Schedule for: 21w5503 - Singularity Formation in Nonlinear PDEs (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will discuss some recent results and formulate some conjectures on singularity formation in the context of geometric wave equations. This comprises joint work with Miao and Schlag and others.

A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid, when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on the existence and asymptotic behaviour of these solutions. We describe, with precise asymptotics, interacting vortices, and travelling helices. We rigorously establish the law of motion of ”leapfrogging vortex rings”, originally conjectured by Helmholtz in 1858. This is joint work with Juan Davila, Monica Musso, and Juncheng Wei.

Whether the 3D incompressible Euler and Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In an effort to provide a rigorous proof of the potential Euler singularity revealed by Luo-Hou's computation, we develop a novel method of analysis and prove that the original De Gregorio model and the Hou-Lou model develop a finite time singularity from smooth initial data. Using this framework and some techniques from Elgindi's recent work on the Euler singularity, we prove the finite time blowup of the 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ initial velocity and boundary. Further, we present some new numerical evidence that the 3D incompressible Euler equations with smooth initial data develop a potential finite time singularity at the origin, which is quite different from the Luo-Hou scenario. Our study also shows that the 3D Navier-Stokes equations develop nearly singular solutions with maximum vorticity increasing by a factor of $10^7$. However, the viscous effect eventually dominates vortex stretching and the 3D Navier-Stokes equations narrowly escape finite time blowup. Finally, we present strong numerical evidence that the 3D Navier-Stokes equations with slowly decaying time-dependent viscosity develop a finite time singularity.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The kinetic wave equation arises in weak wave turbulence theory. In this talk we are interested in its derivation as an effective equation from dispersive waves with quadratic nonlinearity for the microscopic description of a system. We focus on the space-inhomogeneous case, which had not been treated earlier. More precisely, we will consider such a dispersive equations in a weakly nonlinear regime, and for highly oscillatory random Gaussian fields with localised enveloppes as initial data. A conjecture in statistical physics is that there exists a kinetic time scale on which, statistically, the Wigner transform of the solution (a space dependent local Fourier energy spectrum) evolve according to the kinetic wave equation. I will present a joint work with Ioakeim Ampatzoglou and Pierre Germain in which we approach the problem of the validity of this kinetic wave equation through the convergence and stability of the corresponding Dyson series. We are able to identify certain nonlinearities, dispersion relations, and regimes, and for which the convergence indeed holds almost up to the kinetic time (arbitrarily small polynomial loss).

We consider quadratic Klein-Gordon equations with an external potential $V$ in $3+1$ space dimensions. We assume that $V$ is generic and decaying, and that the operator $H= - \Delta+ V+ m^2$ has an eigenvalue $\lambda^2 < m^2$. This is a so-called ‘internal mode’ and gives rise to time-periodic localized solutions of the linear flow. We address the question of whether such solutions persist under the full nonlinear flow. Our main result shows that all small nonlinear solutions slowly decay as the energy is transferred from the internal mode to the continuous spectrum, provided a natural Fermi golden rule holds. This extends the seminal work of Soffer-Weinstein for cubic nonlinearities to the case of any generic perturbation. This is joint work with T. L\'eger (Princeton University).

We will describe the recent asymptotic analysis with Jonas Luehrmann of the Sine-Gordon evolution of odd data near the kink. We do not rely on the complete integrability of the problem in a direct way, in particular we do not use the inverse scattering transform.

In this talk, I will introduce the spherical droplet problem in the Landau-de Gennes theory. With a novel bifurcation diagram, we find solutions with ring and split-core disclinations. This work theoretically confirms the numerical results of Gartland and Mkaddem in 2000.

The question of singularity formation vs global regularity for the 3D Euler equation is a major open problem. Several years ago, Hou and Luo proposed a new scenario for singularity formation based on extensive numerical simulations. Several 1D models of the scenario have been analyzed rigorously and they all lead to finite time blow up for some initial data. In this work, we explore a 2D model that aims to gain insight into the mechanics of boundary layer where extreme growth of vorticity is observed. We isolate a regularization mechanism and build a simplified model around it which is globally regular. For a more realistic model, we prove finite time blow up. This is a joint work with Siming He (Duke University).

In the talk, we will review some recent work on nonlinear asymptotic stability of the two dimensional incompressible Euler equations, with a focus on shear flows and vortices. Some open problems will also be discussed.

