Upcoming seminars
- Hideyuki Miura. Tokyo Institute of Technology.
Mar 20(Wed), 2024 (15:00 ~ 16:30), 27-116
Critical norm blow-up for the energy supercritical nonlinear heat equation
We consider the critical norm blow-up problem for the nonlinear heat equation with power type nonlinearity |u|^{p-1}u in R^n.
In the Sobolev supercritical range p>(n+2)/(n−2), we show that if the maximal existence time T is finite, the scaling critical L^q norm of the solution becomes infinite at t=T. The range of p is optimal in view of known examples of blow-up solutions with the bounded critical norm for the Sobolev critical case. This is a joint work with Jin Takahashi (Tokyo institute of technology).
Past seminars 2023
- Soonsik Kwon. KAIST.
Nov 22, 3:30 pm (27-325)
Blow up dynamics of nonlinear dispersive equations
In the talk, I will give an overview on blow-up dynamics of nonlinear dispersive equations. This will include blow-up construction, instability and rigidity of blow up phenomena. I will give a schematic explanation on the topic and also briefly present our recent works on Chern-Simons-Schrödinger equations, which are joint works with Kihyun Kim and Sung-Jin Oh.
- Dongho Chae. CAU.
Sep 21, 3:00 pm (27-325)
Liouville problems in the Fluid Mechanics
Abstract TBA
- Jaemin Park. University of Basel.
August 14, 3-5 pm (127-104) and August 16, 3-5 pm
Quasi-periodic property is a commonly observed property in many Hamiltonian systems.
While such a property is easily observed in a linear Hamiltonian system, it is much more complicated to prove whether such solutions can exist in a nonlinear Hamiltonian system. The KAM theory is a classical method used to construct quasi-periodic solutions in a nonlinear/perturbed system. In this lecture, I will outline a proof of an application of the KAM theory to the generalized-surface quasi-geostrophic equations, constructing quasi-periodic solutions near a Rankine vortex.
References:
Lecture notes on nonlinear oscillations of Hamiltonian PDEs (Chapter: A tutorial in Nash-Moser theory) by M. Berti
KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation by P. baldi, M. Berti and R. Montalto
KAM for autonomous quasi-linear perturbations of KdV by P. baldi, M. Berti and R. Montalto.
Quasiperiodic solutions of the generalized SQG equation by J. Gomez-Serrano, A. Ionescu, J. Park
- Joonhyun La. KIAS.
August 14, 1:30 pm (127-104)
A uniform bound for solutions to a thermo-diffusive system
We obtain uniform in time L^infty -bounds for the solutions to a class of thermo-diffusive systems. The nonlinearity is assumed to be at most sub-exponentially growing at infinity and have a linear behavior near zero. This is a joint work with Lenya Ryzhik and Jean-Michel Roquejoffre.
- Sam Krupa. Max Planck Leipzig
Jul 28, 2-5 pm (27-325)
Systems of Conservation Laws and Stability of BV and Large L2 Data Solutions
For hyperbolic systems of conservation laws in one space dimension, existence of globally-in-time small-BV solutions is known when the initial data has small BV. Furthermore, it is known that these solutions are unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. One difficulty for the well-posedness theory of conservation laws is that many systems only admit a single entropy condition. In this lecture, I will present a framework for proving well-posedness of solutions to conservation laws using only one entropy. I will use the Burgers equation as a simple example to clarify ideas. Then, I will present a result for 2x2 systems which gives the uniqueness and stability of these globally small-BV solutions, amongst solutions which might have very large data for positive time (and in particular, might not even be BV). This result shows that the Tame Oscillation Condition, or the Bounded Variation Condition are not necessary to ensure the uniqueness of solutions with small BV data. My lecture will not assume strong familiarity with hyperbolic systems. The lecture will be based on joint work with G. Chen, W. Golding, and A. Vasseur.
