A selfsimilar solution to the focusing problem for the porous medium equation

Aronson, D. G.; Graveleau, J. L.

doi:10.1017/s095679250000098x

Cited by 104 publications

(96 citation statements)

References 15 publications

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“…In Aronson & Graveleau [6], equation (2.1 a) is transformed into a two-dimensional autonomous system and the AG profiles are identified as corresponding to a connection between a saddle and a saddle-node which is shown to exist for a unique β, which lies between 1 2 and 1. The analysis here is based on a different transformation which also leads to a two-dimensional quadratic autonomous system.…”

Section: The Ag Solution In the Phase Planementioning

confidence: 99%

“…We briefly recall the phase-plane analysis of the system (2.3) [17,19,6]. There are two finite critical points:…”

Section: The Ag Solution In the Phase Planementioning

confidence: 99%

“…In terms of (2.3) the result in Aronson & Graveleau [6] may now be reformulated as Theorem 2.1 Let δ > 0. For every 0 < ε < 2 there is a unique orbit of (2.3) coming out of infinity in the first quadrant with Y ∼ δX and a unique orbit going into infinity in the first quadrant with X → ε.…”

Section: The Ag Solution In the Phase Planementioning

confidence: 99%

“…As we explain in § 2, there is only one (positive) value of β, for which the solution of (1.5) has this property. The phase plane reduction we use to analyse (1.5 a) is different from the one used in Aronson & Graveleau [6], where the existence of a unique β and corresponding self-similar solution was first proved.…”

Section: Introductionmentioning

confidence: 99%

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Parametric dependence of exponents and eigenvalues in focusing porous media flows

2003

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We study the hole-filling problem for the porous medium equation u t = 1 m ∆u m with m > 1 in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution u = t 2β−1 F(|x|t −β ), and we establish that the self-similarity exponent β is a monotone function of the parameter m. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any m > 1 against perturbations with 2-fold symmetry. In addition, we prove that as m is varied there are bifurcations from the radial solution to self-similar solutions with k-fold symmetry for each k = 3, 4, 5, . . . . These bifurcations are simple and occur at values m 3 > m 4 > m 5 > · · · → 1.

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Section: The Ag Solution In the Phase Planementioning

confidence: 99%

“…We briefly recall the phase-plane analysis of the system (2.3) [17,19,6]. There are two finite critical points:…”

Section: The Ag Solution In the Phase Planementioning

confidence: 99%

Section: The Ag Solution In the Phase Planementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Parametric dependence of exponents and eigenvalues in focusing porous media flows

2003

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“…To study this problem more precisely, Aronson et al (see [2,7]) constructed a interesting radially symmetric solution u(r, t) to the focusing problem for the equation of (1. …”

Section: ) C = O(t γ ) For Large T Andmentioning

confidence: 99%

Regularity and geometric character of solution of a degenerate parabolic equation

Pan

2016

Bull. Math. Sci.

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This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation u t = u m . Our main objective is to improve the Hölder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, for the weak solution u(x, t), the present work will show that:1. The function φ = (u(x, t)) β ∈ C 1 (R n ) for given t > 0 if β is large sufficiently; 2. The surface φ = φ(x, t) is tangent to R n at the boundary of the positivity set of u(x, t); 3. The function φ(x, t)is a classical solution to another degenerate parabolic equation.Moreover, some explicit derivative estimates and expressions about the speed of propagation of u(x, t) and the continuous dependence on the nonlinearity of the equation are obtained.

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A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence

Grebenev

Oberlack

2011

Z Angew Math Mech

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Key words Two-point correlation tensor, Lagrangian, von Kármán-Howarth equation, initial-boundary value problem, solvability, asymptotic behavior.Considering the metric tensor ds 2 (t), associated with the two-point velocity correlation tensor field (parametrized by the time variable t) in the space K 3 of correlation vectors, at the first part of the paper we construct the Lagrangian system (M t , ds 2 (t)) in the extended space K 3 × R+ for homogeneous isotropic turbulence. This allows to introduce into the consideration common concept and technics of Lagrangian mechanics for the application in turbulence. Dynamics in time of (M t , ds 2 (t)) (a singled out fluid volume equipped with a family of pseudo-Riemannian metrics) is described in the frame of the geometry generated by the 1-parameter family of metrics ds 2 (t) whose components are the correlation functions that evolve according to the von Kármán-Howarth equation. This is the first step one needs to get in a future analysis the physical realization of the evolution of this volume. It means that we have to construct isometric embedding of the manifold M t equipped with metric ds 2 (t) into R 3 with the Euclidean metric. In order to specify the correlation functions, at the second part of this paper we study in details an initial-boundary value problem to the closure model [19,26] for the von Kármán-Howarth equation in the case of large Reynolds numbers limit.

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A selfsimilar solution to the focusing problem for the porous medium equation

Cited by 104 publications

References 15 publications

Parametric dependence of exponents and eigenvalues in focusing porous media flows

Parametric dependence of exponents and eigenvalues in focusing porous media flows

Regularity and geometric character of solution of a degenerate parabolic equation

A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence

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