A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma
M.
Escobedo;
M.
A.
Herrero;
J.
J. L.
Velazquez
3837-3901
Abstract: This work deals with the problem consisting in the equation \begin{equation*}{\frac{\partial f}{\partial t}} ={\frac{1}{x^{2}}}{\frac{\partial }{\partial x}} [x^{4}({\frac{\partial f}{\partial x}}+f+f^{2})], \quad \hbox {when}\quad x\in (0,\infty ), t>0, \tag*{(1)}\end{equation*} together with no-flux conditions at $x=0$ and $x=+\infty$, i.e. \begin{equation*}x^{4}({\frac{\partial f}{\partial x}}+f+f^{2})=0\quad \hbox {as}\hskip 0.2cm x\mathop{\longrightarrow } 0 \hskip 0.2cm\hbox {or}\hskip 0.2cmx\mathop{\longrightarrow } +\infty . \tag*{(2)}\end{equation*} Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution $f(x, t)$ in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of $(1)$, $(2)$ which develop singularities near $x=0$ in a finite time, regardless of how small the initial number of photons $N(0)=\int _{0}^{+\infty }x^{2}f(x, 0)dx$ is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition $(2)$ is lost at $x=0$ when the singularity unfolds. The corresponding blow-up pattern is shown to be asymptotically of a shock wave type. In rescaled variables, it consists in an imploding travelling wave solution of the Burgers equation near $x=0$, that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing $(2)$ near $x=0$ as determined by the manner of blow-up, such solutions can be continued for all times after the onset of the singularity.
Stability results on interpolation scales of quasi-Banach spaces and applications
Nigel
Kalton;
Marius
Mitrea
3903-3922
Abstract: We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented.
Local and global properties of limit sets of foliations of quasigeodesic Anosov flows
Sérgio
R.
Fenley
3923-3941
Abstract: A nonsingular flow is quasigeodesic when all flow lines are efficient in measuring distances in relative homotopy classes. We analyze the quasigeodesic property for Anosov flows in $3$-manifolds which have negatively curved fundamental group. We show that this property implies that limit sets of stable and unstable leaves (in the universal cover) vary continuously in the sphere at infinity. It also follows that the union of the limit sets of all stable (or unstable) leaves is not the whole sphere at infinity. Finally, under the quasigeodesic hypothesis we completely determine when limit sets of leaves in the universal cover can intersect. This is done by determining exactly when flow lines in the universal cover share an ideal point.
Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields
Wilfried
H.
Paus
3943-3966
Abstract: In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space. We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat. These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.
On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces
Athanassios
G.
Kartsatos
3967-3987
Abstract: A more systematic approach is introduced in the theory of zeros of maximal monotone operators $T:X\supset D(T)\to 2^{X^{*}}$, where $X$ is a real Banach space. A basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator $T$. These conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the Yosida resolvents of $T$. Furthermore, several interesting corollaries are given, and the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined, demicontinuous, monotone mappings is demonstrated. A result of Guan, about a pathwise connected set lying in the range of a monotone operator, is improved by including non-convex domains. A partial answer to Nirenberg's problem is also given. Namely, it is shown that a continuous, expansive mapping $T$ on a real Hilbert space $H$ is surjective if there exists a constant $\alpha \in (0,1)$ such that $\langle Tx-Ty,x-y\rangle \ge -\alpha \|x-y\|^{2},~x,~y\in H.$ The methods for these results do not involve explicit use of any degree theory.
Rigidity and topological conjugates of topologically tame Kleinian groups
Ken'ichi
Ohshika
3989-4022
Abstract: Minsky proved that two Kleinian groups $G_1$ and $G_2$ are quasi-conformally conjugate if they are freely indecomposable, the injectivity radii at all points of $\bold{H}^3/G_1$, $\bold{H}^3/G_2$ are bounded below by a positive constant, and there is a homeomorphism $h$ from a topological core of $\bold{H}^3/G_1$ to that of $\bold{H}^3/G_2$ such that $h$ and $h^{-1}$ map ending laminations to ending laminations. We generalize this theorem to the case when $G_1$ and $G_2$ are topologically tame but may be freely decomposable under the same assumption on the injectivity radii. As an application, we prove that if a Kleinian group is topologically conjugate to another Kleinian group which is topologically tame and not a free group, and both Kleinian groups satisfy the assumption on the injectivity radii as above, then they are quasi-conformally conjugate.
