The paper proves entropy formulae and Harnack estimates for porous medium equations on Riemannian manifolds.
problem Analyzing solutions to porous medium equations on Riemannian manifolds.
method Proves entropy formulae and differential Harnack estimates.
result Derives Harnack inequalities and Laplacian estimates as applications.
Paper extends Aronson-Bénilan estimates for porous medium equations on manifolds with negative curvature.
problem Estimating gradients for porous medium equations on manifolds with negative curvature.
method Develops Aronson-Bénilan gradient estimates for porous medium equations under lower bounds of N-weighted Ricci curvature with N<0. result Generalizes gradient estimates for porous medium equations to manifolds with negative curvature.
Paper derives gradient estimates for porous medium equations on Riemannian manifolds.
problem Gradient estimates for porous medium equations on Riemannian manifolds.
method Employing cutoff functions and the maximum principle.
result Derives Hamilton-Souplet-Zhang type gradient estimates for porous medium type equations.
The paper analyzes gradient estimates for a nonlinear heat equation on graphs.
problem Gradient estimates for the weighted porous medium equation on graphs.
method Analyzes the gradient estimates for the positive solutions of the weighted porous medium equation on graphs.
result Derives gradient estimates and Harnack inequality for the porous medium equation on graphs.
The paper establishes new gradient estimates for porous medium equations under Ricci flow.
problem Gradient estimates for porous medium equations under Ricci flow.
method Local Aronson-Benolan type gradient estimates for positive solutions of the porous medium equation under Ricci flow.
result Generalizes known gradient estimates to the Ricci flow context.
Researchers study fractional porous medium equation on hyperbolic space.
problem Analyzing the fractional porous medium equation on hyperbolic space.
method Existence results for solutions in weak sense, using fractional Laplacian and Green's function.
result Proves different smoothing effects for solutions.
Study nonnegative solutions on Riemannian manifolds using fractional porous medium equation.
problem Analyzing solutions to fractional porous medium equation on noncompact Riemannian manifolds.
method Existence and smoothing estimates for weak solutions in L1 and weighted spaces. result Results hold for Euclidean and hyperbolic spaces, including larger data classes.
Study improves Harnack estimates for porous medium equation under geometric flow.
problem Improving Harnack estimates for solutions to the porous medium equation under evolving metrics.
method Differential Harnack estimates for positive solutions to the porous medium equation with potential on time-dependent Riemannian metrics evolving by geometric flow.
result New Harnack estimates for the porous medium equation under geometric flow.
The study examines blow-up and global existence of solutions for porous medium equation on curved manifolds.
problem Analyzing blow-up and global existence of solutions for porous medium equation on negatively curved manifolds.
method Examined the porous medium equation with power-type reaction terms on negatively curved Riemannian manifolds.
result For p>m, small data give global solutions; for p<m, large data blow up in infinite time. In this paper we study gradient estimates for the positive solutions of the porous medium equation: ut=Δum where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li-Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curv…
Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.
problem Solving porous medium equation on noncompact manifolds with nonnegative Ricci curvature.
method Constructing a space X of functions larger than L1, in which the Green function on M appears as a weight, to solve the PME.
result The porous medium equation admits a solution in the weak dual sense for certain initial data.
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
Study of long-time behavior of solutions on negatively curved manifolds.
problem Long-time behavior of solutions to the Porous Medium Equation on Cartan-Hadamard manifolds with negative curvature.
method Analysis of long-time behavior, proving existence and uniqueness of solutions, using comparison principles.
result Unexpected separate-variable behavior, reminiscent of Dirichlet problems on bounded Euclidean domains.
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: ut=Δφ(up) associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the m-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which…
In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discove…
In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the…
Study shows existence and uniqueness of solutions for porous medium equation on curved manifolds.
problem Existence and uniqueness of solutions for porous medium equation on curved manifolds.
method Very weak solutions, Cauchy problem, Cartan-Hadamard manifolds, Ricci curvature bounds, sectional curvature bounds.
result Sharp growth rate of initial data for existence, maximal existence time estimate, blow-up for specific manifolds and data.
We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on negatively curved Riemannian manifolds. We show existence of solutions taking as initial condition a finite Radon measure, not necessarily positive. We then establish uniqueness in the class of nonnegative …
Study porous medium equation on curved spaces, proving long-term behavior bounds.
problem Analyzing long-time behavior of solutions on curved Riemannian manifolds.
method Classified manifolds into types based on curvature, proved bounds on solution behavior.
result Sharp upper and lower bounds on long-time behavior of solutions.
Study shows conditions for nonexistence of solutions in Riemannian geometry.
problem Nonexistence of global solutions for quasilinear parabolic problems.
method Test function argument to identify parameter ranges for nonexistence.
result Explicit parameter ranges where nonexistence holds.
Global solutions and smoothing effects for reaction-diffusion equations on manifolds.
problem Global existence and smoothing effects for reaction-diffusion equations on Riemannian manifolds.
method Functional analytic methods, Sobolev and Poincaré inequalities.
result Existence of global solutions under certain conditions on the manifold.
Study eigenvalue problems on Riemannian manifolds with specific curvature conditions.
problem Finding eigenvalues and eigenfunctions for quasilinear equations on Riemannian manifolds.
method Generalized Cheng--Yau gradient estimate for quasilinear equations on complete Riemannian manifolds.
result Eigenvalues give rise to unbounded eigenfunctions under certain conditions.
