*Philip Isett*

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691174822
- eISBN:
- 9781400885428
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174822.003.0009
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a ...
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This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a notion of frequency energy levels is introduced. This notion is meant to accurately record the bounds which apply to the (v, p, R) coming from the previous stage of the construction. The chapter presents an example of a candidate definition for frequency and energy levels. Based on this definition, the effect of one iteration of the convex integration procedure can be summarized in a single lemma, which states that there is a solution to the Euler-Reynolds equations with new frequency and energy levels. The chapter also considers the High–Low Interaction term and the Transport term.Less

This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a notion of frequency energy levels is introduced. This notion is meant to accurately record the bounds which apply to the (*v*, *p*, *R*) coming from the previous stage of the construction. The chapter presents an example of a candidate definition for frequency and energy levels. Based on this definition, the effect of one iteration of the convex integration procedure can be summarized in a single lemma, which states that there is a solution to the Euler-Reynolds equations with new frequency and energy levels. The chapter also considers the High–Low Interaction term and the Transport term.

*Luis Caffarelli and Luis Silvestre*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0004
- Subject:
- Mathematics, Numerical Analysis

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. ...
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This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels K(t, x, s, y) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.Less

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels *K*(*t*, *x*, *s*, *y*) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.