Graduate Thesis Or Dissertation
 

A local perturbation problem of the 3D Navier-Stokes equations

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/cv43p406v

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  • This dissertation explores mathematical theory of the 3 dimensional incompressible Navier-Stokes equations that consists a set of partial differential equations which govern the motion of Newtonian fluids and can be seen as Newton's second law of motion for fluids. The main interest of this work focuses on how local perturbation of the initial condition influence the evolution of the solution to the Navier-Stokes equations, which is non-local in nature. Specifically, we establish an upper bound for the growth rate of the local $L^2$ norm of the difference of two suitable weak solutions $\mathbi{u}_1$ and $\mathbi{u}_2$ that arise from the initial conditions $\mathbi{u}_{1,0}$ and $\mathbi{u}_{2,0}$. The technical difficulty that lies in the lack of regularity of weak solutions makes the analysis challenging, and there are currently very few known results in this direction. Nevertheless, our estimate indicates that the local growth rate $\|\mathbi{u}_1(t)-\mathbi{u}_2(t)\|_{L_{\text{loc}}^2}$ is, given the condition that their global (in space and time) kinetic energy is bounded, approximately $O(t^{1/4})$ when $t$ is small. The main merit of this estimate is the exploitation of the spatial localization technique and Sobolev embedding theorem. Moreover, our results confirm that the right-hand-side continuity of the strong local norm $\|\mathbi{u}_1(t)-\mathbi{u}_2(t)\|_{L_{\text{loc}}^2}$ at $t=0$ holds in the class of suitable weak solutions.
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  • Intellectual Property (patent, etc.)
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  • 2020-12-14 to 2023-01-15

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