This talk presents two examples of the smoothing and stabilizing phenomenon for coupled PDE systems that prevents potential finite-time singularity formation. The 3D incompressible Euler equation can potentially develop finite-time singularities, as indicated by recent numerical simulations and theoretical results. However, when the Euler equation is coupled with the equation of the non-Newtonian stress tensor via the Oldroyd-B model, small data global well-posedness can be established and the coupling prevents the potential singularity. A 2D incompressible Euler-like equation with an extra Riesz transform term is not known to be globally well-posed. But, when coupled with the magnetic field via the magneto-hydrodynamic (MHD) system, we can show the global well-posedness near a background magnetic field with explicit decay rates. The magnetic field stabilizes the fluid.

The first mathematical result of inviscid damping was done in Gevrey spaces for the Couette flow. A natural question is whether Gevrey regularity is really necessary and whether one can consider more general flow than Couette. We will review some recent developments in the inviscid damping theory of 2D Euler. These new results include the optimality of the Gevrey regularity required to get the nonlinear damping as well as the generalization to the asymptotic stability around more general monotone flows.

This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.

The Ginzburg-Landau model is a phenomenological description of superconductivity. A crucial feature is the occurrence of vortex lines, which appear above a certain value of the strength of the applied magnetic field called the first critical field. In this talk I will present a sharp estimate of this value and report on a joint work with Etienne Sandier and Sylvia Serfaty in which we study the onset of vortex lines and derive an interaction energy for them.

We study a hyperbolic version of the Navier-Stokes equations obtained by using Cattaneo heat transfer law instead of Fourier law, evolving in a thin strip $\RR\times (0,\varepsilon)$. The formal limit of these equations is a hyperbolic Prandtl type equation. We prove the existence and uniqueness of a global solution to these equations under a uniform smallness assumption on the data in Gevrey 2 class. Then we justify the limit from the anisotropic hyperbolic Navier-Stokes system to the hydrostatic hyperbolic Navier-Stokes system with Gevrey 2 data. We also exhibit smallness assumptions on the data in Gevrey 2 class, under which the solutions are global in time.

Motivated by the non-corotational Beris-Edwards $Q$-tensor system modeling the hydrodynamic of nematic liquid crystal materials, we consider the corresponding Ericksen vectorial model that Includes kinematic transport parameters for molecules of various shapes and show that there exists a global weak solution in dimension three, which is smooth away from a closed set with Hausdorff dimension at most $15/7$. This is a joint work with Hengrong Du.

I will present some recent results concerning non-degeneracy, existence and multiplicity of solutions to a Lane-Emden critical system obtained in collaboration with R.Frank and S.Kim.

The p-widths of a Riemannian manifold were introduced by Gromov as a nonlinear version of the eigenvalues of the Laplacian (replacing the Dirichlet energy on functions with the area functional on submanifolds). I will discuss recent work with C. Mantoulidis (Rice) concerning the p-widths on surfaces, using in particular Liu—Wei’s analysis of entire solutions to the sine-Gordon equation on the plane. In particular, we prove that the p-widths on a surface correspond to immersed geodesics (instead of geodesic nets) and we compute the entire p-width spectrum of $ S^2$ yielding the constant in the Liokumovich—Marques—Neves Weyl law in this dimension.

We will discuss some recent Liouville type theorems and their applications. The problems under consideration include the Lane-Emden equation and the diffusive Hamilton-Jacobi equation. Joint works with L. Dupaigne, R. Filippucci, P. Pucci and B. Sirakov.

We discuss why the mean curvature flow of generic closed surfaces in ${\mathbb R}^3$ avoids asymptotically conical and non-spherical compact singularities. We also discuss why the mean curvature flow of generic closed low-entropy hypersurfaces in ${\mathbb R}^4$ is smooth until it disappears in a round point. This is joint work with O. Chodosh, K. Choi, and F. Schulze.

present a recent existence result for multi-phase Brakke flow starting from arbitrary partition with locally finite co-dimension 1 Hausdorff measure which improves on my own work with Lami Kim in 2017. The new aspect is that the flow has a character of BV solution, a notion introduced by Luckhaus-Sturzenhecker in 1995, in addition to being a Brakke flow. This is a joint work with Salvatore Stuvard.

Some formal asymptotic results will be presented.

We construct solutions of the fast diffusion equation, which exist for all $t\in {\mathbb R}$ and are singular on the set $\Gamma(t):= \{ \xi(s) ; s \leq ct \}$, $c>0$, where $\xi \in C^3({\mathbb R};{\mathbb R}^n)$ , $n\geq 2$. We also give a precise description of the behavior of the solutions near $\Gamma(t)$. This is a joint work with John King, Jin Takahashi and Eiji Yanagida.

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace leads to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap) or algebraically slow (which is only possible in the presence of zero modes). In the first case, we identify the leading order asymptotics. Our results improve various results in the literature, while shortening their proofs. Joint work with Beomjun Choi and Robert J. McCann.