- Jinmyeong Seok. SNU
June 15, 11 am, 129-309
On uniformly rotating binary stars
We study the asymptotic profiles, uniqueness and orbital stability of MCCANN’s uniformly rotating binary stars (Houston J Math 32(2):603–631, 2006) governed by the Euler–Poisson system. A new uniqueness result will be importantly used in stability analysis. Moreover, we apply our framework to the study of uniformly rotating binary galaxies of the Vlasov–Poisson system through REIN’s reduction (Handbook of differential equations: evolutionary equations, vol III, pp 383–476, 2007).
- Minjun Jo. UBC
June 1, 10 am, 27-325
On well-posedness of singular vortex patches
Symmetries often allow us to simplify the PDE dynamics that could be enigmatic in general, revealing the hidden structures the dynamics of our interest particularly bears. By imposing the m
-fold symmetry on certain singular vortex patches within the 2D Euler system, we prove the global well-posedness of such patches in the H"older spaces for mgeq3
. This lecture is to revisit the existing result of Mem. Amer. Math. Soc. 283(1400):1–102 (2023) by T. Elgindi and I.-J. Jeong in detail with the core concepts.
- Franck Sueur. Institut de Mathématiques de Bordeaux
March 14, 3:30 pm (129-301)
On the transport-Stokes system
In this talk, I will give an overview of some recent results on the so-called transport-Stokes system which describes sedimentation of inertialess particles in a viscous flow. This system couples a transport equation and the steady Stokes equations in the full 3D space. I will address the Cauchy problem, the regularity of the particle trajectories and the controllability issue. We will see that the system has some similarity to the incompressible Euler system. Some of the exposed results have been obtained in collaboration with Amina Mecherbet.
- Wojciech Ożański. Florida State University.
Feb 21, 10 am (zoom)
Instantaneous gap loss of Sobolev regularity of solutions to the 2D incompressible Euler equations
We will discuss classical well-posedness results of the
incompressible Euler equations, and recent results concerning
ill-posedness. We will then discuss, in the $2$D case, the first
result of instantaneous gap loss of Sobolev regularity. Namely we will
describe a construction of initial vorticity in the Sobolev space
$H^\beta$, $\beta \in (0,1)$ which gives rise to a unique
global-in-time solution of the $2$D Euler equations that
instantaneously leaves $H^{\beta'}$ for every $\beta' >(2-\beta )\beta
/(2-\beta^2)$. This is joint work with Diego C\'ordoba and Luis
Mart\'inez-Zoroa.
- Choi, Woocheol.
Jan 26, 2 pm.
Optimization problems in applications of fluid equations
Shape optimization and optimal control problems involving PDEs have been largely studied during last decades. In this talk, I will introduce several cases of such problems involving fluid equations. Two fundamental approach will be introduced: One is the traditional approach related to the constrained optimization methods, and the other one is a relatively new approach using the reinforcement learning.
Past seminars 2022
- Franck Sueur. Institut de Mathematiques de Bordeaux
Nov 24, 4:30 pm via zoom
2D incompressible Euler system in presence of sources and sinks
This talk is devoted to the 2D incompressible Euler system in presence of sources and sinks. This model dates back to Viktor Yudovich in the sixties and is an interesting example of nonlinear open system which has been widely used in controllability theory within the scope of smooth solutions. In this talk we will review how the classical issues of existence and uniqueness of weak solutions are challenged by the presence of incoming and exiting vorticity. This talk relies on two works, respectively with Marco Bravin and Florent Noisette.
- Hyungjun Choi. Princeton University.
Nov 3, 10 am.
Global well-posedness of slightly supercritical SQG equation and exponential gradient estimate
We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol defined as a power of logarithm is added to both velocity and dissipation terms to penalize the equation's criticality. Our primary tool is the nonlinear maximum principle which provides transparent proofs of global regularity for nonlinear dissipative equations. In addition, we prove an exponential gradient estimate for the critical surface quasi-geostrophic equation which improves the previous double exponential bound "Kiselev et al. 2007".
