The paper deals with positive solutions of the initial-boundary value problem for \(u_t=f(u) (\Delta u+\lambda_1 u) \qquad (*)\) with zero Dirichlet data in a smoothly bounded domain \(\Omega \subset \mathbb{R}^{n}, n\ge 1\). Here \(f \in C^{0}([0,\infty)) \cap C^{1}((0,\infty))\) is positive on (0,∞) with f(0) = 0, and λ1 is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence of a sufficiently strong degeneracy. More precisely, writing \(H(s):=\int_{1}^{s} \frac{d\sigma}{f(\sigma)}\), it is shown that if \(\int_{0} sH(s)ds=-\infty\) then there exist global classical solutions of (*) satisfying \(\limsup_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty\) and \(\liminf_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(\Omega)}=0\). Under the additional structural assumption \(\frac{sf'(s)}{f(s)}\ge\kappa > 0\), s > 0, this result can be sharpened: If \(\int_0 sH(s)ds=-\infty\) then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition \(\int_0 sH(s)ds > -\infty\) ensures that all solutions of (*) will stabilize to a single equilibrium.

Oscillating Solutions and Large ω-limit Sets in a Degenerate Parabolic Equation

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L2-bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.

Optimal estimate for the gradient of Green functions on degenerating surfaces and applications

Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier–Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck’s constant, while the volume density of entropy is proportional to Boltzmann’s constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier–Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).

https://www.mdpi.com/1099-4300/18/1/34/pdf

Schroedinger vs. Navier-Stokes

Quantum mechanics has been argued to be a coarse-graining of some underlying deterministic theory. Here we support this view by establishing a map between certain solutions of the Schroedinger equation, and the corresponding solutions of the irrotational Navier-Stokes equation for viscous fluid flow. As a physical model for the fluid itself we propose the quantum probability fluid. It turns out that the (state-dependent) viscosity of this fluid is proportional to Planck's constant, while the volume density of entropy is proportional to Boltzmann's constant. Stationary states have zero viscosity and a vanishing time rate of entropy density. On the other hand, the nonzero viscosity of nonstationary states provides an information-loss mechanism whereby a deterministic theory (a classical fluid governed by the Navier-Stokes equation) gives rise to an emergent theory (a quantum particle governed by the Schroedinger equation).

We show that for a system of two entangled particles, there is a dual description to the particle equations in terms of classical theory of conformally stretched spacetime. We also connect these entangled particle equations with Finsler geometry. We show that this duality translates strongly coupled quantum equations in the pilot-wave limit to weakly coupled geometric equations.

/pdf/two-particle-entanglement-and-its-geometric-duals-2bl51sw1ea.pdf

Two particle entanglement and its geometric duals

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 Hamilton–Jacobi equation of the mechanics so defined.

a Mechanics for the Ricci Flow

We establish a 1-to-1 relation between metrics on compact Riemann surfaces without boundary, and mechanical systems having those surfaces as configuration spaces.

The Ricci flow on Riemann surfaces

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 Hamilton-Jacobi equation of the mechanics so defined.

S. Abraham

Papers

a Mechanics for the Ricci Flow

The Ricci flow on Riemann surfaces

A mechanics for the Ricci flow