In 1993, Ilmanen proved a global existence of the weak solution to the mean curvature flow in the sense of the Brakke flow by using the phase field method. On the other hand, if we try to apply Ilmanen's proof to the volume preserving mean curvature flow, difficulties arise in the $L^2$-estimates of the non-local term. In this talk, we show a global existence of the weak solution in the sense of $L^2$-flow, in the $2$ or $3$-dimensional torus. We consider the Allen-Cahn equa+on with non-local term studied by Golovaty, and show the $L^2$-estimates of the non-local term and the monotonicity formula. As a recent result, we also prove a global existence of the weak solution to the mean curvature flow with forcing term in the suitable Sobolev space, by developing the method.

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We found a family of \(\mathbb{Z}_2\times O(2)\)-symmetric 3D steady gradient Ricci solitons. We show that these solitons are all flying wings. This confirms a conjecture by Hamilton.

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By employing the method of moving planes in a novel way, we extend some classical symmetry and rigidity results for smooth minimal surfaces to surfaces that have singularities of the sort typically observed in soap films.

This talk is based on joint work with Jacob Bernstein at Johns Hopkins University, appeared in preprint form on arXiv 2003.01784.

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The optimal transportation of one measure into another, leading to the notion of their Wasserstein distance, is a problem in the calculus of variations with a wide range of applications. The regularity theory for the optimal map is subtle and was pioneered by Caffarelli. This approach relies on the fact that the Euler-Lagrange equation of this variational problem is given by the Monge-Ampère equation. The latter is a prime example of a fully nonlinear (degenerate) elliptic equation, amenable to comparison principle arguments.

We present a purely variational approach to the regularity theory for optimal transportation, introduced with M. Goldman. Following De Giorgi's philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient through the construction of a variational competitor. This leads to a one-step improvement lemma, and feeds into a Campanato iteration on the \(C^{1,\alpha}\)-level for the optimal map, capitalizing on affine invariance.

On the one hand, this allows to reprove the \(\varepsilon\)-regularity result (Figalli-Kim, De Philippis-Figalli) bypassing Caffarelli's celebrated theory. This also extends to boundary regularity (Chen-Figalli), which is joint work with T. Miura, and to general cost functions, which is joint work with M. Prodhomme and T. Ried.

On the other hand, due to its robustness, it can be used as a large-scale regularity theory for the problem of matching the Lebesgue measure to the Poisson measure in the thermodynamic limit.

This is joint work with M. Goldman and M. Huesmann.

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In this talk, we discuss the insulated conductivity problem with multiple inclusions embedded in a bounded domain in n-dimensional Euclidean space. The gradient of a solution may blow up as two inclusions approach each other. The optimal blow up rate was known in dimension \(n=2\). It was not known whether the established upper bound of the blow up rates in higher dimensions were optimal. We answer this question by improving the previously known upper bound of the blow up rates in dimension \(n>2\).

This is a joint work with Zhuolun Yang.

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I will present a series of far-reaching new existence and structure results for convex ancient solutions to mean curvature flow. An interesting picture (conjectured by Huisken and Sinestrari) emerges: convex ancient and translating solutions in slab regions decompose into certain canonical configurations of Grim Reapers, subject to certain necessary constraints. In particular, we construct new families of solutions with discrete symmetries as well as families of solutions which possess only a single reflection symmetry. These include many eternal solutions which do not evolve by translation (resolving a conjecture of White in the negative). Prior to these results, the only known examples were the rotationally symmetric ancient pancake and the rotationally symmetric flying wing translators. Several interesting questions remain open, however.

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In a 2004 CR note and a 2007 JEMS paper, Jean Bourgain and Haim Brezis established the validity of some surprising estimates for elliptic systems in the \(L^1\) regime. It was an open problem in their 2007 paper whether one has an optimal Lorentz scale estimate for the Div-Curl system they consider.

In this talk I will discuss the resolution of this problem, which has recently been obtained in collaboration with Felipe Hernandez.

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We study the mean field limit of large systems of interacting particles. Classical mean field limit results require that the interaction kernels be essentially Lipschitz. To handle more singular interaction kernels is a longstanding and challenging question but which now has some successes. Joint with P.-E. Jabin, we use the relative entropy between the joint law of all particles and the tensorized law at the limit to quantify the convergence from the particle systems towards the macroscopic PDEs. This method requires to prove large deviations estimates for non-continuous potentials modified by the limiting law. But it leads to explicit convergence rates for all marginals. This in particular can be applied to the Biot-Savart law for 2D Navier-Stokes. To treat more general and singular kernels, joint with D. Bresch and P.-E. Jabin, we introduce the modulated free energy, combination of the relative entropy that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the kernels. Our modulated free energy allows to treat gradient flows with singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as the Patlak-Keller-Segel system in the subcritical regimes, is obtained.