Infinite type homeomorphisms of the circle and convergence of Fourier series
Antônio
Zumpano
4023-4040
Abstract: We consider the problem of convergence of Fourier series when we make a change of variable. Under a certain reasonable hypothesis, we give a necessary and sufficient condition for a homeomorphism of the circle to transform absolutely convergent Fourier series into uniformly convergent Fourier series.
Test ideals in quotients of $F$-finite regular local rings
Janet
Cowden
Vassilev
4041-4051
Abstract: Let $S$ be an $F$-finite regular local ring and $I$ an ideal contained in $S$. Let $R=S/I$. Fedder proved that $R$ is $F$-pure if and only if $(I^{[p]}:I) \nsubseteq \mathfrak{m}^{[p]}$. We have noted a new proof for his criterion, along with showing that $(I^{[q]}:I) \subseteq (\tau ^{[q]}:\tau )$, where $\tau$ is the pullback of the test ideal for $R$. Combining the the $F$-purity criterion and the above result we see that if $R=S/I$ is $F$-pure then $R/\tau$ is also $F$-pure. In fact, we can form a filtration of $R$, $I \subseteq \tau = \tau _{0} \subseteq \tau _{1} \subseteq \ldots \subseteq \tau _{i} \subseteq \ldots$ that stabilizes such that each $R/\tau _{i}$ is $F$-pure and its test ideal is $\tau _{i+1}$. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let $R=T/I$, where $T$ is either a polynomial or a power series ring and $I= P_{1} \cap \ldots \cap P_{n}$ is generated by monomials and the $R/P_{i}$ are regular. Set $J = \Sigma (P_{1} \cap \ldots \cap \hat {P_{i}} \cap \ldots \cap P_{n})$. Then $J=\tau =\tau _{par}$.
Regularity of solutions to a contact problem
Russell
M.
Brown;
Zhongwei
Shen;
Peter
Shi
4053-4063
Abstract: We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.
Self-similar measures and intersections of Cantor sets
Yuval
Peres;
Boris
Solomyak
4065-4087
Abstract: It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-$\alpha$ Cantor set and $\alpha \in ({1 \over 3}, \frac 12)$. We show that for any compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic sum of $K$ and the middle-$\alpha$ Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential supremum, as the translation parameter $t$ varies, of the dimension of the intersection of $K+t$ with the middle-$\alpha$ Cantor set. We also establish a new property of the infinite Bernoulli convolutions $\nu _\lambda^p$ (the distributions of random series $\sum _{n=0}^\infty \pm \lambda^n ,$ where the signs are chosen independently with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$. For $p \neq \frac 12$ near $\frac 12$ and for a.e. $\lambda$ in some nonempty interval, $\nu _\lambda^p$ is absolutely continuous and its density is in $L^{q_1}$ but not in $L^{q_2}$. We also answer a question of Kahane concerning the Fourier transform of $\nu _\lambda ^{\scriptscriptstyle 1/2}$.
An inverse problem for scattering by a doubly periodic structure
Gang
Bao;
Zhengfang
Zhou
4089-4103
Abstract: Consider scattering of electromagnetic waves by a doubly periodic structure $S=\{x_3=f(x_1, x_2)\}$ with $f(x_1+n_1\Lambda _1, x_2+n_2\Lambda _2)=f(x_1, x_2)$ for integers $n_1$, $n_2$. Above the structure, the medium is assumed to be homogeneous with a constant dielectric coefficient. The medium is a perfect conductor below the structure. An inverse problem arises and may be described as follows. For a given incident plane wave, the tangential electric field is measured away from the structure, say at $x_3=b$ for some large $b$. To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain $\Omega$: \begin{displaymath}\left\{ \begin{array}{l} - \triangle u = \lambda u \quad \text{in} \quad \Omega, \nabla \cdot u = 0 \quad \text{in} \quad \Omega, n \times u = 0 \quad \text{on} \quad \partial \Omega. \end{array} \right. \end{displaymath}
On asymptotic approximations of the residual currents
Alekos
Vidras;
Alain
Yger
4105-4125
Abstract: We use a ${\cal D}$-module approach to discuss positive examples for the existence of the unrestricted limit of the integrals involved in the approximation to the Coleff-Herrera residual currents in the complete intersection case. Our results also provide asymptotic developments for these integrals.