- DRIVAS, Theodore. SUNY, Stony Brook (New York, USA).
Oct 18, 10 am.
Remarks on the long-time dynamics of 2D Euler
We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.
- Jacky Chong. UT Austin
Oct 12, 11:00 am
Derivation of the Vlasov equation from quantum many-body Fermionic systems with singular interaction
We consider the combined mean-field and semiclassical limit for a system of the N fermions
interacting through singular potentials. We prove the uniformly in the Planck constant
h propagation of regularity for the Hartree–Fock equation with singular pair interaction
potential of the form $|x-y|^{-a}$, including the Coulomb interaction. Using these estimates, we
obtain quantitative bounds on the distance between solutions of the many-body Schr ̈odinger
equation and solutions of the Hartree–Fock and the Vlasov equations in Schatten norms.
For $a \in (0,1/2)$, we obtain global-in-time results when $N^{-1/2} \ll h \le N^{-1/3}$. In particular, it
leads to the derivation of the Vlasov equation with singular potentials. For $a \in (1/2,1]$, our results hold only on a small time scale, or with an N-dependent cutoff. The talk is based on our recent works in [1, 2, 3]. This is a joint work with Laurent Lafleche and Chiara Saffirio. The talk will be delivered in English and is meant for the general audience.
- Ayman R. Said. Duke University.
Sep 28, 3:30 pm
On the long-time behavior of scale-invariant solutions to the 2d Euler equation
In these 2 lectures we will give a brief introduction to the 2d the Euler equations
and discuss some of the key open questions in the field today. Then we will present
our recent findings, in which we will give a complete description of the long-time
behavior of uniformly bounded scale-invariant solutions the 2d Euler equation
satisfying a discrete symmetry. We show that all such solutions relax in infinite
time to rigidly rotating or steady states, which are fully classified and shown to
be piece-wise constant profiles with countably many jumps. Consequently, all
sufficiently symmetric non-constant scale-invariant solutions that are smooth on
S1 become singular in infinite time. On R2, this corresponds to generic infinite time
spiral and cusp formation. In the process, we also show that for scale invariant
solutions, the measure (on S1) of particles moving away from the origin and toward
spatial infinity is a strictly increasing function of time. This increasing function
of time generalizes to general solutions of the 2d Euler equation that are bounded
and m-fold symmetric (m ≥ 4) .
- CHOI, Kyudong. UNIST (Ulsan, Korea).
Sep 22, 4 pm (Department Colloquium)
Stability/instability of exact solutions of Euler equations
The incompressible Euler equations model flow of ideal fluids. We review a few given exact solutions in vorticity form and their stability issues. Here, the vorticity, which is defined by the curl of the flow velocity, describes the local spinning motion of the fluid. As an application of stability, we observe that some flow generates vortex filaments.
- Wojciech Ożański. Florida State University.
August 23, 10 am (zoom)
Well-posedness of logarithmic spiral vortex sheets
We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl \emph{(Vortr\"age aus dem Gebiete der Hydro- und Aerodynamik, 1922)} and by Alexander \emph{(Phys. Fluids, 1971)}. We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spirals: a \emph{velocity matching} condition and a \emph{pressure matching} condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also discuss well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary, as well as existence of nonsymmetric spirals.
- Xinliang An. National University of Singapore.
August 24, 10 am (zoom)
Low regularity ill-posedness for elastic waves and ideal compressible MHD in 3D and 2D
We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three and two spatial dimensions (3D and 2D). For 3D, inspired by the recent works of Christodoulou, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$, respectively. Both elastic waves and MHD are physical systems with multiple wave-speeds. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. In particular, when the magnetic field is absent in MHD, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations, we reveal their hidden subtle structures. With them we give a complete description of solutions’ dynamics up to the earliest singular event, when a shock forms. If time permits, we will also present the corresponding results in 2D. This talk is based on joint works with Haoyang Chen and Silu Yin.
- KWON, Bongsuk. UNIST.
