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1.
Hou, Qi.
Rough Hypoellipticity for *Local* *Weak* Solutions to the Heat Equation in Dirichlet Spaces.

Degree: PhD, Mathematics, 2019, Cornell University

URL: http://hdl.handle.net/1813/67578

This thesis studies some qualitative properties of local weak solutions of the heat equation in Dirichlet spaces. Let ≤ ft(X,𝓔,𝓕)) be a Dirichlet space where X is a metric measure space, and ≤ ft(𝓔,𝓕)) is a symmetric, local, regular Dirichlet form on L^{2} ≤ ft(X)). Let -P and ≤ ft(H_{t}))_{t>0} denote the corresponding generator and semigroup. Consider the heat equation ≤ ft(\partial_{t+P}))u=f in ℝ × X. Examples of such heat equations include the ones associated with (i) Dirichlet forms associated with uniformly elliptic, second order differential operators with measurable coefficients on ℝ^{n}, and Dirichlet forms on fractal spaces;\ (ii) Dirichlet forms associated with product diffusions and product anomalous diffusions on infinite products of compact metric measure spaces, including the infinite dimensional torus, and the infinite product of fractal spaces like the Sierpinski gaskets.\ We ask the following qualitative questions about local weak solutions to the above heat equations, which in spirit are generalizations of the notion of hypoellipticity: Are they locally bounded? Are they continuous? Is the time derivative of a local weak solution still a local weak solution? Under some hypotheses on existence of cutoff functions with either bounded gradient or bounded energy, and sometimes additional hypotheses on the semigroup, we give (partially) affirmative answers to the above questions. Some of our key results are as follows. Let u be a local weak solution to ≤ ft(\partial_{t+P}))u=f on some time-space cylinder I × Ω.\ (i) If the time derivative of f is locally in L^{2} ≤ ft(I × Ω)), then the time derivative of u is a local weak solution to ≤ ft(\partial_{t+P}))\partial_{t} u=\partial_{t} f.\ (ii) If the semigroup H_{t} is locally ultracontractive, and satisfies some Gaussian type upper bound, and if f is locally bounded, then u is locally bounded.\ (iii) Besides satisfying local contractivity and some Gaussian type upper bound, if the semigroup H_{t} further admits a locally continuous kernel h ≤ ft(t,x,y)), then u is locally continuous.\ (iv) If the semigroup is locally ultracontractive and satisfies some Gaussian type upper bound, then it admits a locally bounded function kernel h ≤ ft(t,x,y)). As a special case, on the infinite torus \mathbb{T}^∞, local boundedness of h ≤ ft(t,x,y)) implies automatically the continuity of h ≤ ft(t,x,y)), and hence of all local weak solutions.\ (v) The needed Gaussian type upper bounds can often be derived from the ultracontractivity conditions. We also discuss such implications under existence of cutoff functions with bounded gradient or bounded energy.\ The results presented in this thesis are joint work with Laurent Saloff-Coste.
*Advisors/Committee Members: Saloff-Coste, Laurent Pascal (chair), Healey, Timothy James (committee member), Cao, Xiaodong (committee member).*

Subjects/Keywords: Dirichlet space; heat equation; heat kernel; heat semigroup; local weak solution; Mathematics

…previous paragraph. And the conclusion is that for any
*local* *weak* *solution* u, its time… …called a
Let I := (a, b) b R, let f ∈ Lloc
P
*local* *weak* *solution* to the heat… …definition of a *local* *weak*
*solution* that we give in Chapter 2, where f is only assumed to be in the… …*local* *weak* *solution* u.
(1) If f is locally in W k,2 I → L2 (Ω) , then… …order k is again a *local* *weak* *solution* to the
heat equation with modified right-hand sides on…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Hou, Q. (2019). Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. (Doctoral Dissertation). Cornell University. Retrieved from http://hdl.handle.net/1813/67578

Chicago Manual of Style (16^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.” 2019. Doctoral Dissertation, Cornell University. Accessed October 20, 2020. http://hdl.handle.net/1813/67578.

MLA Handbook (7^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.” 2019. Web. 20 Oct 2020.

