**Research**

I work on the analysis of partial differential equations
arising in fluid mechanics. Below I will divide most of my papers into a few
different groups based on the general areas they fall into: (1) ** singularity
formation**, (2)

*(1)
Singularity Formation:*

** Finite-time Singularity Formation for Strong Solutions to
the 3D Euler Equations, I, II**, with I. Jeong.

arXiv preprints: https://arxiv.org/abs/1708.09372
and https://arxiv.org/abs/1711.03089.

In this work, we gave examples of finite-energy strong
solutions of the incompressible 3D Euler equations in the exterior of a cone which become singular in finite time. These solutions
initially have jump discontinuities in the gradient of the velocity field (and
the vorticity); however, they retain their regularity
having bounded velocity gradient and vorticity until
some maximal time T* at which time the maximum of both become unbounded. We
also show that this cannot happen in the 2D Euler equation nor
in the 3D axisymmetric Euler equation without swirl. This is the first result
of its kind for the 3D Euler equation. The constructions here rely upon
scale-invariant solutions which we introduced and which satisfy a lower-order
PDE which is much easier to handle than the 3D Euler system and where it is
possible to establish singularity formation in finite time. Two other important
features are the local well-posedness theorem in a
critical class which allows for scale-invariant
solutions as well as the cut-off argument which allows us to cut the infinite
energy scale-invariant solutions to produce finite-energy strong solutions
which become singular in finite time. The cut-off argument is possible mainly
due to the high symmetry of the solutions we construct which
is necessary since the problem is highly non-local. As explained in the
articles, the domain and regularity level play an important role in these works
but we do not believe them to be essential.

*On singularity formation in a Hele-Shaw
model**, *with P. Constantin, H. Nguyen, and V. Vicol.

arXiv preprint: https://arxiv.org/abs/1708.08490.

Here we considered a lubrication approximation model which
was derived in 1993 by Constantin et al. to model two
immiscible viscous fluids which are placed in a narrow gap between two plates
and prove that if the gap is narrow enough all initial profiles pinch either in
finite or infinite time (think of a drop of water pinching off a faucet).
Mathematically, we are considering a fourth-order degenerate dissipative PDE in
1+1 dimensions which models the thickness of a thin
neck of fluid *h* (a non-negative scalar). We show that *h* remains
smooth if and only if it remains strictly positive. This is done using energy
estimates, an iteration scheme, and maximal regularity for fourth order linear
parabolic problems. Then we show that the only stationary solutions to the
system (if the gap is short enough) vanish and are singular—in fact we also
establish a quantitative version of this fact. Thereafter, if a solution were
to be global (i.e. no finite time pinching), then there is a sequence of times
along which the dissipation in the problem vanishes. We then show that this
actually implies uniform convergence to the singular stationary profiles and
this then implies pinching in infinite time. In particular, *any* solution
must become singular either in finite or infinite time.

*Finite-time Singularity Formation for Strong Solutions to
the Boussinesq System**,
*with I. Jeong.

arXiv preprint: https://arxiv.org/abs/1708.02724.

This work has many similarities and actually appeared before
the 3D Euler works above. We prove finite-time singularity formation for
finite-energy strong solutions to the Boussinesq
system in a sector. The angle of the sector can be any value less than 180. One can see this work as the
simplest situation where one can use scale-invariant solutions to establish
finite time singularity formation for finite energy strong solutions. Also,
since the Boussinesq system is a pretty good model of
the axisymmetric 3D Euler system and also models temperature-driven convection,
it is important to understand the dynamics of this system.

*On the Effects of Advection and
Vortex Stretching**, *with
I. Jeong.

arXiv preprint: https://arxiv.org/abs/1701.04050.

