An Introduction to Parabolic Moving Boundary Problems

Lunardi, Alessandra

doi:10.1007/978-3-540-44653-8_4

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…For classical solutions of semi-linear extensions of (1.1) see e. g. [11,24]. In addition to the theory of classical and weak solutions, the corresponding evolution equations have been studied in the framework of maximal L p -regularity, see [10,29] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…This problem and various extensions have been studied extensively in the second half of the 20th century, see [Vui93] for a review of the literature. For classical solutions of semi-linear extensions of (1.1) see e. g. [FP79], [Lun04]. In addition to the theory of classical and weak solutions, the corresponding evolution equations have been studied in the framework of maximal L p -regularity, see [EPS03], [PSS07] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Recent empirical studies of the dependency of price change on the imbalance of the order book (see [CKS14] and [LPS13]) suggest linear behaviour for balanced order books and non-linear behaviour when imbalance is large. ⊲ ⊲ ⊲ In addition to mild and weak solutions as in [KZS12] we obtain solutions in the analytically strong sense and make the transformation from free to fixed boundary, that is introduced in a deterministic setting in [Lun04] and used in [KZS12], rigorous in a stochastic setting. ⊲ ⊲ ⊲ We combine tools from the SPDE framework of da Prato and Zabczyk (cf.…”

Section: Introductionmentioning

confidence: 99%

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A Stefan-type stochastic moving boundary problem

Keller‐Ressel

Müller

2016

Stoch PDE: Anal Comp

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Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform the problem from a moving boundary problem into a stochastic evolution equation with fixed boundary conditions. Using results from interpolation theory we obtain existence and uniqueness of local strong solutions, extending results of Kim, Zheng and Sowers. In addition, we formulate conditions for existence of global solutions and provide a refined analysis of possible blow-up behavior in finite time.Comment: 34 page

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Stefan-type stochastic moving boundary problem

Keller‐Ressel

Müller

2016

Stoch PDE: Anal Comp

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“…In addition, only few contributions have reported the study of parabolic PDEs with time‐varying domain, in which main results are focused on establishing existence and regularity properties of the solution. These include development of transformations to map the PDE onto a new time‐invariant spatial domain and evolution of continuously differentiable diffeomorphisms . Among contributions along this line, a design of nonlinear distributed state observers for systems with moving boundaries using stochastic methods is notable.…”

Section: Introductionmentioning

confidence: 99%

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

Izadi

Dubljević

2014

AIChE Journal

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in Wiley Online Library (wileyonlinelibrary.com) Observer and optimal boundary control design for the objective of output tracking of a linear distributed parameter system given by a two-dimensional (2-D) parabolic partial differential equation with time-varying domain is realized in this work. The transformation of boundary actuation to distributed control setting allows to represent the system's model in a standard evolutionary form. By exploring dynamical model evolution and generating data, a set of time-varying empirical eigenfunctions that capture the dominant dynamics of the distributed system is found. This basis is used in Galerkin's method to accurately represent the distributed system as a finite-dimensional plant in terms of a linear timevarying system. This reduced-order model enables synthesis of a linear optimal output tracking controller, as well as design of a state observer. Finally, numerical results are prepared for the optimal output tracking of a 2-D model of the temperature distribution in Czochralski crystal growth process which has nontrivial geometry. V C 2014 American Institute of Chemical Engineers AIChE J, 61: 494-502, 2015

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“…While the deterministic problems have been extensively studied in the second half of the past century -see e. g. [30] and references therein -the stochastic equations are much less understood. In the past decade, several authors have started to study stochastic extensions of the classical Stefan problem.…”

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confidence: 99%

A stochastic Stefan-type problem under first-order boundary conditions

Müller¹

2018

Ann. Appl. Probab.

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Moving boundary problems allow to model systems with phase transition at an inner boundary. Motivated by problems in economics and finance, we set up a price-time continuous model for the limit order book and consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension. Here, the paths of the moving interface might have unbounded variation, which introduces additional challenges in the analysis. Working on the distribution space, Itō-Wentzell formula for SPDEs allows to transform these moving boundary problems into partial differential equations on fixed domains. Rewriting the equations into the framework of stochastic evolution equations and stochastic maximal L p -regularity we get existence, uniqueness and regularity of local solutions. Moreover, we observe that explosion might take place due to the boundary interaction even when the coefficients of the original problem have linear growths.

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An Introduction to Parabolic Moving Boundary Problems

Cited by 7 publications

References 12 publications

A Stefan-type stochastic moving boundary problem

A Stefan-type stochastic moving boundary problem

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

A stochastic Stefan-type problem under first-order boundary conditions

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