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An investigation of the forced Navier-Stokes equations in two and three dimensions.

機(jī)譯：在二維和三維中研究強(qiáng)制Navier-Stokes方程。

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This dissertation is devoted to expanding the classical theory of the forced Navier-Stokes equations. First, we study the regularity of solutions to the two dimensional Navier-Stokes equations with a singular or ``fractal'' forcing term. The classical theory tells us that the two dimensional Navier-Stokes equations gain two derivatives on a sufficiently smooth force. Following these classical methods we extend this result to spaces with negative fractional derivatives. However, these methods break down at a critical value. In this case, we show that one can still gain two derivatives locally in time. Next, we investigate the long-term behavior of both the two dimensional and three dimensional Navier-Stokes equations with a time-dependent force. When the force is independent of time, it is known that the long-term behavior of the Navier-Stokes equations is encapsulated within a set called the global attractor. The global attractor has a nice characterization, even in the three dimensional case, where we still do not know if there exists unique solutions. We present a framework for studying the existence of an analogous object, the pullback attractor, when the force depends on time. We study the existence and structure of these pullback attractors as well as the relationship between the pullback attractor and other existing notions of attractors. Finally, we apply our framework to the two dimensional and three dimensional Navier-Stokes equations with an appropriate time-dependent force. We also study the effect that the size of the force has on the size of the pullback attractor. Finally, we show that if the force is sufficiently small and periodic, there must exist a unique, smooth, periodic solution to the three dimensional Navier-Stokes equations.
機(jī)譯：本文致力于擴(kuò)展經(jīng)典的Navier-Stokes方程組理論。首先，我們研究帶有奇異或``分形''強(qiáng)迫項(xiàng)的二維Navier-Stokes方程解的正則性。經(jīng)典理論告訴我們，二維Navier-Stokes方程在足夠平滑的力下獲得兩個(gè)導(dǎo)數(shù)。遵循這些經(jīng)典方法，我們將此結(jié)果擴(kuò)展到具有負(fù)分?jǐn)?shù)導(dǎo)數(shù)的空間。但是，這些方法分解為臨界值。在這種情況下，我們表明一個(gè)人仍然可以在本地及時(shí)獲得兩個(gè)導(dǎo)數(shù)。接下來(lái)，我們研究具有時(shí)變力的二維和三維Navier-Stokes方程的長(zhǎng)期行為。當(dāng)力與時(shí)間無(wú)關(guān)時(shí)，已知Navier-Stokes方程的長(zhǎng)期行為被封裝在稱(chēng)為全局吸引子的集合中。即使在三維情況下，全局吸引子也具有很好的特征，在這種情況下，我們?nèi)匀徊恢朗欠翊嬖讵?dú)特的解決方案。我們提供了一個(gè)框架，用于研究當(dāng)力取決于時(shí)間時(shí)類(lèi)似對(duì)象（回拉吸引子）的存在。我們研究了這些拉回吸引子的存在和結(jié)構(gòu)，以及拉回吸引子與其他現(xiàn)有吸引子概念之間的關(guān)系。最后，我們將框架應(yīng)用于具有適當(dāng)時(shí)變力的二維和三維Navier-Stokes方程。我們還研究了力的大小對(duì)回拉吸引子的大小的影響。最后，我們表明，如果力足夠小且具有周期性，則必須存在一個(gè)針對(duì)三維Navier-Stokes方程的唯一，平滑，周期性的解。

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  • 作者

    Kavlie, Landon James.;

  • 作者單位

    University of Illinois at Chicago.;

  • 授予單位 University of Illinois at Chicago.;
  • 學(xué)科 Applied mathematics.
  • 學(xué)位 Ph.D.
  • 年度 2015
  • 頁(yè)碼 131 p.
  • 總頁(yè)數(shù) 131
  • 原文格式 PDF
  • 正文語(yǔ)種 eng
  • 中圖分類(lèi) 遙感技術(shù);
  • 關(guān)鍵詞

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