Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, &amp; nonradial focusing geometries

Denzler, Jochen; McCann, Robert J.

doi:10.1016/j.anihpc.2007.05.002

Cited by 13 publications

(48 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For m = n−2 n+2 , which lies below this range, it is known that solutions of finite mass vanish identically after a finite time, and that their asymptotic behavior is governed by a rotationally symmetric solution with profile v ∼ (1 + |x| 2 ) − n+2 2 [10,8,29]. In contrast, the closed-form solutions of [26,25,19,22,9], including the Barenblatt profile v ∼ (1 + |x| 2 )…”

mentioning

confidence: 88%

“…We use the method of [9] to construct the Ricci flow on two-dimensional cigar manifolds. If u solves equation (6), then π = 1 u satisfies ∂π ∂t = π∆π − |∇π| 2 .…”

Section: Ricci Flow On Two-dimensional Asymmetric Cigarsmentioning

confidence: 99%

“…that was first discovered some twenty years ago by Titov and Ustinov [26] and Tartar [25] (see [30]), and later rediscovered independently several times [19,22,9]. They are relevant for the Yamabe flow, because solutions of equation (2) in dimension n > 2 are transformed into positive solutions of equation (4) with m = n−2 n+2 by setting u = v 4 n+2 .…”

mentioning

confidence: 99%

“…The analysis of Denzler and McCann [9] does not directly apply to the log-diffusion equation in two dimensions, but the quadratic ansatz in equation (5) remains valid, and again leads to a system of ordinary differential equations that can be solved by quadratures. We are interested in the geometric evolution of the cigar-shaped manifolds that correspond to these closed-form solutions of the porous medium equation.…”

mentioning

confidence: 99%

“…As in two dimensions, the family of cigar manifolds is preserved by the Yamabe flow. Closed-form solutions M P (t) in n > 2 dimensions are provided by positive solutions of the porous medium equation (4) within the family of [26,25,19,22,9]. These solutions exist on a maximal time interval that depends on the initial value of P .…”

mentioning

confidence: 99%

See 4 more Smart Citations

Explicit Yamabe Flow of an Asymmetric Cigar

Burchard¹,

McCann

Smith³

2008

Methods and Applications of Analysis

Self Cite

View full text Add to dashboard Cite

Abstract. We consider the Yamabe flow of a conformally Euclidean manifold for which the conformal factor's reciprocal is a quadratic function of the Cartesian coordinates at each instant in time. This leads to a class of explicit solutions having no continuous symmetries (no Killing fields) but which converge in time to the cigar soliton (in two-dimensions, where the Ricci and Yamabe flows coincide) or in higher dimensions to the collapsing cigar. We calculate the exponential rate of this convergence precisely, using the logarithm of the optimal bi-Lipschitz constant to metrize distance between two Riemannian manifolds.Key words. Exact Yamabe flows, Ricci flow, conformally flat non-compact manifold, quadratic conformal factor, cigar soliton, attractor, basin of attraction, rate of convergence, Lyapunov exponent, biLipschitz.

show abstract

mentioning

confidence: 88%

“…We use the method of [9] to construct the Ricci flow on two-dimensional cigar manifolds. If u solves equation (6), then π = 1 u satisfies ∂π ∂t = π∆π − |∇π| 2 .…”

Section: Ricci Flow On Two-dimensional Asymmetric Cigarsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Explicit Yamabe Flow of an Asymmetric Cigar

Burchard¹,

McCann

Smith³

2008

Methods and Applications of Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

Matthes

Osberger

2015

Found Comput Math

View full text Add to dashboard Cite

Abstract. A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. We prove that the discrete solutions are strictly positive and mass conserving. Further, they dissipate both the Fisher information and the logarithmic entropy. Numerical experiments illustrate the practicability of the scheme.Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary non-negative initial data with finite entropy, without any CFL type condition. The key ingredient in the proof is a discretized version of the classical entropy dissipation estimate.

show abstract

Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator

Seis

2014

Journal of Differential Equations

View full text Add to dashboard Cite

We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor.More precisely, the inspection of the eigenfunctions allows to identify symmetries in R N with flows whose rates of convergence are faster than the uniform, translation-governed bound. The present work complements the analogous study of Denzler & McCann for the fast-diffusion equation.Solutions to (1) feature different phenomena depending on the degree of the nonlinearity ρ m . In the case m = 1, equation (1) is the ordinary diffusion (or heat) equation. For 0 < m < 1, the diffusion flux mρ m−1 diverges as ρ vanishes and thus, for suitable initial data, the solution spreads over the whole R N immediately. In this situation, equation (1) is often called the fast diffusion equation. In the porous-medium range m > 1, the diffusion flux increases with the density and degenerates where ρ = 0. Consequently, solutions will preserve a compact support and hence this type of propagation goes by the name slow diffusion. In the present paper, we restrict our attention exclusively to the latter case. For a study of the fast diffusion equation and more general evolutions of porous-medium type, we refer to Vázquez [31] and references therein.The Cauchy problem for the porous medium equation is solved in various settings. As the equation is degenerate parabolic, that is, it is parabolic only where the solution is positive, solutions are in general not classical. More precisely, if the initial datum is zero in some open subset of R N , then there is a slowly propagating free boundary that separates the region where the solution is positive $ The author acknowledges support through NSERC grant 217006-08.

show abstract

Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries

Cited by 13 publications

References 55 publications

Explicit Yamabe Flow of an Asymmetric Cigar

Explicit Yamabe Flow of an Asymmetric Cigar

A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator

Contact Info

Product

Resources

About