Free boundary propagation and noise: analysis and numerics of stochastic degenerate parabolic equations

• Applicant: Professor Dr. Günther Grün
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 397495103

In a series of papers, Barbu, Da Prato, Gess, Kim, Röckner, and others recently studied existence and nonnegativity aspects of stochastic versions of second order degenerate parabolic equations. For stochastic porous-medium equations, finite propagation of the solution's support could be established as well - thus implicitly, these equations constitute free boundary problems. It is the scope of this proposal to investigate analytically and numerically the impact of noise on the propagation of free boundaries in stochastic variants of degenerate parabolic equations.It is based on our recent qualitative results about finite propagation and waiting time phenomena for stochastic porous-medium equations, our existence results for stochastic thin-film equations, and our convergence results for numerical schemes for stochastic porous-medium equations.As model equations, we intend to study stochastic porous-medium equations, stochastic parabolic p-Laplace equations, and stochastic thin-film equations. To guarantee the existence of almost surely globally nonnegative solutions, only multiplicative noise will be considered. It may arise inside a source-term or inside a convective term. Physically, the stochastic thin-film equation has been derived from stochastic Navier-Stokes equations to model the effects of thermal fluctuations on droplet spreading and on the dewetting of unstable liquid films. In particular on nano-scales, stochastic thin-film equations turn out to capture phenomena which cannot be described by their deterministic counterparts. Analytically, the investigation of second order equations is an important first step. In fact, in the deterministic setting, unifying analytical methods are available to obtain optimal results on propagation rates and on the size of waiting times for large classes of second and higher order degenerate parabolic equations. Accordingly, studies on stochastic versions of second order degenerate parabolic equations are expected to provide important methodological insight. In this spirit, we strive for quantitative estimates on the expected values of propagation rates and on the size of waiting times for second order equations. In situations where finite propagation and occurrence of waiting time phenomena are still open problems, we first look for qualitative results.Conceptually, the analytical approach is to adapt energy methods based on functional inequalities (like versions of Stampacchia's lemma) or differential inequalities to the stochastic setting.For stochastic thin-film equations for which so far only existence results for strictly positive solutions are known, we study convergent numerical schemes and we use them for Monte-Carlo simulations to obtain empirical evidence on the noise impact on the spreading of bulk droplets.

• DFG Programme: Research Grants
• International Connection: Austria, United Kingdom
• Cooperation Partners: Professor Dr. Nicolas Dirr; Professor Dr. Julian Fischer