In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.

Purpose: The purpose of this study was to determine if it is possible for the macula to remain attached if a bullous retinal detachment blocks the examiner's view to the macula. Methods: A mathematical analysis compared the arc length of the attached retina versus the length of a detached retina necessary to obscure the macula (hang over the visual axis). The shape (oblate ellipsoid) and dimensions of the retina were based on a published study. The complete path of the hanging retina was calculated as a static catenary so as to depict the lowest possible position ("worst case scenario"). Results: Themeasured and calculated angle between the fovea and ora serrata was 105 degrees. When considering a catenary shape of the hanging retina, the macula could, mathematically, still be attached despite the retina hanging down 1.03 mm below the visual axis for an emmetropic eye. The maximal distance calculated was 1.095 mm for a -12 diopter (D) myopic eye. Conclusions: If the macular center cannot be viewed due to a bullous superior retinal detachment hanging into the examiner's view, it is unlikely but possible that the macula remains attached. If the view is obscured by at least 1 mm below the fovea, it is not mathematically possible for the fovea to be attached. Translational Relevance: The status of the macula being detached is subject to mathematical constraints, which, explored herein, offer a higher certainty of clinical decision making that could inform management for better clinical results.

The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.

A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semi-processes in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.

We obtain a number of Hardy type inequalities for continuous and discrete operators.

Given a map $\Phi$ defined on bounded subsets of the (base) metric space $X$ and with bounded sets as its values, one can follow the orbits $A$, $\Phi(A)$, $\Phi^2(A)$, $\ldots$, of nonempty, closed, and bounded sets $A$ in $X$. This is the system $(\Phi, X)$. On the other hand, the same orbits can be viewed as trajectories of points in the hyperspace $X^\sharp$ of nonempty, closed, and bounded subsets of $X$. This is the system $(\Phi, X^\sharp)$. We study the existence and properties of global attractors for both $(\Phi, X)$ and $(\Phi, X^\sharp)$. We give very basic conditions on $\Phi$, stated at the level of the base space $X$, that are necessary and sufficient for the existence of a global attractor for $(\Phi, X)$. Continuity is not among those conditions, but if $\Phi$ is continuous in a certain sense then the attractor and the $\omega$-limit sets are $\Phi$-invariant. If $(\Phi, X)$ has a global attractor, then $(\Phi, X^\sharp)$ has a global attractor as well. Every point of the global attractor of $(\Phi, X^\sharp)$ is a compact set in $X$, and the union of all the points of that attractor is the global attractor of $(\Phi, X)$.

We are interested in time evolution of systems that switch their modes of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes may be finite or infinite. The mathematical setting for such systems is variable time step dynamics with choice. We have used this setting previously to study the long term behavior of such systems. In this paper, we define and study the continuous time dynamics whose trajectories are limits of trajectories of discrete systems as time step goes to zero. The limit dynamics is multivalued. In the special case of a switched system, when the dynamics is generated by switching between solutions of a finite number of systems of ODEs, we show that our continuous limit solution set coincides with the solution set of the relaxed differential inclusion.

We are interested in systems that can and possibly do switch between different regimes; the durations of staying in each regime (dwell times) may vary. We work with the deterministic picture and study global attractors both in the state space and, one of the novelties, in the hyperspace. We give an example of a simple ODE system where the existence or non-existence of attractors depends on the intervals from which the dwell times are chosen. Another novelty is the description of restricted iterated function systems and their attractors. An unexpected corollary is that for the restricted to a subshift hyperbolic IFS, its attractor is the image under Hutchinson’s map \(p\) of a different, in general, subshift (we call it dual to the original one).

Mathematical setting for discrete dynamics is a state space, X, and a map S : X ! X (the evolution operator) which denes the change of a state over one time step. Dynamics with choice, as we dene it in [2], is a generalization of discrete dynamics where at every time step there is not one but several available maps that can transform the current state of the system. Many real life processes, from autocatalytic reaction systems to switched systems to cellular biochemical processes, exhibit the properties described by dynamics with choice. We are interested in the long-term behavior of such systems. In [2] we studied dynamics with choice with a nite number of available maps,

In this paper I describe a complete set of homotopy invariants for maps from three-manifolds to the two-sphere. This description was obtained jointly with D. Auckly. It is analytic in nature and extends to discontinuous maps with finite Faddeev energy and maps in suitable Sobolev spaces. For smooth maps, our description is proved to be equivalent to Pontrjagin's original homotopy classification from the 1930's. For the finite energy maps, the invariants take on exactly the same values as for smooth maps. I conclude with a brief discussion of skyrmions and Faddeev's hopfions coupled with gravity.

Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes, even when the formulas defining the invariants make sense, they may not have expected properties (e.g., there are maps having non-integral degree). In this paper we define a complete set of homotopy invariants for maps from 3-manifolds to the 2-sphere and show that these invariants extend to finite Faddeev energy maps and maps in suitable Sobolev spaces. For smooth maps, our description is proved to be equivalent to Pontrjagin's original homotopy classification from the 1930's. We further show that for the finite energy maps the invariants take on exactly the same values as for smooth maps. We include applications to the Faddeev model. The techniques that we use would also apply to many more problems and/or other functionals. We have tried to make the paper accessible to analysts, geometers and mathematical physicists.

Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real-life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behaviour in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of the practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability or a strategy or Nature's intervention, the long-term behaviour of the system may not be correctly described by a system with 'averaged' values for the parameters. There may be a Gestalt effect.

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L2loc connection; the local developing maps for such connections need not be continuous.

In this paper we consider a generalization of the Faddeev model for the maps from a closed three-manifold into the two-sphere. We give a novel representation of smooth S 2 -valued maps based on flat connections. This representation allows us to obtain an analytic description of the homotopy classes of S 2 -valued maps that generalizes to Sobolev maps. It also leads to a new proof of an old theorem of Pontrjagin. For the generalized Faddeev model, we prove the existence of minimizers in every homotopy class.

In this paper we consider a generalization of the Faddeev model for the maps from a closed three-manifold into the two-sphere. We give a novel representation of smooth $ S^2$-valued maps based on flat connections. This representation allows us to obtain an analytic description of the homotopy classes of $ S^2$-valued maps that generalizes to Sobolev maps. It also leads to a new proof an old theorem of Pontrjagin. For the generalized Faddeev model, we prove the existence of minimizers in every homotopy class.

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L 2 loc connection; the local developing maps for such connections need not be continuous.

We discuss matching control laws for underactuated systems. We previously showed that this class of matching control laws is completely characterized by a linear system of first order partial differential equations for one set of variables ($\,\lambda$) followed by a linear system of first order partial differential equations for the second set of variables ($\,{\widehat g}$, $\,{\widehat V}$). Here we derive a new first order system of partial differential equations that encodes all compatibility conditions for the $\,\lambda$-equations. We give four examples illustrating different features of matching control laws. The last example is a system with two unactuated degrees of freedom that admits only basic solutions to the matching equations. There are systems with many matching control laws where only basic solutions are potentially useful. We introduce a rank condition indicating when this is likely to be the case.

In this paper we will discuss problems and techniques related to underactuated systems. We give a mathematical formulation of several problems arising from applications, review some standard and new techniques, and pose some interesting and challenging open questions.