We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton–Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.
We study the heat equation associated to the Hodge Laplacian on simplicial complexes. Using recently developed techniques for magnetic Schrödinger operators, we prove Davies-Gaffney-Grigoryan type estimates for the kernel of the heat semigroup on ℓ^2, which we then use to extend the semigroup to ℓ^p for p∈[1,∞] under suitable curvature and volume growth conditions. Furthermore, we establish p-independence of the Hodge Laplacian spectrum under the assumption of form bounded curvature and uniform subexponential volume growth. While the main focus of the paper is the Hodge Laplacian on simplicial complexes, the results are indeed proven for general positive magnetic Schrödinger operators on graphs.
We introduce horizontal and vertical motivic invariants of birational maps between rational dominant maps and study their basic properties. As a first application, we show that the (usual) motivic invariants vanish for birational automorphisms of threefolds over algebraically closed fields of characteristic zero. On the other hand, we prove that the motivic invariants of the birational automorphism group of many types of varieties, including projective spaces of dimension at least four over a field of characteristic zero, do not form a bounded family, even after extending scalars to the algebraic closure of the field. For such varieties, we further show that their birational automorphism groups are not generated by maps preserving a conic bundle or a rational surface fibration structure, and their abelianizations do not stabilize.
We study long time behavior of some nonlinear discrete velocity kinetic equations in the one and three dimensions with periodic boundary conditions. We prove the exponential time decay of solutions towards the global equilibrium in the L^2 space. Our result holds for a wide class of interaction rates including the Goldstein-Taylor and Carleman equations, and the estimates on the rate of convergence are explicit and constructive. The technique is based on the construction of suitable Lyapunov functionals by modifying Boltzmann's entropy.
In this article we investigate the relationship of the 3D incompressible, primitive equations with wind-driven boundary conditions and Coriolis force and the Ekman spiral. We identify three distinct regimes. In the first one, we show that every solution converges exponentially fast to the Ekman spiral as t → + ∞. In particular, this implies that the Ekman spiral is the unique equilibrium of the system. In the second regime, we prove nonlinear stability of the Ekman spiral, and in the third one, we show nonlinear instability of the Ekman spiral.
