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

Ambrose, David M.

doi:10.1112/blms.12283

Cited by 19 publications

(43 citation statements)

References 12 publications

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“…Moreover, we observe that there recently have been several research works to apply the Wiener algebra A./, or in the notation of this paper L 1 k , to study the global-in-time existence and gain of analyticity for solutions to some evolution equations with diffusions, for instance, Constantin-Córdoba-Gancedo-Strain [22], Constantin-Córdoba-Gancedo-Rodríguez-Piazza-Strain [21], Patel-Strain [70], Gancedo-García-Juárez-Patel-Strain [37], Liu-Strain [62], Granero-Belinchón-Magliocca [42], and Ambrose [7]. See also a recent work by Lei-Lin [60] for the global well-posedness of mild solutions to the three-dimensional, incompressible Navier-Stokes equations if the initial data satisfy r x f 0 P L 1 k .…”

Section: Motivation Of the Current Workmentioning

confidence: 99%

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

Duan

Liu

Sakamoto

et al. 2020

Comm Pure Appl Math

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This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the LT∞Lv2 norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc.

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Section: Motivation Of the Current Workmentioning

confidence: 99%

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

Duan

Liu

Sakamoto

et al. 2020

Comm Pure Appl Math

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“…Let us also mention that when the exponential in (1.1) is linearized and the Laplacian is replaced by the p−Laplacian, the resulting equation has been studied by Giga & Kohn [11] (see also the recent preprint by Xu [20]). Short after this paper was posted, two new papers studying (1.1) appeared, one by Jian-Guo Liu & Robert Strain [15] and another by David Ambrose [1]. We refer to the discussion section below for more details about these works and some words comparing our results and theirs.…”

mentioning

confidence: 96%

“…After the completion of this work, two new papers studying (1.1) appeared, one by Jian-Guo Liu & Robert Strain [15] and another by David Ambrose [1].…”

mentioning

confidence: 99%

“…Ambrose [1] showed that in the case of the d−dimensional torus, the solutions become analytic at any positive time, with the radius of analyticity growing linearly for all time (in Liu & Strain, due to the fact that they consider the free space, the growth was like t 0.25 ). Furthermore, using different estimates, Ambrose proved that the size restriction |v 0 | 0 < 0.25 is enough to guarantee global existence, thus, improving the conditions in Theorem 2.2 and in [15].…”

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confidence: 99%

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Global existence and decay to equilibrium for some crystal surface models

Granero-Belinchón¹,

Magliocca²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equationsThese two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771(Phys. D, 240, -1784(Phys. D, 240, , 2011, respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.

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“…Development of singularity and finite extinction of solutions were considered in [8]. Also see [2] for the existence of analytic solutions.…”

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

Cited by 19 publications

References 12 publications

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

Global existence and decay to equilibrium for some crystal surface models

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

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