Differential and Integral Equations

Parabolic stochastic partial differential equations with dynamical boundary conditions

Igor Chueshov and Björn Schmalfuss

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We consider systems of nonlinear parabolic stochastic partial differential equations with dynamical boundary conditions. These boundary conditions are qualitatively different from the standard, like Dirichlet, or Neumann, or Robin boundary conditions. Such conditions contain a time derivative and can be used to describe mathematical models with a dynamics on the boundary. In our model the noise is acting in the domain but also on the boundary and is presented as the temporal generalized derivative of an infinite-dimensional Wiener processes. In addition, we have coefficients for the spatial differential operators depending on space and time. We prove existence and uniqueness of mild solutions to these stochastic partial differential equations and study the properties of these solutions. The stochastic equation on the boundary contains a parameter. If this parameter becomes small, then the equation on the boundary has an interpretation in a fast time scale. We prove the relative compactness of the distribution of the solution if the parameter mentioned tends to 0.

Article information

Differential Integral Equations, Volume 17, Number 7-8 (2004), 751-780.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 35Q35: PDEs in connection with fluid mechanics 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]


Chueshov, Igor; Schmalfuss, Björn. Parabolic stochastic partial differential equations with dynamical boundary conditions. Differential Integral Equations 17 (2004), no. 7-8, 751--780. https://projecteuclid.org/euclid.die/1356060328

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