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In this talk, we survey recent developments in higher-order curvature flow, with a focus on the most prominent fourth-order curvature flow: Willmore flow, surface diffusion flow, and Chen’s flow. We discuss these new developments in the context of big early successes, with a perspective to the current important open problems.

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The mean curvature flow is a parabolic PDE for hypersurfaces. As like semi-linear heat equations, it develops singularities where we can find ancient solutions by blowing-up. Therefore, we can observe what happens at a singularity if we classify all ancient solutions from the singularity and figure out their nice properties. For example, we can show the well-posedness around stable singularities by proving that all ancient flows from them always have positive speed. Also, we can find a way to avoid unstable singularities by studying one-sided ancient flows.

In this talk, we discuss some applications of ancient mean curvature flows. Also, we talk about how Angenent-Daskalopoulos-Sesum applied singularity analysis of semi-linear heat equations for classification of ancient flows.

This is based on several joint works with Brendle, Chodosh-Mantoulidis-Schulze, and Haslhofer-Hershkovits-White.

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The talk is about the so-called \(H\)-Theorem for a class of nonlinear Fokker-Planck equations which are of porous media type on the whole Euclidean space perturbed by a transport term. We first construct a solution in the sense of mild solutions on \(L^1\) through a nonlinear semigroup of contractions. Then we study the asymptotic behavior of the solutions when time tends to infinity. For a large class \(M\) of initial conditions we show their relative compactness with respect to local \(L^1\)-convergence, while all limit points belong to \(L^1\). Under an additional assumption, we obtain that we in fact have convergence in \(L^1\), if the initial condition is a probability density. The limit is then identified as the unique stationary solution in \(M\) to the nonlinear Fokker-Planck equation. This solution is thus an invariant measure of the solution to the corresponding distribution dependent SDE whose time marginals converge to it in \(L^1\). It turns out that under our conditions the underlying nonlinear Kolmogorov operator is a (both in the second and first order part) nonlinear analog of the generator of a distorted Brownian motion. The solution of the above mentioned distribution dependent SDE can thus be interpreted as a nonlinear distorted Brownian motion. Our main technique for the proofs is to construct a suitable Lyapunov function acting nonlinearly on the path in \(L^1\), which is given by the nonlinear contraction semigroup applied to the initial condition, and then adapt a classical technique of Pazy to our situation. This Lyapunov function is given by a generalized entropy function (which in the linear case specializes to the usual Boltzmann-Gibbs entropy) plus a mean energy part.

This is based on joint work with Viorel Barbu (Romanian Academy, Iasi).

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Our main interest in this talk is about the pointwise convergence problem for the Schrödinger propagator as time \(t\to 0\) initiated by Carleson. We address some generalisation of this problem, namely the pointwise convergence problem for infinitely many particles (Fermion).

In order for discussing this problem, we also generalise Kenig-Ponce-Vega’s maximal in time estimate for the Schrödinger propagator and one-dimensional endpoint Strichartz estimate. The later one provides almost sharp answer to an endpoint problem raised by Frank-Sabin in their work on Strichartz estimates for orthonormal system of data.

This talk is based on the joint work with Professors Neal Bez (Saitama University) and Sanghyuk Lee (Seoul National University).

The Alfvén waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfvén waves are governed by a system of nonlinear partial differential equations called the MHD system. In the talk, we will focus on the rigidity aspects of the scattering problem for the MHD equations: We prove that the Alfven waves must vanish if their scattering fields vanish at infinities. The proof is based on a careful study of the null structure and a family of weighted energy estimates.

This is based on the joint work with Mengni Li.

In this talk, I will report recent progress on global behaviors for solutions of energy subcritical defocusing semilinear wave equations with pure power nonlinearity. We prove that in space dimension 1 and 2, the solution decays in time with an inverse polynomial rate, hence giving an affirmative answer to a conjecture raised by Lindblad and Tao. In higher dimension, we obtain improved scattering results for the solutions. The proof is based on vector field method with new multipliers.

These works are jointed with Dongyi Wei.

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The Bernstein problem asks whether entire minimal graphs in $\mathbb{R}^{n+1}$ are necessarily hyperplanes. This problem was completely solved by the late 1960s in combined works of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti.

We will discuss the analogue of this problem for more general elliptic functionals, and some recent progress in the case $n = 6$.

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Mean curvature flow (MCF) is the negative gradient flow for the area functional in Euclidean spaces, or more generally in Riemannian manifolds. Over the past few decades, there have been substantial progress towards our knowledge on the analytic and geometric properties of MCF. For compact surfaces without boundary, we have a fairly good understanding of the convergence and singularity formation under the flow. The corresponding boundary value problems, however, are relatively less studied.