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
We introduce a fractional Yamabe flow involving nonlocal conformally invariant operators on the conformal infinity of asymptotically hyperbolic manifolds, and show that on the conformal spheres $(\Sn, [g_{\Sn}])$, it converges to the standard sphere up to a Möbius diffeomorphism. This result allows us to obtain extinct…
We investigate the m-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement K-convexity of the m-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the K-convexity of the weig…
CNNs predict porosity, permeability, and tortuosity from porous media images.
problem Predicting key properties of porous media from images.
method Convolutional neural networks (CNNs) trained with lattice Boltzmann simulations.
result CNNs accurately predict porosity, permeability, and tortuosity.
Large particle systems' fluctuations converge to SPDE with additive noise.
problem Understanding large equity markets and investment strategies.
method Hydrodynamic limit and SPDE analysis of rank-based models.
result Fluctuations of empirical cumulative distribution functions converge to SPDE.
We study the limiting behaviour of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that in a wide range of cases the evolution of the cum…
Using the characterization of last multipliers as solutions of the Liouville's transport equation, new results are given in this approach of ODE by providing several new characterizations, e.g. in terms of Witten and Marsden differentials or adjoint vector field. Applications to Hamiltonian vector fields on Poisson man…
In this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric g0 of constant scalar curvature, and and for Porous media equation on a bounded regular domain in the plane. We show that with a suitable assumption about $g(0)…
Study models fractures in porous media using geometric analysis.
problem Analyzing fluid flow in fractures with complex geometries.
method Developed a geometric model using Riemannian manifold and Laplace Beltrami operators.
result Reduced model accurately approximates flow in complex fractures.
Fractional porous media equations yield q-Gaussian solutions for stock price returns.
problem Modeling stock price returns using fractional porous media equations.
method Analyzed three types of fractional extensions of the porous media equation.
result Local and non-local fractional extensions fit S&P 500 data better than classical models.
The paper characterizes stochastic incompleteness in Riemannian manifolds.
problem Stochastic incompleteness of Riemannian manifolds and its characterization.
method Characterization through solutions to nonlinear parabolic equations.
result Stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions to certain nonlinear parabolic equations.
Develops geometric framework for dissipative field equations.
problem Dissipative field equations and their geometric analysis.
method Canonical k-contact manifolds, k-contactifications, splitting results, regularity conditions, criteria for PDEs. result Explicit Hamiltonian descriptions for various nonlinear PDEs.
Stock market returns follow q-Gaussian distributions with super-diffusion.
problem Characterizing stock market price returns.
method Used q-Gaussian distributions and porous media equation to model stock market returns.
result Stock market returns follow q-Gaussian distributions with super-diffusion.
The article proves long-time existence and convergence of the edge Yamabe flow.
problem Analyzing the normalized Yamabe flow on incomplete edge singularities.
method Novel maximum principle results and uniform bounds established without barrier functions or Krylov-Safonov estimates.
result Long-time existence and convergence of the edge Yamabe flow.
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: ut=ΔF(u), with F′(u)>0, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equ…
CNNs reconstruct medium properties from wave probing responses.
problem Determining medium properties from wave responses.
method Deep convolutional neural networks (CNNs) for nonlinear wave equations.
result Quantitative dependence of network depth and units on medium complexity.
The paper studies Yamabe metrics and stability in Riemannian manifolds.
problem Existence of complete Yamabe metrics with zero scalar curvature.
method Yamabe flow and local L1-stability analysis. result Local L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature. Global existence and smoothing effects for reaction-diffusion equations with blowup in infinite time.
problem Analyzing reaction-diffusion equations with power-type nonlinearity and slow diffusion.
method Functional analytic methods based on Sobolev and Poincaré inequalities.
result Solutions corresponding to large initial data blow up everywhere in infinite time on Cartan-Hadamard manifolds.
Purely real space versions of the differential equations describing the kinematics of a dislocated crystalline medium are considered. The differential geometric structures associated with them are revealed.
New method learns PDE solutions from low-fidelity data.
problem Challenges in learning PDE surrogates with scarce data.
method Flow matching in infinite-dimensional space with conditional neural operators.
result Accurately learns PDE solutions across different resolutions and fidelities.
Efficiently quantifies uncertainty in subsurface flow using neural networks guided by theory.
problem Uncertainty in dynamic subsurface flow predictions.
method Theory-guided Neural Network (TgNN) for efficient uncertainty quantification.
result TgNN surrogate improves efficiency of uncertainty quantification compared to MC method.
A new hybrid approach combines physics and machine learning for porous media transport.
problem Simulating 2-phase immiscible transport in porous media.
method Physics-informed deep learning with adversarial neural networks and automatic differentiation.
result The model accurately simulates shock and rarefaction phenomena with limited data.
We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the do…
In this paper, we consider three typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. We use the Bochner formula to derive the Bernstein type estimate of the heat equation. The second is to derive the Reilly t…
The paper introduces new uniformity and homogeneity concepts for Cosserat media.
problem Characterizing uniformity and homogeneity in Cosserat media.
method Using groupoids and smooth distributions, the authors derive three canonical equations to characterize uniformity and homogeneity.
result The paper provides a unique and maximal division of Cosserat media into uniform and second-grade parts.