Related papers: Smooth Hamilton-Jacobi solutions for the Horocycle…
This paper defines symplectic scale manifolds based on Hofer-Wysocki-Zehnder's scale calculus. We introduce Hamiltonian vector fields and flows on these by narrowing down sc-smoothness to what we denote by strong sc-smoothness, a concept…
This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature…
In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton--Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods.…
In the present work the classical problem of harmonic oscillator in the hyperbolic space $H_2^2$: $z_0^2+z_1^2-z_2^2-z_3^2=R^2$ has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator…
We study the structure of the Mather and Aubry sets for the family of lagrangians given by the kinetic energy associated to a riemannian metric $ g$ on a closed manifold $ M$. In this case the Euler-Lagrange flow is the geodesic flow of…
We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton-Jacobi-Bellman (HJB) equations on a bounded domain with oblique boundary conditions. These equations appear naturally in…
A novel routine to investigate the scalar fields in a cosmological context is discussed in the framework of the Hamiltonian formalism. Starting from the Einstein-Hilbert action coupled to a Lagrangian density that contains two components -…
We establish homogenization for nondegenerate viscous Hamilton-Jacobi equations in one space dimension when the diffusion coefficient $a(x,\omega) > 0$ and the Hamiltonian $H(p,x,\omega)$ are general stationary ergodic processes in $x$. Our…
A new Hamiltonian formulation for the fully nonlinear water-wave problem over variable bathymetry is derived, using an exact, vertical series expansion of the velocity potential, in conjunction with Luke's variational principle. The…
Let ({\Sigma}, {\omega}) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on {\Sigma} provides a smooth path in Ham({\Sigma}), the group of…
We analyze gradient flows with jumps generated by a finite set of complete vector fields in involution using some Radon measures $u\in \mathcal{U}_a$ as admissible perturbations. Both the evolution of a bounded gradient flow $\{x^u(t,\l)\in…
We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent…
Particle-laden effects in high-speed flows require a coupled Euler and Lagrangian prediction technique with varying fidelity of thermochemical models, depending on the simulation conditions of interest. This requirement makes the…
This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces.…
By using the Hamilton-Jacobi [HJ] framework the topological theories associated with Euler and Pontryagin classes are analyzed. We report the construction of a fundamental $HJ$ differential where the characteristic equations and the…
In this paper, we consider the mean curvature flow of entire Lagrangian graphs with initial data in the pseudo-Euclidean space, which is related to the special Lagrangian parabolic equation. We show that the parabolic equation \eqref{11}…
We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the…
We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of…
In this paper, we find various analytic (1+3)D solutions to relativistic ideal hydrodynamic equations based on embedding of known low-dimensional scaling solutions. We first study a class of flows with 2D Hubble Embedding, for which a…
We present qualitative and quantitative homogenization results for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. When there is only one such signal and the Hamiltonian is convex, we show that the equation,…