On the regularity and uniqueness of conically self-similar free-vortex solutions to the Navier-Stokes equations

In this paper the real analyticity of all conically self-similar free-vortex solutions to the Navier-Stokes equations is proven. Furthermore, it is mathematically established that such solutions are uniquely determined by the values of three derivatives on the symmetry axis, and hence a numerical method, invented and successfully used by Shtern & Hussain (1993,1996), is justified mathematically. In addition, it is proven that these results imply that for any conically self-similar free-vortex solution to the Navier--Stokes equations there exists a second order non-swirling correction term. For this term it is also shown that the second order contribution to the total axial flow force vanishes in the cases of the entire space and a half-space, but that it need not vanish for general conical domains. In doing so an old claim by Burggraf & Foster (1977) is established mathematically, however not for Long's problem but for Shtern & Hussain's (1996) extension of this problem to the full Navier-Stokes equations and the entire space.     

Non-Constant Positive Steady States of a Predator-Prey System with Non-Monotonic Functional Response and Diffusion

This paper deals with non-constant positive steady-state solutionsof a predator-prey system with non-monotonic functional response,also called Holling type-IV interaction terms, and diffusionunder the homogeneous Neumann boundary condition. We first establishpositive upper and lower bounds for such solutions, and thenstudy their non-existence, global existence and bifurcation.2000 Mathematics Subject Classification 35J55, 92D25.     

Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class

Let us consider the problem whether there does exist a finite-time self-similar solution of the backward type to the semilinear Keller-Segel system. In the case of parabolic-elliptic type for n?3 we show that there is no such a solution with a finite mass in the scaling invariant class. On the other hand, in the case of parabolic-parabolic type for n?2, non-existence of finite-time self-similar solutions is proved in a larger class of a finite mass with some local bounds.     

Critical curves of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this paper, we consider the existence and non-existence of global solutions of the non-Newtonian polytropic filtration equations with nonlinear boundary conditions. We first obtain the critical global existence curve by constructing various self-similar supersolutions and subsolutions. And then the critical Fujita curve is conjectured with the aid of some new results.     

A New Construction of the Baby Monster and Its Applications

In this paper we show how to construct 4370 x 4370 matricesover GF(2), generating Fischer's {3,4}- transposition groupB, popularly known as the Baby Monster.This for the first timegives a construction which permits effective calculations tobe performed in this very large simple group.We use this togive a new existence proof, and describe briefly applicationsto subgroup structure, geometry, 2-modular characters, and genusactions     

A criterion for the non-existence or non-uniqueness of solutions of self-similar problems in the mechanics of a continuous medium

Systems of hyperbolic partial differential equations expressing conservation laws are considered. A sufficient condition is formulated under which the self-similar problem of the disintegration of an arbitrary discontinuity (or the “piston” problem) either has no solution or the solution is not unique. This sufficient condition is determined by the existence of non-evolutionary discontinuities which may be considered as a sequence of two evolutionary discontinuities moving at the same velocity, if such a representation is unique. The condition is more general than that formulated previously, which was based on the existence of a non-proper Jouguet point. The new criterion is satisfied by weak quasitranverse shock waves in elastic media, whatever the sign of the coefficient of the non-linear deformation term. It also enables one to draw conclusions as to the non-existence or non-uniqueness of solutions of problems of the theory of elasticity in the case of finite-amplitude waves.     

Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

In this paper, we consider the semilinear wave equation with a power nonlinearity in one space dimension. We exhibit a universal one-parameter family of functions which stand for the blow-up profile in self-similar variables at a non-characteristic point, for general initial data. The proof is done in self-similar variables. We first characterize all the solutions of the associated stationary problem, as a one parameter family. Then, we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm, in the non-characteristic case, and to a finite decoupled sum of such a solution in the characteristic case. Finally, in the case where this sum is reduced to one element, which is the case for non-characteristic points, we use modulation theory coupled with a nonlinear argument to show the exponential convergence (in the self-similar time variable) of the various parameters and conclude the proof. This step provides us with a result of independent interest: the trapping of the solution in self-similar variables near the set of stationary solutions, valid also for non-characteristic points. The proof of these results is based on a new analysis in the self-similar variable.     

非线性抛物椭圆方程组的正则解和奇异解

该文用单模方法在Lorentz空间研究了抛物椭圆方程组奇异解和正则解的存在性, 其中初值属于Lorentz空间L^{n/2,∞ ₍}Rⁿ), n≥ 3. 利用时间加权的Lorentz空间, 还得到了其正则解. 此外, 如果初值满足自相似结构, 也得到了自相似解的存在性.     

