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Singular stochastic partial differential equations on the real plane under critical regime
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Kiedrowski, Jacek (2021) Singular stochastic partial differential equations on the real plane under critical regime. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3880526
Abstract
In this thesis, the fractional incompressible Stochastic Navier-Stokes (SNS) equation on R2 and the fractional Anisotropic Kardar–Parisi–Zhang (AKPZ) equation on R2 are studied, formally defined as
∂tv = −12 (−Δ)θv − λv ・ ∇v + ∇p − ∇⊥(−Δ)θ−1 2 ξ, ∇ ・ v = 0 , (1)
and
∂th = −12 (−Δ)θh − λ((∂1h)2 − (∂2h)2) + (−Δ)θ+1 2 ξ , (2)
respectively, where θ ∈ (0, 1], ξ is the space-time white noise on R+ × R2 and λ is the coupling constant. For any value of θ both equations are ill-posed due to the singularity of the noise, are critical for θ = 1 and supercritical for θ ∈ (0, 1). For θ = 1, the weak coupling regime for both of the equations is shown, i.e. regularisation at scale N and coupling constant λ = ˆλ/ √logN, is meaningful in that the sequences {vN}N of regularised solutions of SNS and the sequences {hN}N of regularised solutions of AKPZ are tight and the corresponding nonlinearities do not vanish as N → ∞. Instead, for θ ∈ (0, 1) it is shown that the large scale behaviour of v and h is trivial, as the nonlinearity vanishes and v is simply converges to the solution of (1) with λ = 0, while h converges to (2) also with λ = 0. In order to further understand the limiting behaviour of AKPZ as regularisation is removed a quantity called bulk diffusivity is investigated numerically on a torus, with the aim of quantifying how different the limit is from stochastic heat equation.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Stochastic processes, Differential equations, Navier-Stokes equations, Gaussian processes | ||||
Official Date: | 2021 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Cannizzaro, Giuseppe ; Zygouras, Nikos | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | 3 unnumbered pages, 81 pages | ||||
Language: | eng |
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