions

The possible continuation of solutions of the nonlinear heat equation in R^N × R₊ u_t = Δu^m + u^p with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation u_t = Δu^m − u^p, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc.     

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

The classic problem of regularity of boundary points for higher-order partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener’s (J. Math. Phys. Mass. Inst. Tech. 3:127–146, 1924) and Petrovskii’s (Math. Ann. 109:424–444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960–1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat’ev and Maz’ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:

Eigenfunctions and Very Singular Similarity Solutions of Odd‐Order Nonlinear Dispersion PDEs: Toward a “Nonlinear Airy Function” and Others

Asymptotic properties of nonlinear dispersion equations (1) with fixed exponents n > 0 and p > n+ 1 , and their (2k+ 1) th‐order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy‐deformation” approach, where the limit in the first equation in ( 1 ) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non‐self‐adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [ 1 ]. Various other nonlinear operator and numerical methods for ( 1 ) are also applied. As a key alternative, the “super‐nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in ( 1 ), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values {p=p_l(n)}_l≥0 of the absorption exponents is discussed.     

A quasilinear heat equation with a source: Peaking,localization, symmetry exact solutions,asymptotics, structures

A survey is given of results of investigating unbounded solutions (regimes with peaking) of quasilinear parabolic equations of nonlinear heat conduction with a source. Principal attention is devoted to the investigation of the property of localization of regimes with peaking. A group classification of nonlinear equations of this type is carried out, properties of a broad set of invariant (self-similar) solutions are investigated, and special methods of investigating the space-time structure of unbounded solutions are developed.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 28, pp. 95–206, 1986.     

Very Singular Solutions for Thin Film Equations with Absorption

The large-time behavior of weak nonnegative and sign changing solutions of the fourth-order thin film equation (TFE-4) with absorption where    n  ∈ (0, 3)   and the absorption exponent  p  belongs to the  subcritical  range is studied. First, the standard free-boundary problem with zero-height, zero contact angle, and zero-flux conditions at the interface and bounded compactly supported initial data is considered. Very singular similarity solutions (VSSs) have the form Here  f  solves the quasi-linear degenerate elliptic equation that becomes an ODE for    N  = 1   or in the radial setting in      . By a combination of analytical, asymptotic, and numerical methods, existence of various branches of similarity profiles  f  parameterized by  p  is established. Secondly, changing sign VSSs of the Cauchy problem are described.
This study is motivated by the detailed VSS results for the second-order porous medium equation with absorption   ( u  ≥ 0)  which have been known since the 1980s.     

Calculation of the strength of reinforced phenolic-foam flexural elements

Recommendations are given for calculating the maximum bending moments in poured phenolic-foam elements reinforced with wood laths, glass-reinforced plastic cables, and resin-saturated paper rods.Leningrad Zonal Scientific-Research and Planning Institute for Standard and Experimental Design of Residential and Public Buildings. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 359–361, March–April, 1969.