Self-similar solutions to the isothermal compressible Navier-Stokes equations

^** Email: guo_zhenhua{at}iapcm.ac.cn^*** Email: jiang{at}iapcm.ac.cn We investigate the self-similar solutions to the isothermalcompressible Navier–Stokes equations. The aim of thispaper is to show that there exist neither forward nor backwardself-similar solutions with finite total energy. This generalizesthe results for the incompressible case in Neas, J., Rika, M.& verák, V. (1996, On Leray's self-similar solutionsof the Navier-Stokes equations. Acta. Math., 176, 283–294),and is consistent with the (unproved) existence of regular solutionsglobally in time for the compressible Navier–Stokes equations.     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

Blowup in the Nonlinear Schrödinger Equation Near Critical Dimension

This article contains an analysis of the cubic nonlinear Schrödinger equation and solutions that become singular in finite time. Numerical simulations show that in three dimensions the blowup is self-similar and symmetric. In two dimensions, the blowup still appears to be symmetric but is no longer self-similar. In the case that the dimension, d, is greater than and exponentially close to 2 in terms of a small parameter associated to the norm of the blow-up solution, a locally unique, monotonically decreasing in modulus, self-similar solution that satisfies the boundary and global conditions associated with the blow-up solution is constructed in Kopell and Landman [1995, SIAM J. Appl., Math.55, 1297-1323]. In this article, it is shown that this locally unique solution also exists for d > 2 and algebraically close to 2 in the same small parameter. The central idea of the proof involves constructing a pair of manifolds of solutions (to the nonautonomous ordinary differential equation satisfied by the self-similar solutions) that satisfy the conditions at r = 0 and the asymptotic conditions respectively and then showing that these intersect transversally. A key step involves tracking one of the manifolds over a midrange in which the ordinary differential equation has a turning point and hence obtaining good control over the solutions on the manifold.     

Radial equivalence for the two basic nonlinear degenerate diffusion equations

In this paper we prove that there exists an explicit correspondence between the radially symmetric solutions of two well-known models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. We establish exact correspondence formulas between these solutions. We also study in detail the application of the results in the important case of self-similar solutions. In particular, we derive the existence of new self-similar solutions for the evolution p-Laplacian equation.     

Non-local boundary value problems of arbitrary order

We give a new unified method of establishing the existence ofmultiple positive solutions for a large number of non-lineardifferential equations of arbitrary order with any allowed numberof non-local boundary conditions (BCs). In particular, we areable to determine the Green's function for these problems withvery little explicit calculation, which shows that studyinga more general version of a problem with appropriate notationcan lead to a simplification in approach. We obtain existenceand non-existence results, some of which are sharp, and givenew results for both non-local and local BCs. We illustratethe theory with a detailed account of a fourth-order problemthat models an elastic beam and also determine optimal valuesof constants that appear in the theory.     

Existence of Solutions for Asymptotically 'Linear' p-Laplacian Equations

Under very general conditions, a proof is given, via Morse theory,of the existence of solutions for asymptotically ‘linear’p-Laplacian equations, where the asymptotic limit may be greaterthan the second eigenvalue. The existence of nonzero solutionsis also considered. 2000 Mathematics Subject Classification35J65, 58E05.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.     

Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong L^p{\cal L}^{p} space is the null solution, infinitely many self-similar solutions do exist in weak- L^p{\cal L}^{p} spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.     

A Class of Dust-Like Self-Similar Solutions of the Massless Einstein�CVlasov System

In this paper the existence of a class of self-similar solutions of the Einstein–Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein–Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein–Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point P ₀ and converges to a stationary solution P ₁ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes. The spacetimes constructed do not constitute a counterexample to cosmic censorship since they are not asymptotically flat. They should be seen as the first step on a possible path towards constructing solutions of importance for understanding the issue of the formation of naked singularities in general relativity.     

Existence and Nonexistence of Global Solutions for a Semi-Linear Heat Equation with Fractional Laplacian

In this paper, we are concerned with the existence and non-existence of global solutions of a semi-linear heat equation with fractional Laplacian. We obtain some extension of results of Weissler who considered the case α = 1, and h ≡ 1.     

Self-similar solutions for dyadic models of the Euler equations

We show existence of self-similar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin [1]. The proof is based on the analysis of certain dynamical systems on the plane.