Related papers: On Equilibrium States for Partially Hyperbolic Hor…

We establish partial regularity for vector-valued solutions to parabolic systems where the coefficients are possibly discontinuous with respect to (x,t). More precisely, we assume a VMO-condition with respect to the (x,t) and continuity…

Given a strictly hyperbolic $n\times n$ system of conservation laws, it is well known that there exists a unique Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation, which are limits of…

We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any…

Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are…

We study a class of two dimensional partially hyperbolic systems, not necessarily skew products, trying to establish the germ of a general theory. To illustrate the scope of the theory, we apply our results to the case of fast-slow…

The dynamics of a periodically driven system whose time evolution is governed by the Schr\"{o}dinger equation with non-Hermitian Hamiltonians can be perfectly stable. This finding was only obtained very recently and will be enhanced by many…

It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the…

This paper presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov cycle where trajectories turn in opposite directions near the two saddle-foci. We show that…

After a short introduction to the characteristic geometry underlying weakly hyperbolic systems of partial differential equations we review the notion of symmetric hyperbolicity of first-order systems and that of regular hyperbolicity of…

For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space $B^{\alpha,\infty}_q$ and satisfying a one-sided bound condition are unique within the class of dissipative…

We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main…

This paper is devoted to hyperbolic systems of balance laws with non local source terms. The existence, uniqueness and Lipschitz dependence proved here comprise previous results in the literature and can be applied to physical models, such…

This paper is concerned with the global dynamics of a hybrid parabolic-hyperbolic model describing populations with distinct dispersal and sedentary stages. We first establish the global well-posedness of solutions, prove a comparison…

In dimension three and under certain regularity assumptions, we construct a renormalisation scheme at the heterodimensional tangency of a non-transverse heterodimensional cycle associated with a pair of saddle-foci whose limit dynamic is a…

Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $\mu$ with positive entropy, and let $\mathcal{A}$ be a H\"older continuous cocycle of injective bounded linear…

We introduce a notion of autonomous dynamical systems and apply it to prove rigidity of partially hyperbolic diffeomorphisms on closed compact three-manifolds under some smoothness hypothesis of their associated framing.

We extend the Kreiss--Majda theory of stability of hyperbolic initial--boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does…

The study of resonances (and well-posedness) for complex systems under time-periodic loading is of broad interest in application. The work of Galdi et al.~(2014) connects asymptotic stability of solutions to an unforced Cauchy problem to…

A {\em singular hyperbolic set} is a partially hyperbolic set with singularities (all hyperbolic) and volume expanding central direction \cite{MPP1}. We study connected, singular-hyperbolic, attracting sets with dense closed orbits {\em and…

We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be…