July 25, 3 pm
Singularities in nonlinear hyperbolic equations
In this talk, I will mostly discuss the singularity formation of Burgers equation. It is well-known that, when the initial data has negative gradient at some point, the solutions blow up in a finite time. We shall study the properties of the blow-up profile of Burgers equation by introducing the self-similar variables and the modulations, which can be used to study the blow-up for general nonlinear hyperbolic systems. If time permits, I will also discuss the singularity formation for the 1D compressible Euler equations.
- SEOK, Jinmyoung. Kyonggi University.
July 25, 1:30 pm
N-equivariant solutions to Landau-LIfshitz equations with the Chern-Simons gauge
In this talk, we propose the Landau-Lifshitz type system augmented with Chern-Simons gauge terms, which can be considered as the geometric analog of so-called the Chern-Simons-Schrodinger equations. We first derive its self-dual equations through the energy minimization so that we provide i$N$-equivariant static solutions. We next deliver basic ideas of constructing $N$-equivariant solitary waves for non-self-dual cases and investigating their qualitative properties.
- KWON, Hyunju. ETH Zurich
July 25, 11 am.
Nonuniqueness of steady-state weak solutions to the surface quasi-geostrophic equations
In this talk, I'll present a construction of non-trivial steady-state weak solutions to both inviscid and dissipative surface quasi-geostrophic equations. The proof uses the convex integration scheme. This is joint work with X. Cheng and D. Li.
- YANG, Jonguk. University of Zurich
July 14, 4 pm
Renormalization of Dynamical Systems
It has been observed that two seemingly very different dynamical systems can bear striking resemblances when viewed at sufficiently small scales. Drawing motivations from physics, Feigenbaum (and independently, Collet and Tresser) introduced renormalization in the mid 1970's as a conjectural explanation of this phenomenon. Since then, this idea has been successfully applied to a wide variety of fundamentally important examples of dynamical systems, leading to deep and rigorous mathematical theories that describe their long-term behaviors.
In this talk, I will outline the general structure of a fully developed renormalization theory. The aim will be to emphasize intuition over formal and technical details, and to avoid giving specifics that excessively narrow the scope of the discussion. At the end, I will conclude with some concrete results (both classical and new) that were obtained through applications of renormalization techniques.
- JEON, Junekey. UCSD
July 6, 10 am
An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches
We prove that splash-like singularities cannot occur for sufficiently regular patch solutions to the generalized surface quasi-geostrophic equation on the plane or half-plane with parameter $\alpha \le 1/4$. This includes potential touches of more than two patch boundary segments in the same location, an eventuality that has not been excluded previously and presents nontrivial complications (in fact, if we do a priori exclude it, then our results extend to all $\alpha \in (0,1)$. As a corollary, we obtain an improved global regularity criterion for H3 patch solutions when $\alpha \le 1/4$, namely that finite time singularities cannot occur while the H3 norms of patch boundaries remain bounded.
- SHKOLLER, Steve. University of California, Davis
June 9, 9 am
The geometry of shock formation for Euler
I will describe a new geometric approach for the shock formation problem for the multi-dimentional Euler equations that provides uniform estimates for the solution along the entire hypersurface on which the shock forms. This, in turn, allows for a complete description of the solution along this hypersurface of first singularities.
- LA, Joonhyun. Stanford University.
May 19, 10 am
Propagation of singularities by Osgood vector field and for 2D inviscid incompressible fluids
We show that certain singular structures (Holderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of log log+(1=jxj) vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set. This is a joint work with Theo Drivas and Tarek Elgindi.
- YAO, Yao. National University of Singapore
Mar 17, 4 pm and Mar 24, 2 pm.