Here we studied a class of one dimensional
models of the 3D Euler equation and similar models. In 1985, Constantin, Lax, and Majda (CLM)
introduced a 1D equation which models vortex stretching in
the 3D Euler equation and proved singularity formation in finite time for a
class of smooth solutions. Thereafter, De Gregorio took the CLM model and added
to it the natural transport term to give a 1D model of the full 3D Euler system
(with advection and vortex stretching). The interesting thing about De
Gregorio’s model is that it has a large class of smooth stationary states like
the 3D Euler equation and also seems to not exhibit finite time blow up for
smooth solutions. This is to say that the transport term mitigates the effects
of the vortex stretching term, at least numerically. This interplay between
transport and vortex stretching has been studied numerically in the 3D setting
and in other models quite extensively. Theoretical works where singularity
formation (or global regularity) is proven when transport and vortex stretching
are working against each other are essentially non-existent. Here, we showed
that if the transport term is weak enough, singularity formation still occurs.
Once even a very weak transport term is added to the system, it becomes very
difficult to prove that finite-time singularity formation still occurs
especially since the problem is non-local. We get around this essentially by
first finding a “blow-up profile” for the case without transport and then using
an implicit function-theorem type argument to deduce the existence of a profile
even when a weak transport term is added. It is expected that in the De
Gregorio case, where there is a stronger transport term, there is global
regularity.

*(2)
Well/ill-posedness Results:*

The following works are focused on issues related to
local-in-time solvability of some of the basic equations of fluid mechanics in
“critical spaces.” There are a number of ways to define “criticality.” One way
to define a critical space is by scaling but this definition is a little flimsy
for inviscid problems where there are multiple scalings. We consider a space to be critical if it is the
“largest” space where existence and uniqueness can “hope” to be established for
a given equation. In larger spaces there could be some sort of ill-posedness and in smaller
spaces, existence and uniqueness is relatively easy to establish. With Nader Masmoudi, we proved ill-posedness for the incompressible Euler equation and
similar incompressible models in the class of C^1 or Lipschitz
velocity fields. With In-Jee Jeong
we then proved ill-posedness
for the incompressible Euler equation in the critical Sobolev
space H^2. Both of these were based on the possibility of having large velocity
gradient while having small vorticity—which is
related to the unboundedness of singular integrals on
L^\infty. Then I showed that this L^\infty ill-posedness
can actually be much worse when the velocity field is conspiring with a
singular integral to produce an optimal L^p
estimates. On
the positive side, I then found a certain cancellation which
allows one to show that singular integrals of a merely bounded function could
still be bounded if the function satisfies a certain symmetry. Thereafter, In-Jee Jeong and I showed well-posedness for the
incompressible Euler and SQG equations in a scale of critical spaces. An
important consequence of this last result is that one can solve the
incompressible Euler equation with scale-invariant data.

** Ill-posedness results in critical
spaces for some equations arising in hydrodynamics**,
with N. Masmoudi.

arXiv preprint: https://arxiv.org/abs/1405.2478.

We established ill-posedness
of the incompressible Euler equation in the class of C^1 and Lipschitz continuous velocity fields. In fact, we showed
that many *incompressible* fluid equations are ill-posed
in L^\infty based spaces. The proof is based on
capitalizing on a linear instability mechanism coming from the non-local
pressure and making sure that non-linear terms do not disrupt the growth
mechanism. This seems to only be possible because we were working in the
“critical regime” so that non-linear and linear terms are about the same size
for short time. Since the blow-up of the C^1 norm of the velocity field is
actually proven by a contradiction argument, an interesting question which
remains open is to give an accurate description of the local-in-time dynamics
of the solutions we construct.

*Ill-posedness for the
incompressible Euler equations in critical Sobolev
spaces**, *with I. Jeong, Annals of PDE (2017).

arXiv preprint: https://arxiv.org/abs/1603.07820.

