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 Navier-Stokes-like equation – A probabilistic perspective.
Abstract: Although it is not known if an L^3-initial datum can produce a blowup solution of
the 3-dimensional Navier-Stokes equations, various results have been established based on the
assumption that there is such a blowup-generating datum. One of the results is the existence of
a blowup-generating datum with minimal norm (Jia-Sverak 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 Ginzburg-Landau
equation and the Schrodinger equation. In 1997, Le Jan and Sznitman constructed solutions of the
Navier-Stokes equations via branching processes, similar to McKean’s approach to the KPP-Fisher equation
in 1975. We employ their idea to study the minimal blowup data of a Navier-Stokes-like equation called
cheap NSE by Montgomery-Smith, 2002. Joint work with Radu Dascaliuc and Chris Orum.
Applied Mathematics and Computation Seminar 1/10/2020:
Title: On the blowup, nonuniqueness, and stochastic explosion of PDE.
Abstract: The blowup phenomena of reaction-diffusion equations were studied by Kaplan,
Fujita, Levine and others in the 1960s. In 1975, McKean found a representation of solution
to the KPP-Fisher 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 Navier-Stokes 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 non-uniqueness of solutions. However,
this remains a challenging problem. In several toy models (e.g. the alpha-Riccati 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.
Title: Stochastic cascade solutions of the Navier-Stokes equations.
Abstract: Branching processes were used by McKean in 1975 to study the KPP-Fisher 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 Navier-Stokes 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 Navier-Stokes equations,
for example the alpha-Riccati equation and the complex Burgers equation. In this talk,
I will present some history and recent results of cascade solutions, with applications to the
Analysis seminar 10/14/2019:
Title: On smallness condition of initial data for Le Jan-Sznitman cascade of the Navier-Stokes
Abstract: The theory of cascade solutions to the Navier-Stokes 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.
Analysis seminar 04/15/2019:
Title: A global regularity criterion for the Navier-Stokes 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 Navier-Stokes 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 scale-invariant
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.
Analysis seminar 11/26/2018:
Title: Local regularity of strong solutions to the Navier-Stokes 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, sw-solutions (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 Navier-Stokes singularities in the half-space.
Analysis seminar 11/19/2018:
Title: Sw-solutions for the Navier-Stokes equations.
Abstract: Sw-solutions 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,
weak-strong uniqueness and persistence of singularities. Sw-solutions work well both in interior domains
and near non-slip boundaries, and seem to be a natural class of solutions for the study of boundary
Department Colloquium 10/29/2018:
Title: Minimal blowup data for potential Navier-Stokes singularities on the half-space.
Abstract: The existence of minimal blowup-generating initial data, under the assumption that
there exists an initial data leading to finite-time 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 better-suited 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 epsilon-regularity criteria and (3) a special decomposition of the
pressure near the boundary due to Seregin (2002). Our method works well for both the half-space and
the whole space. Joint work with Vladimir Sverak.
Department of Mathematics
Oregon State University
This page was last modified on Friday, February 28, 2020.