Lecture I: Small scale formations in the incompressible porous media equation
The incompressible porous media (IPM) equation describes the evolution of density transported by an incompressible velocity field given by Darcy’s law. Here the velocity field is related to the density via a singular integral operator, which is analogous to the 2D SQG equation. The question of global regularity vs finite-time blow-up remains open for smooth initial data, although numerical evidence suggests that small scale formation can happen as time goes to infinity. In this talk, I will discuss rigorous examples of small scale formations in the IPM equation: we construct solutions to IPM that exhibit infinite-in-time growth of Sobolev norms, provided that they remain globally smooth in time. As an application, this allows us to obtain nonlinear instability of certain stratified steady states of IPM. This is a joint work with Alexander Kiselev.
Lecture II: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations
In this talk, I will discuss some results on radial symmetry property for stationary and uniformly-rotating solutions for the 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.
- PARK, Jung-Tae KIAS (Seoul, Korea)
Jan 19, 10 am
Marcinkiewicz regularity for singular parabolic $p$-Laplace type equations with measure data
In this talk, we consider a parabolic $p$-Laplace type equation when the right-hand side is a signed Radon measure with finite total mass, whose model is
$$u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = \mu \quad \textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R}.$$
In the singular range $\frac{2n}{n+1} < p \le 2-\frac{1}{n+1}$, we discuss integrability results for the spatial gradient of a solution in the Marcinkiewicz space, under a suitable density condition of the right-hand side measure $\mu$.
- KIM, Chanwoo. University of Wisconsin-Madison
Jan 17, 10 am and Jan 18, 10 am.
From Boltzmann to incompressible Euler
We talk about a convergence of kinetic vorticity of Boltzmann toward the vorticity of incompressible Euler in 2D. When the Euler vorticity is below Yudovich, we prove a weak convergence toward Lagrangian solutions, while for the Yudovich class we have a strong convergence toward a unique solution with a rate. The talk would be self-contained covering necessary background in basic Boltzmann theory, asymptotic expansion (Hilbert expansion), and Lagrangian solutions of Euler.
- SHIN, Jinwoo KIAS (Seoul, Korea)
Feb 21, 3 pm
Introduction to the Yamabe Problem
A natural question in Riemannian geometry is whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This is called the Yamabe Problem. In 1960, Yamabe attempted to solve this problem using techniques of calculus of variations and elliptic partial differential equations. In this talk, we will discuss some basic facts related to the Yamabe problem and the Yamabe flow, which is another way to prove the Yamabe problem.
- KIM, Daehwan. Daegu University
Feb 21, 4 pm
Half-space type theorem of a translating soliton for the mean curvature flow
Mean curvature flow is that the hypersurface deforms in a normal direction and its speed equals to the mean curvature at each point. It is well-known that any closed hypersurface occurs singularities in finite time under the mean curvature flow. These singularities distinguishes into two types of hypersurfaces as blow up models: translating solitons and self-similar solitons that are also special solutions to the flow. In this talk, we introduce a half-space type theorem for a translating soliton. We define a half-space as one of the two parts into which a hyperplane divides R^(n+1). We prove that any complete translating soliton can not be contained in a half-space that is opposite direction of the direction of a translating soliton under the mean curvature flow.
Past seminars 2021
- DRIVAS, Theodore. SUNY, Stony Brook (New York, USA).
July 1, 9 am.
Some remarks on Kolmogorov's problem
We will discuss the stability and instability of Kolmogorov
flow on two-dimensional flat tori (Meshalkin-Sinai) and a related
example of non-uniqueness of smooth steady states of the
Navier-Stokes equations (Yudovich). Destabilization of this laminar
regime relates to the transition to turbulence. This talk will mainly be a survey of past
work, although we will finish with some speculative remarks (partially
based on numerical simulation) about the infinite Reynolds number limit.
- KWON, Hyunju. Institute for Advanced Study (Princeton, USA).
July 2, 9 am.
On Hölder continuous globally dissipative Euler flows
In the theory of turbulence, a famous conjecture of Onsager asserts that the threshold Hölder regularity for the total kinetic energy conservation of (spatially periodic) Euler flows is 1/3. In particular, there are Hölder continuous Euler flows with Hölder exponent less than 1/3 exhibiting strict energy dissipation, as proved recently by Isett. In light of these developments, I'll discuss Hölder continuous Euler flows which not only have energy dissipation but also satisfy a local energy inequality.
