Note for “Some Exact Blowup Solutions to the pressureless Euler equations in R” [Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2993-2998]

In this supplementary note, we can generalize the exact solutions for the pressureless Euler equations in [Yuen MW. Some exact blowup solutions to the pressureless Euler equations in R^N, Commun. Nonlinear Sci. Numer. Simulat. 2011;16:2993-8]. Here, the solutions are constructed in implicit or explicit forms.     

Geometry of 2×2 Hermitian matrices over any division ring

Let D be a division ring with an involution-,H2(D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A-B) be the arithmetic distance between A,B ∈ H2(D) . In this paper,the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D(char(D) = 2) is proved:if  :H2(D) → H2(D) is the adjacency preserving bijective map,then  is of the form (X) = tP XσP +(0) ,where P ∈ GL2(D) ,σ is a quasi-automorphism of D. The quasi-automorphism of D is studied,and further results are obtained.     

Invariant Metrics and Laplacians on Siegel-Jacobi Disk

Let Dn be the generalized unit disk of degree n. In this paper, Riemannian metrics on the Siegel-Jacobi disk Dn × C（^m,n） which are invariant under the natural action of the Jacobi group are found explicitly and the Laplacians of these invariant metrics are computed explicitly. These are expressed in terms of the trace form.     

The fundamental theory of abstract majorization inequalities

Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ → Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1...     

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.     

Quasilinear elliptic equations with Neumann boundary and singularity

Let Ω be a bounded domain with a smooth C2 boundary in RN（N ≥ 3）, 0 ∈Ω, and n denote the unit outward normal to ЭΩ.We are concerned with the Neumann boundary problems： -div（｜x｜α｜△u｜p-2△u）=｜x｜βup（α,β）-1-λ｜x｜γup-1,u（x）〉0,x∈Ω,Эu/Эn=0 on ЭΩ,where 1〈p〈N and α〈0,β〈0 such that p（α,β）△=p（N＋β）/N-p＋α〉p,y〉α-p.For various parameters α,βorγ,we establish certain existence results of the solutions in the case 0∈Ω or 0∈ЭΩ.     

Boundedness of commutators on homogeneous Morrey-Herz spaces over locally compact Vilenkin groups

This paper studies some boundedness results of commutators on a class of new spaces M K˙pα,,qλ (G) named as homogenous Morrey-Herz spaces over locally compact Vilenkin groups.     

The Behavior of Solutions of Multidimensional Aggregation Equations with Mildly Singular Interaction Kernels

The authors consider the multidimensional aggregation equation tp-div（p K＊ ρ） = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin （Lipschitz or better）, and review recent results on this problem concerning well-posedness of nonnegative solutions and finite time blowup in multiple space dimensions depending on the behavior of the kernel at the origin. The problem with bounded initial data, data in L^p ∩ L^1, and measure solutions are also considered.     

An Erdös-Ko-Rado theorem for restricted signed sets

A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.     

Elementary characterizations of generalized weighted Morrey-Campanato spaces

Let α∈ (0,∞), p, q ∈ [1,∞), s be a nonnegative integer, and ω∈ A1(Rn) (the class of Muckenhoupt's weights). In this paper, we introduce the generalized weighted Morrey-Campanato space L(α, p, q, s, ω; Rn) and obtain its equivalence on different p ∈ [1,β) and integers s ≥ nα (the integer part of nα), where β = (1q - α)-1 when α 1q or β = ∞ when α≥ 1q. We then introduce the generalized weighted Lipschitz space ∧(α, q, ω; Rn) and prove that L(α, p, q, s, ω; Rn)  ∧(α, q, ω; Rn) when α∈ (0,∞), s ≥ nα , and p ∈ [1,β).     

Analytical Blowup Solutions to the Compressible Euler Equations with Time-depending Damping

In this paper, the analytical blowup solutions of the N-dimensional radial symmetric compressible Euler equations are constructed. Some previous results of the blowup solutions for the compressible Euler equations with constant damping are generalized to the time-depending damping case. The generalization is untrivial because that the damp coefficient is a nonlinear function of time t.

     

Optimal estimates for blowup rate and behavior for nonlinear heat equations

We first describe all positive bounded solutions of where \input amstex \loadmsbm $(y,s)\in \Bbb R^N\times \Bbb R$ , 1 < p, and (N − 2)p ≤ N + 2. We then obtain for blowup solutions u(t) of uniform estimates at the blowup time and uniform space-time comparison with solutions of u′ = u^p. © 1998 John Wiley & Sons, Inc.     

On stably -monotone Banach couples

The $ \vec E $ \vec E = (E, E(2^-k )) is a stably $ \vec E $ \vec E is $ \vec E $ \vec E is $ \vec E $ \vec E is -monotone, then either E = l _p (1 ≤ p < ∞) or E = c ₀.     

Energy criterion of global existence for energy-critical Schrödinger equations with subcritical perturbations

We present a variational approach to study the energy-critical Schrödinger equations with subcritical perturbations. Through analysing the Hamiltonian property we establish two types of invariant evolution flows, and derive a new sharp energy criterion for blowup of solutions for the equation. Furthermore, we answer the question: how small are the initial data such that the solutions of this equation are bounded in H ¹(R ^N )?     

A Blow-up Criterion of Strong Solutions to the Compressible Viscous Heat-Conductive Flows with Zero Heat Conductivity

We study the compressible Navier-Stokes equations of viscous heat-conductive fluids in a periodic domain \mathbbT³\mathbb{T}^{3} with zero heat conductivity k=0. We prove a blow-up criterion for the local strong solutions in terms of the temperature and positive density, similar to the Beale-Kato-Majda criterion for ideal incompressible flows.     

Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R². Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case (γ=1), the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.     

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.     

Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density

We present a sufficient condition on the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, it is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier-Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.     

Blow up of incompressible Euler solutions

We present analytical and computational evidence of blowup of initially smooth solutions of the incompressible Euler equations into non-smooth turbulent solutions. We detect blowup by observing increasing L ₂-residuals of computed solutions under decreasing mesh size. AMS subject classification (2000)  35Q30, 65M60