Inspired by studies on the regularity of solutions to the fractional Navier-Stokes system and the impact of noise on singularity formation in hydrodynamic models, we
investigated these issues within the framework of the fractional 1D Burgers equation.
Initially, our research concentrated on the deterministic scenario, where we conducted
precise numerical computations to understand the dynamics in both subcritical and
supercritical regimes. We utilized a pseudo-spectral approach with automated resolution refinement for discretization in space combined with a hybrid Crank-Nicolson/
Runge-Kutta method for time discretization.We estimated the blow-up time by analyzing the evolution of enstrophy (H1
seminorm) and the width of the analyticity
strip. Our findings in the deterministic case highlighted the interplay between dissipative and nonlinear components, leading to distinct dynamics and the formation of
shocks and finite-time singularities.
In the second part of our study, we explored the fractional Burgers equation under
the influence of linear multiplicative noise. To tackle this problem, we employed the
Milstein Monte Carlo approach to approximate stochastic effects. Our statistical
analysis of stochastic solutions for various noise magnitudes showed that as noise
amplitude increases, the distribution of blow-up times becomes more non-Gaussian.
Specifically, higher noise levels result in extended mean blow-up time and increase its
variability, indicating a regularizing effect of multiplicative noise on the solution. This
highlights the crucial role of stochastic perturbations in influencing the behavior of
singularities in such systems. Although the trends are rather weak, they nevertheless
are consistent with the predictions of the theorem of [41]. However, there is no
evidence for a complete elimination of blow-up, which is probably due to the fact
that the noise amplitudes considered were not sufficiently large. This highlights the
crucial role of stochastic perturbations in influencing the behavior of singularities in
such systems. / Thesis / Master of Science (MSc)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/29851 |
| Date | January 2024 |
| Creators | Masud, Sadia |
| Contributors | Protas, Bartosz, Mathematics and Statistics |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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