This is joint work with Camillo De Lellis.
- CHOI, Kyudong. UNIST (Ulsan, Korea).
July 6 and 7, 10 am
Stability of vortex patches of 2D Euler equations
We consider the incompressible Euler equations in R2 when the initial vorticity is of circular patch type. We show that it is stable in some weighted norm related to the angular impulse. This talk is designed to be introductory. Joint work w/ D. Lim.
- CHOI, Kyudong and LIM, Deokwoo. UNIST (Ulsan, Korea).
July 8 and 9, 10 am
Stability of radially symmetric, monotone vorticities of 2D Euler equations
We consider the incompressible Euler equations in R2 when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity produces stability in some weighted norm related to the angular impulse. For instance, it covers the cases of circular vortex patches and Gaussian distributions. Our stability does not depend on L∞-bound or support size of perturbations. The proof is based on the fact that such a radial monotone distribution minimizes the impulse of functions having the same level set measure. Joint work w/ D. Lim.
- ELGINDI, Tarek. Duke University.
July 15, 7:30 pm
Infinite time singularity formation in incompressible fluids
We discuss generic instability properties of stationary solutions to the 2d Euler equation, filamentation of vorticity, and the infinite time growth of the gradient of vorticity.
- OH, Sung-Jin. University of California Berkeley (Berkeley, USA).
July 26, 27, 29, and 30, 9 am
On the Cauchy problem for quasilinear dispersive PDEs
Quasilinear dispersive PDEs often arise in fluid dynamics and plasma physics as effective models. The goal of this lecture series is to provide an introduction to the theory of local well-/ill-posedness of the Cauchy problem for such equations. In the first part, I will cover classical concepts that are relevant to the wellposedness theory of quasilinear evolution equations, such as hyperbolicity, energy estimate, and the continuity of the solution map. In the second part, I will discuss illposedness mechanisms in the dispersive case, and techniques for proving wellposedness in the absence of such obstructions. An emphasis will be given on the phenomenon of degenerate dispersion, which is a strong instability mechanism for conservative quasilinear dispersive PDE.
- GIE, Gung-Min. University of Louisville (Louisville, USA).
July 29, 2 pm
Singular perturbations in fluid mechanics: Analysis and computations
Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers near the boundary of a domain, called boundary layers, where many important physical phenomena occur. In fluid mechanics, the Navier-Stokes equations, which describe the behavior of viscous flows, appear as a singular perturbation of the Euler equations for inviscid flows, where the small perturbation parameter is the viscosity. In general, verifying the convergence of the Navier-Stokes solutions to the Euler solution (known as the vanishing viscosity limit problem) remains an outstanding open question in mathematical physics. Up to now, it is not known if this vanishing viscosity limit holds true or not, even in 2D for which the existence, uniqueness, and regularity of solutions for all time are known for both the Navier-Stokes and Euler. In this talk, we discuss a recent result on the boundary layer analysis for the Navier-Stokes equations under a certain symmetry where the complete structure of boundary layers, vanishing viscosity limit, and vorticity accumulation on the boundary are investigated by using the method of correctors. We also discuss how to implement effective numerical schemes for slightly viscous fluid equations where the boundary layer correctors play essential roles. This is a joint work in part with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes, and with C.-Y. Jung and H. Lee.
- CHEN, Jiajie. Caltech (USA).
August 31 (Tue), 10 am
On the competition between advection and vortex stretching
Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. The presence of vortex stretching is the primary source of a potential finite-time singularity. However, to obtain a singularity, the effect of the advection is one of the obstacles. In this talk, we will first talk about some examples in incompressible fluids about the competition between advection and vortex stretching. Then we will discuss the De Gregorio (DG) model on a circle, which was proposed in 1990, and generalized the Constantin-Lax-Majda model to model this competition. The regularity of the DG model on a circle remains an open problem. For initial data with specific sign and symmetry properties, which provides the most promising candidate for a potential blowup solution up to now, we will show that the DG model develops a finite-time singularity if the advection is "weaker" than the vortex stretching , and it is globally well-posed if the advection is "stronger". In particular, our results rule out the potential blowup from smooth initial data in such a class.
