Bounded $H_\infty $-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator

Escher, Joachim; Seiler, Jörg

doi:10.1090/s0002-9947-08-04589-3

Cited by 16 publications

(26 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…be its realization on H 1 2 (T 1 ). Then there is some λ0 ≥ 0 such that λ0 + A(t0) possesses a bounded H ∞ -calculus due to [20,Theorem 4.10]. This implies that for every A(t0) has maximal L p -regularity on every finite time interval and for every 1 < p < ∞ due to [19,Theorem 3.2].…”

Section: )mentioning

confidence: 99%

Sharp Interface Limit for a Stokes/Allen–Cahn System

Abels

Liu

2018

Arch Rational Mech Anal

111

View full text Add to dashboard Cite

We consider the sharp interface limit of a coupled Stokes/Allen-Cahn system, when a parameter ε > 0 that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. For sufficiently small times we prove convergence of the solutions of the Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature equation with an additional convection term coupled to a two-phase Stokes system with an additional contribution to the stress tensor, which describes the capillary stress. To this end we construct a suitable approximation of the solution of the Stokes/Allen-Cahn system, using three levels of the terms in the formally matched asymptotic calculations, and estimate the difference with the aid of a suitable refinement of a spectral estimate of the linearized Allen-Cahn operator. Moreover, a careful treatment of the coupling terms is needed. Mathematics Subject Classification (2000): Primary: 76T99; Secondary: 35Q30, 35Q35, 35R35, 76D05, 76D45 Key words: Two-phase flow, diffuse interface model, sharp interface limit, Allen-Cahn equation, Free boundary problems 1 INTRODUCTION, MAIN RESULT, AND OVERVIEW is the jump of a function u : Ω × [0, T0] → R 2 at Γt in direction of nΓ t , HΓ t and VΓ t are the curvature and the normal velocity of Γt, both with respect to nΓ t . Furthermore, Dv = 1 2 (∇v + (∇v) T ) and σ = R θ ′ 0 (ρ) 2 dρ, where θ0 is the so-called optimal profile that is the unique solution ofIf the material derivative ∂tvε +vε ·∇vε is added to the right-hand side of (1.5) (i.e., the Navier-Stokes equations are considered), the system (1.5)-(1.9) was already suggested by Liu and Shen in [31] as an alternative approximation of a classical sharp interface model for a two-phase flow of viscous, incompressible, Newtonian fluids, which has advantages for numerical simulations since the Allen-Cahn equation is of second order and not of fourth order as the Cahn-Hilliard equation. On the other hand, for solutions of (1.5)-(1.9) the total mass Ω cε(x, t)dx is in general not preserved in time, in contrast to solutions of (1.1)-(1.4), which is a disadvantage if the model is used to approximate a two-phase flow without phase transitions. However, (1.5)-(1.9) can be considered as a simplified model for a two-phase flow with phase transitions. Such models can yield systems of Navier-Stokes/Allen-Cahn type, cf. e.g. Blesgen [11]. Finally, let us mention that in [31] a rigorous result on the sharp interface limes of (1.5)-(1.9) was announced, which was not published so far to the best of the author's knowledge.The limit system (1.11)-(1.16) was also studied by Liu, Sato, and Tonegawa in [30] if the Stokes equation on the right-hand side is replaced by a modified Navier-Stokes equation for a shear thickening non-Newtonian fluid of power-law type. They constructed weak solutions for this system using a Galerkin approximation by a corresponding Navier-Stokes/Allen-Cahn system. In the proof they pass to the limit in the Ga...

show abstract

Section: )mentioning

confidence: 99%

Sharp Interface Limit for a Stokes/Allen–Cahn System

Abels

Liu

2018

Arch Rational Mech Anal

111

View full text Add to dashboard Cite

show abstract

“…e.g. Escher and Seiler , can be or has been included for these boundary value problems. (It was possible to include

C_{*}^{s}

in the regularity study for the restricted fractional Laplacian in using Johnsen .)…”

Section: Further Developmentsmentioning

confidence: 99%

Regularity of spectral fractional Dirichlet and Neumann problems

Grubb

2015

Mathematische Nachrichten

View full text Add to dashboard Cite

Received XXXX, revised XXXX, accepted XXXX Published online XXXX MSC (2000) 35P99, 35S15, 47G30Consider the fractional powers (ADir) a and (ANeu) a of the Dirichlet and Neumann realizations of a secondorder strongly elliptic differential operator A on a smooth bounded subset Ω of R n . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of H s p -Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on H ∞ -calculus. We also include an overview of the various Dirichlet-and Neumann-type boundary problems associated with the fractional Laplacian.

show abstract

“…Remark A.3. In fact, by applying the general results of [8] for pseudo-differential operators with nonsmooth symbols, it follows that −iµ(·, D x ) admits a bounded H ∞ calculus in H s (R d ), s ∈ [0, 1). In this sense, the properties of iµ(·, D x ) stated in Theorem A.2 themselves are not essentially new.…”

Section: A Appendixmentioning

confidence: 99%

On Poisson operators and Dirichlet-Neumann maps in $H^s$ for divergence form elliptic operators with Lipschitz coefficients

Maekawa

Miura

2015

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We consider second order uniformly elliptic operators of divergence form in R d+1 whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space H s (R d ) for each s ∈ [0, 1]. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.A ′ = (a i,j ) 1≤i,j≤d , b = a d+1,d+1 , r 1 = (a 1,d+1 , · · · , a d,d+1 ) ⊤ , r 2 = (a d+1,1 , · · · , a d+1,d ) ⊤ .

show abstract

Bounded $H_\infty $-calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator

Cited by 16 publications

References 21 publications

Sharp Interface Limit for a Stokes/Allen–Cahn System

Sharp Interface Limit for a Stokes/Allen–Cahn System

Regularity of spectral fractional Dirichlet and Neumann problems

On Poisson operators and Dirichlet-Neumann maps in $H^s$ for divergence form elliptic operators with Lipschitz coefficients

Contact Info

Product

Resources

About