In Ginzburg-Landau theory, the presence of a strong magnetic field allows for the existence of stable vortex states. The study of global minimizers of the Ginzburg-Landau energy in $2d$ and a characterization of their vorticities is the focus of a series of works by Sandier and Serfaty in the$\varepsilon \to 0$ limit, where $\varepsilon$is the inverse of the Ginzburg-Landau parameter. However, the full range of existence of stable configurations with prescribed vorticity, different from the optimal one, remains an open problem. In particular, it is expected that local minimizers with $1\ll N\sim 1/\varepsilon^\alpha,$ for some $\alpha>0$ should exist, provided the magnetic field is strong enough. The best partial results until recently could only cover very slowly diverging ($N\lesssim |\log \varepsilon|$)numbers of vortices. In joint work with R. L. Jerrard, we prove the existence of local minimizers with prescribed vorticity for a wide range of external fields and treat for the first timea number ofvortices comparable to a power of $1/\varepsilon.$

Harmonic maps with free boundary are rather old objects in geometry which has been used recently in several results related to the co-dimension one conjecture, extremal metrics of Steklov eigenvalues or liquid crystal flows. I will report on recent results on a new approximation of these maps which allows to better capture the boundary behavior and construct weak solutions of the associated heat flow. I will also give a small energy criterion which allows to prove partial regularity of the solutions.

For the Stokes system in the half space, Kang [Math.Ann.2005] showed that a solution generated by a compactly supported, H\"older continuous boundary flux may have unbounded normal derivatives near the boundary. We first prove explicit global pointwise estimates of a slightly revised solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary. This is a joint work with Kyungkeun Kang, Baishun Lai and Chen-Chih Lai.

The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Very recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,\alpha}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,\alpha}$ finite time blow-up solution for Boussinesq equations where temperature has diffusion and finite energy. The main difficulty is that the Laplace operator of temperature equation is not coercive under Sobolev weighted norm which is introduced by Elgindi. We introduced a new time scaling formulation and new weighted Sobolev norms, under which we obtain the coercivity estimate. The new norm is well-coupled with the original norm, which enable us to finish the proof.

We consider the Keller-Segel system in the plane with an initial condition with suitable decay and critical mass 8 pi. We find a function u0 with mass 8 pi such that for any initial condition sufficiently close to u0 and mass 8 pi, the solution is globally defined and blows up in infinite time. We also find the profile and rate of blow-up. This result answers affirmatively the question of the nonradial stability raised by Ghoul and Masmoudi (2018). This is joint work with Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).

In 1994 Velázquez constructed a smooth \( O(4)\times O(4)\) invariant Mean Curvature Flow that forms a type-II singularity at the origin in space-time. Stolarski very recently showed that the mean curvature on this solution is uniformly bounded. Earlier, Velázquez also provided formal asymptotic expansions for a possible smooth continuation of the solution after the singularity. Jointly with S. Angenent and N. Sesum we establish the short time existence of Velázquez' formal continuation, and we verify that the mean curvature is also uniformly bounded on the continuation. Combined with the earlier results of Velázquez--Stolarski we therefore show that there exists a solution \(\{M_t^7\subset\RR^8 \mid -t_0 (Online)

This lecture is devoted to two inequalities: (1) Reverse Hardy-Littlewood-Sobolev inequalities, (2) Two-dimensional logarithmic inequalities. None of these inequalities is classical. Both raise interesting open questions, with applications to nonlinear diffusions and Schroödinger equations This corresponds to joint results obtained with: (1) Jose A. Carrillo, Matias G. Delgadino, Rupert L. Frank, Franca Hoffmann (2) Rupert L. Frank, Louis Jeanjean.

We consider the energy critical heat equation $$ u_t=\Delta u+ u^{\frac{n+2}{n-2}}, u(x,0)= u_0 $$ We prove that if $n\geq 7$ and $ u_0\geq 0$, then any blow-up must be of Type I. (In the radially symmetric case, $n\geq 5$). The proof uses some ideas from geometric measure theory and a reverse inner-outer gluing mechanism.

The primitive equation is an important model for large scale fluid model including oceans and atmosphere. While solutions to the viscous model enjoy global regularity, inviscid solutions may develop singularities in finite time. In this talk, I will review the methods to show blowup, and case share more recent progress on qualitative properties of singularity formation.

After a brief discussion for harmonic map problems from a three-ball into the two-sphere, we review on an open problem posed by R.Schoen, the notions of relaxed energy, minimal connection and some results in the late 1980s by several groups. We then focus on a higher dimensional version of these studies. And we shall present a solution to an open problem proposed by Brezis-Mironescu recently.