- MIURA, Hideyuki. Tokyo Institute of Technology (Tokyo, Japan).
Sep 6 (Mon), 4 pm
Local regularity conditions on initial data for local energy solutions
of the Navier-Stokes equations
We study the regular sets of local energy solutions to the
Navier-Stokes equations in terms of conditions on the initial data.
It is shown that if a weighted L^2 norm of the initial data is finite,
then all local energy solutions are regular in a region confined by
space-time hypersurfaces determined by the weight.
This result refines and generalizes Theorems C and D of Caffarelli,
Kohn and Nirenberg (1982).
This is a joint work with Kyungkeun Kang and Tai-Peng Tsai.
- ABE, Ken. Osaka City University (Osaka, Japan).
Sep 15 (Wed), 10 am
Rigidity of Beltrami fields with a non-constant proportionality factor
Beltrami fields curl u=f u, div u=0, appear as steady states of ideal incompressible flows or plasma equilibria. I will discuss existence and non-existence issues on them for non-constant factors f.
- BAE, Junsik. NCTS.
Sep 28 (Tue), 2 pm
Global existence of smooth solutions to the 3D Vlasov-Poisson system
We study global existence of smooth solutions to the 3D Vlasov-Poisson system. We introduce "the good, the bad and the ugly" decomposition following the proof given in Glassey's book, a modification of Schaeffer's simplification (1991) of Pfaffelmoser's original proof (1992).
Reference: Robert T. Glassey, The Cauchy Problem in Kinetic Theory
- OH, Sung-Jin. University of California Berkeley (Berkeley, USA).
Nov 25, 4 pm
A tale of two tails
In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-time tails are in general different(!) from the stationary case in the presence of dynamical and/or nonlinear perturbations of problem. This is joint work with Jonathan Luk (Stanford).
- CHOI, Kyeongsu. KIAS (Seoul, Korea)
Dec 01, 3:30 pm
Asymptotic behavior at singularities of mean curvature flow
The mean curvature flow is an evolution of hypersurfaces under a geometric heat equation. As a solution to a parabolic PDE, the flow converges to a self-similar solution at its singularity after rescaling. Hence, we can find a scalar-valued function defined over the self-similar solution whose graph is the rescaled flow. Therefore, the function is a solution to a parabolic PDE, and thus we can study the fine asymptotic behavior by using the spectrum of the linearized operator. Indeed, we can also apply this theory for the classification of ancient flows.
In this talk, we discuss it applications to the optimal regularity of arrival time, namely solutions to the 1-Laplace equation. Also, we talk about a potential application to the generic mean curvature flow and the knot theory.
- LA, Joonhyun. Stanford University.
Dec 20, 4 pm
The Navier-Stokes-wall-grafted polymer system, global well-posedness, and polymer drag reduction.
The problem of reducing energy dissipation and wall drag in turbulent pipe and channel flows is a classical one which is important in practical engineering applications.
Remarkably, the addition of trace amounts of polymer into a turbulent flow has a pronounced effect on reducing friction drag. To study this mathematically, we introduce a new boundary condition for Navier-Stokes equations which models the situation where polymers are grafted to the wall. For engineering applications, the effects of polymer on drag reduction are thought to be most pronounced near the boundary and therefore such wall-grafted polymer chains are sometimes employed as drag-reducing agents. Our boundary condition - derived from a fluid-polymer stress balance - closes in the macroscopic fluid variables and becomes an evolution equation for the vorticity along the solid walls. We prove global well-posedness for the resulting system in two spatial dimensions and show that it captures the drag reduction effect in the sense that the vanishing viscosity limit holds with a rate. Consequently, we obtain bounds on energy dissipation rate which qualitatively agree with some experiments.