We established ill-posedness
for the 2D incompressible Euler equation in the (critical) Sobolev
space of H^2 velocity fields on the torus. Our approach follows, in spirit,
some of the ideas of my previous work with Masmoudi
on ill-posedness in the
class of C^1 velocity fields. In particular, both results are based on choosing
the right initial data and trying to prove that one can essentially do a Taylor
expansion in time. The right initial data here should have unbounded velocity
gradient and should have a hyperbolic stagnation point exactly at the point
where the gradient of the velocity field becomes unbounded. To make this
rigorous, we also use an idea which originates in work
of Kiselev and Sverak which
allows us to get a good control on the velocity field near a hyperbolic
stagnation point. An important question which remains open
after this work is what happens in the limit of vanishing viscosity. In
particular, if we take the data for which the 2D Euler equation is ill-posed in
H^1, how can we get lower bounds on the dissipation rate for the solution to
the 2D Navier-Stokes equation with the same data?

*Sharp L^p estimates for singular
transport equations**, *to
appear in Advances in Mathematics.

arXiv preprint: https://arxiv.org/abs/1407.2286.

I showed sharp estimates for a certain class of transport
equations with a singular integral forcing. In fact, the upper bounds are
trivial but the lower bounds are established via an intricate construction
where we carefully found the very worst possible instability
which could not be stopped by any secondary instabilities.

*Remarks on functions with bounded Laplacian.** *

arXiv preprint: https://arxiv.org/abs/1605.05266.

It is well known that a function with bounded and compactly
supported Laplacian does not necessarily have locally
bounded Hessian. In this work, I showed that this problem can
be overcome in a number of interesting situations. Namely, it is shown that
functions which have bounded Laplacian and which
satisfy a discrete symmetry around a given point must be quadratically
flat near the point of symmetry while there are explicit counterexamples (due
to an extra logarithmic singularity) without symmetry. This result is actually
the key to several of the finite-time singularity results described above. An
interesting question here would be to prove a similar result for general
elliptic operators with nice enough coefficients. We extended it in some cases
in our work on singularity-formation for strong solutions to the 3D Euler
equation above but a more general theory would be desirable.

*Symmetries and Critical Phenomena
in Fluids***,** with I. Jeong.

arXiv preprint: https://arxiv.org/abs/1610.09701.

This work focuses on the 2D Euler and SQG equations, though
the ideas are general and applicable in a number of other settings. Our first
observation, based on my work above “Remarks on functions with bounded Laplacian”, is that when the vorticity
of an ideal fluid satisfies a discrete symmetry, the full velocity field can be
recovered uniquely without any decay assumption on the vorticity.
This allows us to get existence and uniqueness of solutions to the 2D Euler
equation with merely bounded vorticity in the class
of discretely symmetric solutions. In a certain sense this result is optimal
since a type of “non-uniqueness” occurs when the symmetry assumption is
dropped. This is essentially because there could be pressure effects “from
infinity” if the vorticity doesn’t decay. An
important corollary is that one can then consider solutions
which have 0-homogeneous vorticity. Such
solutions then are shown to satisfy a 1D active scalar equation on S^1. We then
move to analyze the dynamics of solutions to this equation. It is shown that
solutions can display a number of interesting dynamical properties. In
particular, we construct time quasi-periodic solutions to the 2D Euler equation which exhibit “pendulum” like behavior. In
particular, a number of interesting ODE systems are found to be embedded in the
full 2D Euler system this way. In my opinion, much work should be done to
understand the dynamics of this simple 1D system. It seems to be a great place
to test many of the questions which we have for the 2D
Euler system. One interesting question is whether there exist solutions to this
1D system which have Sobolev
norms which grow exponentially fast. We have already given a large class of
solutions where the growth is linear (and no faster) without a boundary while
in the presence of a boundary solutions can grow exponentially fast.

** Osgood’s Lemma and results on the slightly supercritical 2-D
Euler equations for incompressible flow**, Archive for Rational
Mechanics and Analysis (2014).

arXiv
preprint: https://arxiv.org/abs/1308.1155.

