Related papers: Unstable manifolds for rough evolution equations
We formulate and consider the problem of an inextensible, unshearable, viscoelastic rod, with evolving natural configuration, moving on a plane. We prove that the dynamic equations describing quasistatic motion of an Eulerian strut, an…
We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable…
An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad \dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert…
There is a recent surge in research activities on modeling the flow of fluids in porous media with complex pore-networks. A prominent mathematical model, which describes the flow of incompressible fluids in porous media with two dominant…
We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we…
We investigate the uniform stability properties of discrete-time linear switched systems subject to arbitrary switching, focusing on the "marginally unstable" regime in which the system is not Lyapunov stable but in which trajectories…
In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian…
A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start…
We investigate the global in time stability of regular solutions with large velocity vectors to the evolutionary Navier-Stokes equation in ${\bf R}^3$. The class of stable flows contains all two dimensional weak solutions. The only…
Rough differential equations are solved for signals in general Besov spaces unifying in particular the known results in H\"older and p-variation topology. To this end the paracontrolled distribution approach, which has been introduced by…
The two-dimensional solitary waves of the Gross-Pitaevskii equation in the Kadomtsev-Petviashvili limit are unstable with respect to three-dimensional perturbations. We elucidate the stages in the evolution of such solutions subject to…
A method is proposed to obtain examples of smooth CR-manifolds whose local stability group is neither a Lie group nor infinite-dimensional.
We study exchange of stability in the dynamics of solitary wave solutions under changes in the nonlinear balance in a 1+1 evolutionary partial differential equation related both to shallow water waves and to turbulence. We find that…
In this paper, we consider a non-linear fourth-order evolution equation of Cahn-Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order…
We establish the global existence of higher-order Sobolev solutions for a non-local integrable evolution equation arising in the study of pseudospherical surfaces and non-linear wave propagation. Under a natural assumption on the initial…
M. Kruskal showed that each nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of each order, near which rapid…
We study the stability and dynamics of traveling-front solutions of a modified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion…
We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch…
We study (3+1)D kinetic (Boltzmann-Vlasov) equations for relativistic plasma particles in a one-dimensionally expanding geometry motivated by ultrarelativistic heavy-ion collisions. We set up local equations in terms of Yang-Mills…
Discrete time evolution of one-dimensional maps is embedded in continuous time by truncating the Taylor series expansion of the time evolution operator to a finite order N. Truncations with N > 4 leads to unconditional instability.…