- SEOK, Jinmyoung. Kyonggi University.
Dec 21, 2:30 pm
Kinetic description of stable white dwarfs
In this talk, I will present some results on fermion ground states of the relativistic Vlasov-Poisson system arising in the semiclassical limit from relativistic quantum theory of white dwarfs. I'd like to discuss the existence and orbital stability of fermion ground states of the three dimensional relativistic Vlasov-Poisson system for subcritical mass. We will also see that the mass density of such fermion ground states satisfies the Chandrasekhar equation for white dwarfs. This is a joint work with Prof. Juhi Jang at USC.
- HONG, Youngjoon. Sungkyunkwan University.
Dec 21, 5 pm
Solving partial differential equations using deep learning
The Neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility to represent the solution of the partial differential equation. In particular, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). In this talk, we propose a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to various differential equations.
- CHOI, Kyeongsu. KIAS (Seoul, Korea)
Dec 22, 2 pm
Asymptotic behavior of solutions to elliptic and parabolic PDEs with compact self-adjoint linearized operator
In this talk, we mainly study asymptotic behavior of solutions to quasilinear parabolic PDEs near self-similar solutions, when the linearized operators of the self-similar solutions are compact and self-adjoint. Then, we talk about the fully nonlinear parabolic equations. We also consider solutions to elliptic PDEs, whose blow-ups or blow-downs are homogeneous functions. Then, we will discuss how the radila parameter in the elliptic case plays the role of time variable in the parabolic case.
- KIM, Seunghyeok. Hanyang University.
Dec 22, 4 pm
Asymptotic analysis on positive solutions of the Lane-Emden system near the critical hyperbola
We are concerned with positive solutions of the Lane-Emden system on a smooth bounded convex domain.
This system appears as an extremal equation of a particular Sobolev embedding and is also closely related to the Calderon-Zygmund estimate.
Given an arbitrary family of solutions, we thoroughly analyze its asymptotic behavior
as the exponents of the nonlinearities tend to the critical ones, establishing a detailed qualitative and quantitative description.
In particular, we derive a priori energy bound for the solutions and prove that the multiple bubbling phenomena may arise.
Also, we observe a dichotomy phenomenon for the asymptotic behavior of the family of solutions.
In one case, the nonlinear structure of the system makes the interaction between bubbles so strong,
so the determination process of the blow-up rates and locations is totally different from that of the classical Lane-Emden equation.
In the other case, the blow-up scenario is relatively close to (but not the same as) that of the classical Lane-Emden equation,
and only one-bubble solutions can exist.
Even in the latter case, the standard approach does not work well, which forces us to devise a new method.
As a by-product of our analysis, we also obtain a general existence theorem valid on any smooth bounded domains.
This is joint work with Sang-Hyuck Moon (National Center for Theoretical Sciences, Taiwan).
- HWANG, Sukjung KIAS (Seoul, Korea)
Dec 28, 10 am and Dec 30, 10 am.
Regularity and existence of weak solutions of porous medium equation with a divergence type of drift I, II
Part I: In this talk, we consider porous medium type equations with the divergence form of drift that can be applied to fluid dynamics and math biology. The first part of the talk is about conditions on the drift concerning continuity of nonnegative weak solutions (joint work with K. Kang and Y. P. Zhang). The second part is about the existence of nonnegative weak solutions in the Wasserstein space where the nonlinear diffusion and initial data affect the scaling invariant classes of the drift (joint work with K. Kang and H. Kim).
Part II: In this talk, we explain the existence of nonnegative weak solutions in the Wasserstein space where the nonlinear diffusion and initial data affect the scaling invariant classes of the drift.
Moreover, we discuss uniqueness result and application for a repulsive Keller-Segel model. This is joint work with K. Kang (Yonsei Univ.) and H. Kim (Hannam Univ.).