Global existence and decay to equilibrium for some crystal surface models

Granero-Belinchón, Rafael; Magliocca, Martina

doi:10.48550/arxiv.1804.09645

Cited by 6 publications

(10 citation statements)

References 17 publications

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“…The proof of this Theorem follows the approach in [18]. We define the new variable vpx, tq " upx, tq ´xu 0 y.…”

Section: Proof Of Theoremmentioning

confidence: 99%

On a nonlocal differential equation describing roots of polynomials under differentiation

Granero-Belinchón¹

2018

Preprint

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In this work we study the nonlocal transport equation derived recently by Steinerberger when studying how the distribution of roots of a polynomial behaves under iterated differentation of the function. In particular, we study the well-posedness of the equation, establish some qualitative properties of the solution and give conditions ensuring the global existence of both weak and strong solutions. Finally, we present a link between the equation obtained by Steinerberger and a one-dimensional model of the surface quasi-geostrophic equation used by Chae, Córdoba, Córdoba & Fontelos.

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“…The proof of this Theorem follows the approach in [18]. We define the new variable vpx, tq " upx, tq ´xu 0 y.…”

Section: Proof Of Theoremmentioning

confidence: 99%

On a nonlocal differential equation describing roots of polynomials under differentiation

Granero-Belinchón¹

2018

Preprint

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“…This equation has been derived in [8], and again more recently in [10], as a model in epitaxial growth of thin films. Two recent works have analyzed this model; Granero-Bellinchon and Magliocca have proved global existence of small solutions on the torus, and decay to equilibrium [6]. On free space rather than the torus, Liu and Strain have demonstrated global existence of small solutions, decay to equilibrium, and analyticity of solutions [9].…”

Section: Introductionmentioning

confidence: 99%

The radius of analyticity for solutions to a problem in epitaxial growth on the torus

Ambrose

2019

Bull. London Math. Soc.

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A certain model for epitaxial film growth has recently attracted attention, with the existence of small global solutions having been proved in both the case of the n‐dimensional torus and free space. We address a regularity question for these solutions, showing that in the case of the torus, the solutions become analytic at any positive time, with the radius of analyticity growing linearly for all time. As other authors have, we take the Laplacian of the initial data to be in the Wiener algebra, and we find an explicit smallness condition on the size of the data. Our particular condition on the torus is that the Laplacian of the initial data should have norm less than 1/4 in the Wiener algebra (in fact, the result holds for data a bit larger than this).

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“…Unfortunately, in multiple space dimensions, none of the above conditions can really be expected. More recently, the authors in [10] introduced the change of variable 3 The equation was then coupled with the initial and periodic boundary conditions. The existence of a "much stronger" weak solution than the one in [14] was obtained, provided that the initial data was suitably small,.…”

Section: Introductionmentioning

confidence: 99%

“…To mention a few, we refer the reader to [7,13,14] for solutions that contain measures. The study of exponential decay of solutions can be found in [10,12]. Development of singularity and finite extinction of solutions were considered in [8].…”

Section: Introductionmentioning

confidence: 99%

Special solutions to a fourth-order nonlinear parabolic equation in non-divergence form

2019

Communications in Mathematical Sciences

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In this paper we study a crystal surface model first proposed by H. Al Hajj Shehadeh, R.V. Kohn, and J. Weare ( Physica D, 240,1771( -1784. By seeking a solution of a particular function form, we are led to a boundary value problem for a fourth-order nonlinear elliptic equation. The mathematical challenge of the problem is due to the fact that the degeneracy in the equation is directly imposed by one of the two boundary conditions. An existence theorem is established in which a meaningful mathematical interpretation of one of the boundary conditions remains open. Our proof seems to suggest that this is unavoidable. We also obtain self-similar solutions to the crystal surface model which are positive and unbounded. This is in sharp contrast with the linear biharmonic heat equation.1991 Mathematics Subject Classification. 35D30, 35J66, 35J40, 35K65, 35K41.

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Global existence and decay to equilibrium for some crystal surface models

Abstract: In this paper we study the large time behavior of the solutions to the following nonlinear fourthorder equationsThese two PDE were proposed as models of the evolution of crystal surfaces by J.

Cited by 6 publications

References 17 publications

On a nonlocal differential equation describing roots of polynomials under differentiation

On a nonlocal differential equation describing roots of polynomials under differentiation

The radius of analyticity for solutions to a problem in epitaxial growth on the torus

Special solutions to a fourth-order nonlinear parabolic equation in non-divergence form

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