This
was my first (successful!) project as a PhD student. Essentially the motivating
question for this project is: how much more singular can the Biot-Savart law be taken to keep global regularity in the
2D vorticity equation? The same can be asked for the
vortex patch problem. Constantin, Chae,
and Wu had shown that if the Biot-Savart law were
more singular by a loglog of a derivative then the
“slightly supercritical” Euler equation remains globally regular. We then
showed that the loglog is not actually optimal and
that there is a (presumably sharp) integral condition
in the spirit of Osgood’s ODE lemma that leads to global regularity. The same
condition also ensures global regularity for vortex patches. The vortex patch
case is quite a bit more difficult and requires us to combine different aspects
of Chemin’s and Bertozzi-Constantin’s
proofs of global regularity for classical vortex patches. In particular, in our
case the velocity gradient is *not* uniformly bounded and it is necessary
to modify several of the steps in the proof accordingly. On the other hand, tangential derivatives of the normal part of the velocity field being
bounded is what saves the day.

*(3)
Dispersive Problems:*

** Sharp decay estimates for
an anisotropic linear semigroup and applications to
the SQG and inviscid Boussinesq
systems**,
with K. Widmayer, SIAM J. of Mathematical Analysis
(2016).

arXiv preprint: https://arxiv.org/abs/1410.1415.

Here we studied an anisotropic (linear) dispersive operator which arises naturally in a number fluid
problems—notably in the *stable* regime of Rayleigh-Bernard convection
when a fluid is being heated from above and cooled from below. In this case
there is a dispersive stabilizing mechanism which leads to linear point-wise
decay of small perturbations of the base (linear) stationary state. Here we
rigorously prove these sharp decay estimates and use them to prove long-time
stability for the linear stationary state in the non-linear Boussinesq
system.

** Long-time stability for solutions of a Beta-plane Equation**,
with K. Widmayer, Communications on Pure and Applied
Mathematics (2017).

arXiv preprint: https://arxiv.org/abs/1509.05355.

Here we prove almost global existence for the Beta-plane
equation. Consider an inviscid fluid on a rotating
sphere. If a disturbance is localized and small amplitude, it is expected that
the disturbance disperse for long time. The beta-plane equation is an
approximation of how an incompressible fluid behaves on a rotating sphere
(though the problem is on the plane, R^2). The linear dispersive effect due to
the rotation is anisotropic and one can show that linear solutions decay like
1/t. One of the difficulties in the problem is the lack of rotational symmetry
since the problem is anisotropic, which significantly reduces the number of
commuting vector fields in the problem. Despite this, we use the linear decay
as well as a careful analysis of resonances to show almost global stability and
point-wise decay of small solutions using the method of space-time resonances.
One important component of the proof is the presence of a null structure which weakens the effects of space resonances. This result was later sharpened by Pusateri
and Widmayer to yield global stability for small
solutions for the same problem.

*(4)
Partially Dissipative Problems:*

We call a linear ODE x’=Ax is partially dissipative if A is
negative semi-definite. That is to say that A has some eigenvalues with
negative real part and possibly some eigenvalues with zero real part. The
directions corresponding to the negative eigenvalues decay exponentially fast
in time while the directions which are are undamped remain constant or oscillate in time. One can
imagine that this sort of scenario shows up in a number of physical problems.
Perhaps there is a preferred “direction” (due to gravity for example) in a
physical system and all other directions are killed off by some dissipative
mechanism. In this case, the long-time behavior of solutions to the linear
problem is that they live on the non-dissipative modes. If there are many
non-dissipative modes this behavior might be complicated but it is still a
“reduced” system. When one introduces a non-linearity to the problem, things
may change drastically. Moreover, while fully dissipative operators can usually
fully neutralize non-linear effects in small solutions, partially dissipative
operators may not be able to stop non-linear effects at all. Some of these
issues are discussed in great detail in the first paper below. For these reasons, understanding
the asymptotic behavior of solutions to non-linear partially dissipative
problems can be quite difficult and may also yield very interesting results. Allow
me to explain one interesting point which is found in
our second paper below with Wenqing Hu and Vladimir Sverak. We show that if we consider the 2D Navier-Stokes equations on the torus T^2 and we modify the
viscous term to damp all but finitely many modes, then all solutions converge
in the long-time limit to a stationary solution to the 2D Euler equation living
on those finitely many modes. Then one can classify those stationary solutions
precisely. For example, if one removes damping from exactly two modes which are
of different frequency and in different directions (like sin(2x)
and cos(y) for example), then solutions *must*
choose one of the two modes and land only on one of them. This “choice” happens
through a non-linear process and it is unclear whether there are even
statistics of which one is most likely chosen (though, one expects that the
stationary solution with the lowest frequency is generically chosen in the
long-time limit).

