Finite Hausdorff Research Articles

Invariant Measures and Global Well Posedness for the SQG Equation

We construct an invariant measure $\mu$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $\mu$ are initial conditions of global, unique solutions of SQG, that depend continuously on the initial data. In addition, we show that the support of $\mu$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measures $\mu$ is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.

L2-boundedness of gradients of single-layer potentials and uniform rectifiability

Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with H\"older continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $\mu$ be a compactly supported $n$-AD-regular measure in $\mathbb R^{n+1}$ and consider the associated operator $$T_\mu f(x) = \int \nabla_x\mathcal E_A(x,y)\,f(y)\,d\mu(y).$$ We show that if $T_\mu$ is bounded in $L^2(\mu)$, then $\mu$ is uniformly $n$-rectifiable. This extends the solution of the codimension $1$ David-Semmes problem for the Riesz transform to the gradient of the single layer potential. Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E\subset\mathbb R^{n+1}$ with finite Hausdorff measure $\mathcal H^n$, if $T_{\mathcal H^n|_E}$ is bounded in $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. Further, as an application we show that if the elliptic measure associated to the above PDE is absolute continuous with respect to surface measure, then it must be rectifiable, analogously to what happens with harmonic measure.

An existence theorem for Brakke flow with fixed boundary conditions

Consider an arbitrary closed, countably n-rectifiable set in a strictly convex (n+1)-dimensional domain, and suppose that the set has finite n-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As t uparrow infty , the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.

Disintegration property of coherent upper conditional previsions with respect to Hausdorff outer measures for unbounded random variables

In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures on the linear space of all absolutely Choquet integrable random variables. They are proven to satisfy the disintegration property on every non-null partition if Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Coherent upper conditional probabilities are obtained as restrictions to the indicators function and it is proven they can be extended as finitely additive probabilities in the sense of Dubins. Examples are given in the discrete metric space and in the Euclidean metric space.

Exact Hausdorff and packing measures for random self-similar code-trees with necks

Random code-trees with necks were introduced recently to generalise the notion of $V$-variable and random homogeneous sets. While it is known that the Hausdorff and packing dimensions coincide irrespective of overlaps, their exact Hausdorff and packing measure has so far been largely ignored. In this article we consider the general question of an appropriate gauge function for positive and finite Hausdorff and packing measure. We first survey the current state of knowledge and establish some bounds on these gauge functions. We then show that self-similar code-trees do not admit a gauge functions that simultaneously give positive and finite Hausdorff measure almost surely. This surprising result is in stark contrast to the random recursive model and sheds some light on the question of whether $V$-variable sets interpolate between random homogeneous and random recursive sets. We conclude by discussing implications of our results.

The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping

In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space E0 and Ek, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor Ak in space Ek. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on Ek via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor Ak.

Quasiconformal Jordan Domains

AbstractWe extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y,dY). We say that a metric space (Y,dY) is aquasiconformal Jordan domainif the completion ̄Yof (Y,dY) has finite Hausdorff 2-measure, theboundary∂Y= ̄Y\Yis homeomorphic to 𝕊1, and there exists a homeomorphismϕ: 𝔻 →(Y,dY) that is quasiconformal in the geometric sense.We show thatϕhas a continuous, monotone, and surjective extension Φ: 𝔻 ̄ →Ȳ. This result is best possible in this generality. In addition, we find a necessary and sufficient condition forΦto be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction ofΦto 𝕊1being a quasisymmetry and to∂Ybeing bi-Lipschitz equivalent to a quasicircle in the plane.

Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"

<p style='text-indent:20px;'>The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence of an exponential attractor for the semigroup associated with the transformed (equivalent) autonomous Kelvin-Voigt fluid flow equations with "fading memory". Finally, we show that the semigroup has Ladyzhenskaya's squeezing property and hence is quasi-stable, which also implies the existence of global as well as exponential attractor having finite fractal dimension.</p>

Multi-dimensional Interpretations of Presburger Arithmetic in Itself

Abstract Presburger arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by Visser (1998, An overview of interpretability logic. Advances in Modal Logic, pp. 307–359). In order to prove the result, we show that all linear orderings that are interpretable in $({\mathbb{N}},+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.

Federer’s characterization of sets of finite perimeter in metric spaces

Federer’s characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension 1. In complete metric spaces that are equipped with a doubling measure and support a Poincare inequality, the “only if” direction was shown by Ambrosio (2002). By applying fine potential theory in the case p=1, we prove that the “if” direction holds as well.

