Seminar lectures
An archive of lecture notes I prepare to give at seminars and colloquia.
Analysis Seminar (2/17/2020) and Math Bio Seminar (2/19/2020):
Title: Blowup solutions of a NavierStokeslike equation – A probabilistic perspective.
Abstract: Although it is not known if an L^3initial datum can produce a blowup solution of
the 3dimensional NavierStokes equations, various results have been established based on the
assumption that there is such a blowupgenerating datum. One of the results is the existence of
a blowupgenerating datum with minimal norm (JiaSverak 2013, Gallagher et al 2013, etc). The NSE is
a coupling of diffusion and transport processes. The diffusion tends to regularize the velocity field,
thus preventing blowup. On the other hand, the transport of the velocity field can build up velocity
in a very small region too quickly, possibly causing blowup. For small initial data, the diffusion
wins. From a control theory perspective, the existence of minimal blowup data implies that there is
a minimal ‘cost’ one has to pay to generate blowup. The existence of minimal blowup data has been
studied in several differential equations such as the harmonic map heatflow, the complex GinzburgLandau
equation and the Schrodinger equation. In 1997, Le Jan and Sznitman constructed solutions of the
NavierStokes equations via branching processes, similar to McKean’s approach to the KPPFisher equation
in 1975. We employ their idea to study the minimal blowup data of a NavierStokeslike equation called
cheap NSE by MontgomerySmith, 2002. Joint work with Radu Dascaliuc and Chris Orum.
Presentation
Applied Mathematics and Computation Seminar 1/10/2020:
Title: On the blowup, nonuniqueness, and stochastic explosion of PDE.
Abstract: The blowup phenomena of reactiondiffusion equations were studied by Kaplan,
Fujita, Levine and others in the 1960s. In 1975, McKean found a representation of solution
to the KPPFisher equation through a branching process, leading to a probabilistic approach
to study the regularity of solutions to a family of semilinear parabolic equations. In 1997,
Le Jan and Sznitman used ideas similar to McKean's for the NavierStokes equations (NSE).
The branching process associated with NSE is known to be explosive in stochastic sense (Dascaliuc,
Pham, Thomann, Waymire 2019). This property appeals the idea of nonuniqueness of solutions. However,
this remains a challenging problem. In several toy models (e.g. the alphaRiccati equation, the cheap NSE),
one can indeed construct more than one solution. I will discuss the nonuniqueness, blowup, and stochastic
explosion of these models. Joint work with Radu Dascaliuc, Enrique Thomann, and Ed Waymire.
Presentation
Title: Stochastic cascade solutions of the NavierStokes equations.
Abstract: Branching processes were used by McKean in 1975 to study the KPPFisher equation.
His ideas were robust enough to apply to a larger class of semilinear parabolic equations.
In 1997, Le Jan and Sznitman used similar ideas for the NavierStokes equations.
They introduced a class of stochastic cascade solutions. Since then, the theory of cascade solutions
has been quite fruitful, producing further insights on the uniqueness/nonuniqueness of solutions.
Cascade solutions can also be defined for various toy models of the NavierStokes equations,
for example the alphaRiccati equation and the complex Burgers equation. In this talk,
I will present some history and recent results of cascade solutions, with applications to the
NavierStokes equations.
Presentation
Analysis seminar 10/14/2019:
Title: On smallness condition of initial data for Le JanSznitman cascade of the NavierStokes
equations.
Abstract: The theory of cascade solutions to the NavierStokes equations was introduced by Le Jan
and Sznitman and later elaborated by Bhattacharya et al. They relied on a pointwise smallness
condition of the initial data in the Fourier domain to construct cascade solutions from a branching
process. We show how smallness of initial data in some global sense can be sufficient to define cascade
solutions. We also show a connection between cascade solutions and mild solutions constructed by
fixed point method. Joint work with Enrique Thomann.
Presentation
Analysis seminar 04/15/2019:
Title: A global regularity criterion for the NavierStokes equations based on approximate solutions.
Abstract: Since the pioneering works by Leray in the 1930s, various sufficient conditions have been
established for global regularity of the 3D NavierStokes equations. These criteria often require
some smallness condition or symmetry structure of the initial condition. We are motivated by
approximation perspective: is it possible to infer global regularity from one approximate solution,
for instance from the size of a numerical solution? By assuming a relatively simple scaleinvariant
relation of the size of the approximate solution, the resolution parameter, and the initial energy,
we show that the answer is affirmative for quite a general class of approximate solutions, including
Leray's mollified solutions. In this talk, I will explain two methods (called global and local pictures)
that lead to essentially the same result. Joint work with Vladimir Sverak.
Presentation
Analysis seminar 11/26/2018:
Title: Local regularity of strong solutions to the NavierStokes equations near blowup time.
Abstract: It is known that near blowup time, the critical norms of strong solutions blow up.
At this time, strong solutions cease to exist. However, swsolutions (or local energy solutions in
general) remain in local energy space and coincide the strong solution up to the blowup time.
In recent years, a large volume of literature on local regularity near blowup time has been published,
including the work of D. Albritton and T. Barker (2018), H. Jia and V. Sverak (2013), W. Rusin and
V. Sverak (2011). I will give a brief overview of these works and their connections to the problem of
minimal data for potential NavierStokes singularities in the halfspace.
Lecture notes
Analysis seminar 11/19/2018:
Title: Swsolutions for the NavierStokes equations.
Abstract: Swsolutions are local energy solutions, introduced by G. Seregin and V. Sverak in 2017.
"S" stands for strong/ Stokes/ split, "w" for weak. This class of solutions inherits good regularity
properties from the class of strong solutions, the energy inequality and global existence from the class
of weak solutions. It helps simplify the proofs of a number of important properties such as compactness,
weakstrong uniqueness and persistence of singularities. Swsolutions work well both in interior domains
and near nonslip boundaries, and seem to be a natural class of solutions for the study of boundary
regularity.
Lecture notes
Department Colloquium 10/29/2018:
Title: Minimal blowup data for potential NavierStokes singularities on the halfspace.
Abstract: The existence of minimal blowupgenerating initial data, under the assumption that
there exists an initial data leading to finitetime singularity, has been studied by Rusin and
Sverak (2011), Jia and Sverak (2013), and Gallagher, Koch and Planchon (2013, 2016) in several
critical spaces on the whole space. Our aim is to study the influence of the boundary on the
existence of minimal blowup data. We introduce a type of weighted critical spaces for the external
force that is bettersuited for our analysis than the usual Lebesgue spaces. Our main tools to treat
regularity near the boundary are (1) the notion of ``split'' weak solutions introduced by Seregin and
Sverak (2017), (2) the boundary epsilonregularity criteria and (3) a special decomposition of the
pressure near the boundary due to Seregin (2002). Our method works well for both the halfspace and
the whole space. Joint work with Vladimir Sverak.
Presentation
Links
Department of Mathematics
Oregon State University
 This page was last modified on Friday, February 28, 2020.