** On the asymptotic stability of stationary solutions of the
inviscid incompressible porous medium equation**,
Archive for Rational Mechanics and Analysis (2017).

arXiv preprint: https://arxiv.org/abs/1411.6958,
also note the errata here (which do not affect the results of the paper): Errata.

** On 2d Incompressible Euler Equations with Partial Damping**,
with W. Hu and V. Sverak, Communications on
Mathematical Physics (2017).

arXiv preprint: https://arxiv.org/abs/1511.02530.

*(5)
The Vanishing Viscosity Limit:*

The “vanishing viscosity limit” problem here refers to the
following question: do solutions to the Navier-Stokes
equation with viscosity v converge to solutions of the Euler equation as v—>
0? If so, is there a rate? These questions are non-trivial in two regimes: when
the initial data are non-smooth OR when the initial datum are
smooth but there are boundaries. When the initial data is smooth and there are
no boundaries, the answer is “yes” and there is a rate which
is known to be sharp. Much less is known when the data is non-smooth or there
are boundaries. In the case when there is a boundary, it is classical that the
convergence holds when the data is spatially analytic while it is still open
whether any sort of convergence holds even for smooth solutions. The reason for
the difficulty is that the Navier-Stokes and Euler
solutions have different boundary conditions. The Navier-Stokes
velocity vanishes identically on the boundary while the tangential component of
the Euler velocity is non-vanishing in general. This already means that the
convergence could never be uniform. However, it doesn’t preclude the
possibility of L^2 convergence or weak convergence, which is what the question
is really about. Along with the analyticity result, there are also some
necessary and sufficient conditions for convergence and these are described in
the second paper below. To now, there is no example of non-convergence and
convergence is only established in very special cases. When the boundary is
allowed to move with the fluid, convergence was established and we discuss this
in the first paper below. In
the case of non-smooth data, the rate of convergence is completely unknown in
2D since it is even possible that the Euler solution is non-unique. For
borderline data, the convergence can be given using the Yudovich
theory and we describe things related to this in the third paper below.

** Uniform Regularity for Free Boundary Navier-Stokes
Equations with Surface Tension**, with Donghyun Lee, to appear in the Journal of Hyperbolic
Differential Equations (2018).

arXiv preprint: https://arxiv.org/abs/1403.0980.

** Remarks on the Inviscid Limit for
the Navier-Stokes Equations for Uniformly Bounded
Velocity Fields**, with Peter Constantin,
Mihaela Ignatova, and Vlad Vicol, SIAM Journal of
Mathematical Analysis (2017).

arXiv preprint: https://arxiv.org/abs/1512.05674.

** On the inviscid limit of the 2D Navier-Stokes equations with vorticity
belonging to BMO-type spaces**,

arXiv preprint: https://arxiv.org/abs/1401.1382.

*(6)
Complex Fluids:*

** On some electroconvection models**,
with P. Constantin, M. Ignatova,
and Vlad Vicol, Journal of
Nonlinear Science (2017).

arXiv preprint: https://arxiv.org/abs/1512.00676.

** Global Regularity for some Oldroyd-B
type Models**, with F. Rousset,
Communications on Pure and Applied Mathematics (2015).

arXiv preprint: https://arxiv.org/abs/1308.0746.