In this talk, we will discuss some recent results on MCF of surfaces with boundary. In the presence of boundary, suitable boundary conditions have to be imposed to ensure the evolution equations are well-posed. Two such boundary conditions are the Dirichlet (fixed or prescribed) and Neumann (free boundary or prescribed contact angle) boundary conditions. We will mention some new phenomena in contrast with the classical MCF without boundary. Using a new perturbation technique, we establish new convexity and pinching estimates for MCF with free boundary lying on an arbitrary convex barrier with bounded geometry. These imply the smooth convergence to shrinking hemispheres along the flow, provided that the surface is initially convex enough. This can be compared to Huisken’s celebrated convergence results for MCF in Riemannian manifolds.

This is joint work with Sven Hirsch. (These works are partially supported by RGC grants from the Hong Kong Government.)

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The theory of linear pseudo-differential operators is now well-known and, in particular, the mapping properties of linear pseudo-differential operators of Hörmander's class $S^m_{\rho, \delta}$ is well understood. Multilinear pseudo-differential operators were introduced by Coifman and Meyer before 1980 but detailed study of their mapping properties were developed after 2000. Some interesting facts peculiar to the multilinear case have been found.

In this lecture, I survey some features of multilinear pseudo-differential operators and introduce some recent results concerning the multilinear version of Hörmander's class.

This talk is based on joint works with Naohito Tomita (Osaka University) and Tomoya Kato (Gunma University).

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In this talk, I present a blow up construction - jointly with Kihyun Kim - on the self-dual Chern-Simons-Schrödinger equation (CSS), also known as a gauged nonlinear Schrödinger equation (NLS). CSS is $L^{2}$-critical, admits solitons, and has the psuedoconformal symmetry. These features are similar to the $L^{2}$-critical NLS. In this work, we consider pseudoconformal blow-up solutions under $m$-equivariance, $m\geq1$. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution $u$ with given asymptotic profile $z^{\ast}$: \[ \Big[u(t,r)-\frac{1}{|t|}Q\Big(\frac{r}{|t|}\Big)e^{-i\frac{r^{2}}{4|t|}}\Big] e^{im\theta}\to z^{\ast}\qquad\text{in }H^{1} \] as $t\to0^{-}$, where $Q(r)e^{im\theta}$ is a static solution. Secondly, we show that such blow-up solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of $u$. We construct a continuous family of solutions $u^{(\eta)}$, $0\leq\eta\ll1$, such that $u^{(0)}=u$ and for $\eta>0$, $u^{(\eta)}$ is a global scattering solution. Moreover, we exhibit a rotational instability as $\eta\to0^{+}$: $u^{(\eta)}$ takes an abrupt spatial rotation by the angle \[ \Big(\frac{m+1}{m}\Big)\pi \] on the time interval $|t|\lesssim\eta$.

We are inspired by works in the $L^{2}$-critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Raphaël, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay $r^{-(m+2)}$. This causes many technical issues for small $m$. Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blow-up ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles.

In the talks, I will present background of the problem, main theorems, and outline of the proof with emphasis on heuristics of main features, such as the long- range interaction between blow up profile and asymptotic profile $z$, and rotational instability mechanism.

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The classical Moser-Trudinger inequality is a borderline case of Soblolev inequalities and has important applications in geometric analysis and PDEs. On the two dimensional sphere, Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be reduced by half if the functions are restricted to a subset of the Sobolev space $H^1$ with mass center of the functions at the origin, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without constraints.

One such inequality, for example, incorporates the mass center deviation (from the origin) into the optimal inequality of Onofri on the sphere. In another view point, this inequality also generalizes to the sphere the Lebedev-Milin inequality and the second inequality in the Szegö limit theorem on the Toeplitz determinants on the circle, which is useful in the study of isospectral compactness for metrics defined on compact surfaces, among other applications.

The talk is based on a joint work with Amir Moradifam (University of California, Riverside) and a recent joint work with Alice Chang (Princeton).

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It's one of common interests in probability group to derive nonlinear PDEs or stochastic PDEs from microscopic interacting systems via scaling limits in space and time, usually by some averaging effect caused by the local ergodic property of the microscopic systems.

In the talk, we discuss the derivation of two objects: Motion by mean curvature (MMC) and coupled KPZ (Kardar-Parisi-Zhang) equation. The microscopic system we take is that of particles which perform random walks with interaction. To derive MMC, we allow creation and annihilation of particles. The system exhibits a phase separation to sparse and dense regions of particles and, macroscopically, the interface separating these two regions evolves under the MMC. We pass through the Allen-Cahn equation with nonlinear Laplacian at an intermediate level. On the other hand, the coupled KPZ equation is a system of nonlinear singular stochastic PDEs. It is ill-posed in a classical sense and requires renormalizations. We derive it under the fluctuation limit of multi-species weakly-asymmetric particle system of interacting random walks without creation and annihilation. The so-called Boltzmann-Gibbs principle plays a fundamental role for both.

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This talk bases on joint work with Liang Jin of Nanjing University of Science and Technology.

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