Vancouver:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. [Internet] [Doctoral dissertation]. Cornell University; 2019. [cited 2020 Oct 20]. Available from: http://hdl.handle.net/1813/67578.

Council of Science Editors:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. [Doctoral Dissertation]. Cornell University; 2019. Available from: http://hdl.handle.net/1813/67578

2. Mungkasi, Sudi. A study of well-balanced finite volume methods and refinement indicators for the shallow water equations .

Degree: 2012, Australian National University

URL: http://hdl.handle.net/1885/10301

This thesis studies solutions to the shallow water equations analytically and numerically. The study is separated into three parts. The first part is about well-balanced finite volume methods to solve steady and unsteady state problems. A method is said to be well-balanced if it preserves an unperturbed steady state at the discrete level. We implement hydrostatic reconstructions for the well-balanced methods with respect to the steady state of a lake at rest. Four combinations of quantity reconstructions are tested. Our results indicate an appropriate combination of quantity reconstructions for dealing with steady and unsteady state problems.
The second part presents some new analytical solutions to debris avalanche problems and reviews the implicit Carrier-Greenspan periodic solution for flows on a sloping beach. The analytical solutions to debris avalanche problems are derived using characteristics and a variable transformation technique. The analytical solutions are used as benchmarks to test the performance of numerical solutions. For the Carrier-Greenspan periodic solution, we show that the linear approximation of the Carrier-Greenspan periodic solution may result in large errors in some cases. If an explicit approximation of the Carrier-Greenspan periodic solution is needed, higher order approximations should be considered. We propose second order approximations of the Carrier-Greenspan periodic solution and present a way to get higher order approximations. The third part discusses refinement indicators used in adaptive finite volume methods to detect smooth and nonsmooth regions. In the adaptive finite volume methods, smooth regions are coarsened to reduce the computational costs and nonsmooth regions are refined to get more accurate solutions. We consider the numerical entropy production and weak local residuals as refinement indicators. Regarding the numerical entropy production, our work is the first to implement the numerical entropy production as a refinement indicator into adaptive finite volume methods used to solve the shallow water equations. Regarding weak local residuals, we propose formulations to compute weak local residuals on nonuniform meshes. Our numerical experiments show that both the numerical entropy production and weak local residuals are successful as refinement indicators.

Subjects/Keywords: finite volume methods; dam break; debris avalanche; Carrier-Greenspan solution; numerical entropy production; weak local residuals; adaptive mesh refinement; shallow water equations

…117
7.2
*Weak* *local* residuals of balance laws . . . . . . . . . . . . . . . . . 118
7.3… …*Weak* *local* residuals of shallow water equations . . . . . . . . . . 120
7.3.1
7.3.2
Wet/dry… …interface treatment for *weak* *local* residuals . . . . 124
7.3.3
7.4
Well-balancing the *weak* *local*… …125
*Weak* *local* residuals in adaptive methods . . . . . . . . . . . . . . 131
7.4.1
7.4.2
A… …with Markus on *weak* *local* residuals and with
Linda on adaptive grid methods.
I am indebted to…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Mungkasi, S. (2012). A study of well-balanced finite volume methods and refinement indicators for the shallow water equations . (Thesis). Australian National University. Retrieved from http://hdl.handle.net/1885/10301

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Mungkasi, Sudi. “A study of well-balanced finite volume methods and refinement indicators for the shallow water equations .” 2012. Thesis, Australian National University. Accessed October 20, 2020. http://hdl.handle.net/1885/10301.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Mungkasi, Sudi. “A study of well-balanced finite volume methods and refinement indicators for the shallow water equations .” 2012. Web. 20 Oct 2020.

Vancouver:

Mungkasi S. A study of well-balanced finite volume methods and refinement indicators for the shallow water equations . [Internet] [Thesis]. Australian National University; 2012. [cited 2020 Oct 20]. Available from: http://hdl.handle.net/1885/10301.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Mungkasi S. A study of well-balanced finite volume methods and refinement indicators for the shallow water equations . [Thesis]. Australian National University; 2012. Available from: http://hdl.handle.net/1885/10301

Not specified: Masters Thesis or Doctoral Dissertation