On the universal completion of pointfree function spaces

This paper approaches the construction of the universal completion of the Riesz space C(L) of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that C(L) and C(M) have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that C(L) is universally complete as almost Boolean frames. The application of this last result to the classical case C(X) of the space of continuous real functions on a topological space X characterizes those spaces X for which C(X) is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.

Long-Time Dynamics for a Nonlinear Viscoelastic Kirchhoff Plate Equation

This paper is devoted to study the long-time dynamics for a nonlinear viscoelastic Kirchhoff plate equation. Under some growth conditions of g and f, the existence of a global attractor is granted. Furthermore, in the subcritical case, this global attractor has finite Hausdorff and fractal dimensions.

Functions of Bounded Variation on Complete and Connected One-Dimensional Metric Spaces

AbstractIn this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda [18]. Furthermore, we study the necessity of conditions on the underlying space in Federer’s characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

Discrete convolutions of $$\mathrm {BV}$$ functions in quasiopen sets in metric spaces

We study fine potential theory and in particular partitions of unity in quasiopen sets in the case $$p=1$$. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique to show that every function of bounded variation ($$\mathrm {BV}$$ function) can be approximated in the $$\mathrm {BV}$$ and $$L^{\infty }$$ norms by $$\mathrm {BV}$$ functions whose jump sets are of finite Hausdorff measure. Our results seem to be new even in Euclidean spaces but we work in a more general complete metric space that is equipped with a doubling measure and supports a Poincare inequality.

Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations

The dynamics of three dimensional Kelvin-Voigt-Brinkman- Forchheimer equations in bounded domains is considered in this work. The existence and uniqueness of strong solution to the system is obtained by exploiting the \begin{document}$ m $\end{document} -accretive quantization of the linear and nonlinear operators. The long-term behavior of solutions of such systems is also examined in this work. We first establish the existence of an absorbing ball in appropriate spaces for the semigroup associated with the solutions of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Then, we prove that the semigroup is asymptotically compact, which implies the existence of a global attractor for the system. Next, we show the differentiability of the semigroup with respect to the initial data and then establish that the global attractor has finite Hausdorff and fractal dimensions. Furthermore, we establish the existence of an exponential attractor and discuss about its fractal dimensions for the associated semigroup of such systems. Finally, we discuss about the inviscid limit of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations to the 3D Navier-Stokes-Voigt system and then to the simplified Bardina model.

A Gradient Flow of Isometric $$\mathrm {G}_2$$-Structures

We study a flow of $G_2$ structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the connection gives a nice diffusion-reaction equation for the torsion along the flow. We define a quantity for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi on the mean curvature flow, we define an entropy functional and after proving an $\epsilon$-regularity theorem, we show that low entropy initial data lead to solutions of the flow which exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We also study the finite-time singularities and show that at the singular time the flow converges to a smooth $G_2$ structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton for the flow.

Removable sets for intrinsic metric and for holomorphic functions

We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.

Quasi-isometries of pairs: Surfaces in graph manifolds

We show there exists a closed graph manifold [Formula: see text] and infinitely many non-separable, horizontal surfaces [Formula: see text] such that there does not exist a quasi-isometry [Formula: see text] taking [Formula: see text] to [Formula: see text] within a finite Hausdorff distance when [Formula: see text].

Singular quasisymmetric mappings in dimensions two and greater

For all n≥2, we construct a metric space (X,d) and a quasisymmetric mapping f:[0,1]n→X with the property that f−1 is not absolutely continuous with respect to the Hausdorff n-measure on X. That is, there exists a Borel set E⊂[0,1]n with Lebesgue measure |E|>0 such that f(E) has Hausdorff n-measure zero. The construction may be carried out so that X has finite Hausdorff n-measure and |E| is arbitrarily close to 1, or so that |E|=1. This gives a negative answer to a question of Heinonen and Semmes.

Stochastic Euler–Bernoulli beam driven by additive white noise: Global random attractors and global dynamics

This paper concerns with the long time dynamical behavior of a stochastic Euler–Bernoulli beam driven by additive white noise. By verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam, the existence of global random attractors that attracts all bounded sets in phase space is proved. Furthermore, the finite Hausdorff dimension for the global random attractors is attained. In light of the relationship between global random attractor and random invariant probability measure, the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors.