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) sharp local well/ill-posedness results, (3) dispersive problems, (4) partially dissipative problems, (5) the vanishing viscosity limit, (6) and complex fluids.

 

(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, with Frederic Bernicot and Sahbi Keraani, Annales de l’Institut Henri Poincare (C), Analyse non lineaire (2016). 

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.