The Ext class of an approximately inner automorphism
Akitaka
Kishimoto;
Alex
Kumjian
4127-4148
Abstract: Let $A$ be a simple unital $A\mathbf{T}$ algebra of real rank zero. It is shown below that the range of the natural map from the approximately inner automorphism group to $KK(A, A)$ coincides with the kernel of the map $KK(A, A) \rightarrow \bigoplus _{i=0}^{1} \operatorname{Hom}(K_i(A), K_i(A))$.
On free actions, minimal flows, and a problem by Ellis
Vladimir
G.
Pestov
4149-4165
Abstract: We exhibit natural classes of Polish topological groups $G$ such that every continuous action of $G$ on a compact space has a fixed point, and observe that every group with this property provides a solution (in the negative) to a 1969 problem by Robert Ellis, as the Ellis semigroup $E(U)$ of the universal minimal $G$-flow $U$, being trivial, is not isomorphic with the greatest $G$-ambit. Further refining our construction, we obtain a Polish topological group $G$ acting freely on the universal minimal flow $U$ yet such that ${\mathcal S}(G)$ and $E(U)$ are not isomorphic. We also display Polish topological groups acting effectively but not freely on their universal minimal flows. In fact, we can produce examples of groups of all three types having any prescribed infinite weight. Our examples lead to dynamical conclusions for some groups of importance in analysis. For instance, both the full group of permutations $S(X)$ of an infinite set, equipped with the pointwise topology, and the unitary group $U(\mathcal{H})$ of an infinite-dimensional Hilbert space with the strong operator topology admit no free action on a compact space, and the circle $\mathbb{S}^{1}$ forms the universal minimal flow for the topological group ${\operatorname {Homeo}\,}_{+}(\mathbb{S}^{1})$ of orientation-preserving homeomorphisms. It also follows that a closed subgroup of an amenable topological group need not be amenable.
The structure of indecomposable injectives in generic representation theory
Geoffrey
M. L.
Powell
4167-4193
Abstract: This paper considers the structure of the injective objects $I_{V_{n}}$ in the category $\mathcal F$ of functors between ${\mathbb F}_2$-vector spaces. A co-Weyl object $J_\lambda$ is defined, for each simple functor $F_\lambda$ in $\mathcal F$. A functor is defined to be $J$-good if it admits a finite filtration of which the quotients are co-Weyl objects. Properties of $J$-good functors are considered and it is shown that the indecomposable injectives in $\mathcal F$ are $J$-good. A finiteness result for proper sub-functors of co-Weyl objects is proven, using the polynomial filtration of the shift functor $\tilde\Delta: \mathcal F \rightarrow \mathcal F$. This research is motivated by the Artinian conjecture due to Kuhn, Lannes and Schwartz.
On the rigidity theorem for elliptic genera
Anand
Dessai;
Rainer
Jung
4195-4220
Abstract: We give a detailed proof of the rigidity theorem for elliptic genera. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level $N$.
Computations in generic representation theory: maps from symmetric powers to composite functors
Nicholas
J.
Kuhn
4221-4233
Abstract: If ${\bold F}_q$ is the finite field of order $q$ and characteristic $p$, let ${\cal F}(q)$ be the category whose objects are functors from finite dimensional ${\bold F}_q$-vector spaces to ${\bold F}_q$-vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in ${\cal F}(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in {\bold F}_q[\Sigma _n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma _n}$,$S^n(V) = V^{\otimes n}/\Sigma _n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$. Fixing $F$, we discuss the problem of computing $\operatorname{Hom}_{{\cal F}(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $\operatorname{Hom}_{{\cal F}(q)}(S_m, G)$ for all $m$. When $q = p$, we get a complete answer for any functor $F$ chosen from the families listed above. Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when $F=S^n$, arose in recent work on the homology of iterated loopspaces.
Universal maps on trees
Carl
Eberhart;
J.
B.
Fugate
4235-4251
Abstract: A map $f:R \to S$ of continua $R$ and $S$ is called a universal map from $R$ to $S$ if for any map $g:R \to S$, $f(x) = g(x)$ for some point $x \in R$. When $R$ and $S$ are trees, we characterize universal maps by reducing to the case of light minimal universal maps. The characterization uses the notions of combinatorial map and folded subedge of $R$.
Chung's law for integrated Brownian motion
Davar
Khoshnevisan;
Zhan
Shi
4253-4264
Abstract: The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.