Past Partial Differential Equations Seminar

Nonlinear wave equations are ubiquitous in physics, and in three spatial dimensions they can exhibit a wide range of interesting behaviour even in the small data regime, ranging from dispersion and scattering on the one hand, through to finite-time blowup on the other. The type of behaviour exhibited depends on the kinds of nonlinearities present in the equations. In this talk I will explore the boundary between "good" nonlinearities (leading to dispersion similar to the linear waves) and "bad" nonlinearities (leading to finite-time blowup). In particular, I will give an overview of a proof of global existence (for small initial data) for a wide class of nonlinear wave equations, including some which almost fail to exist globally, but in which the singularity in some sense takes an infinite time to form. I will also show how to construct other examples of nonlinear wave equations whose solutions exhibit very unusual asymptotic behaviour, while still admitting global small data solutions.

It the talk, various conditions of local regularity of axisymmetric suitable weak solutions, including the so-called slightly supercritical ones, will be discussed.

The soap bubble theorem says that a closed, embedded surface of the Euclidean space with constant mean curvature must be a round sphere. Especially in real-life problems it is of importance whether and to what extent this phenomenon is stable, i.e. when a surface with almost constant mean curvature is close to a sphere. This problem has been receiving lots of attention until today, with satisfactory recent solutions due to Magnanini/Poggesi and Ciraolo/Vezzoni.
The purpose of this talk is to discuss further problems of this type and to provide two approaches to their solutions. The first one is a new general approach based on stability of the so-called "Nabelpunktsatz". The second one is of variational nature and employs the theory of curvature flows. 

Nonuniform Ellipticity is a classical topic in PDE, and regularity of solutions to nonuniformly elliptic and parabolic equations has been studied at length. I will present some recent results in this direction, including the solution to the longstanding issue of the validity of Schauder estimates in the nonuniformly elliptic case obtained in collaboration with Cristiana De Filippis. 

The aim of this talk is to present some recent developments of Geometric Measure Theory on non smooth spaces with lower Ricci Curvature bounds, mainly related to the first and second variation formula for the area, and their applications to the isoperimetric problem on non compact manifolds. The reinterpretation of some classical results in Geometric Analysis in a low regularity setting, combined with the compactness and stability theory for spaces with lower curvature bounds, leads to a series of new geometric inequalities for smooth, non compact Riemannian manifolds. The talk is based on joint works with Andrea Mondino, Gioacchino Antonelli, Enrico Pasqualetto and Marco Pozzetta.

The aim of this talk is to understand the qualitative properties that emerge from a PDE model inspired from neurosciences, in order to understand what are the key processes that lead to mathematical complex patterns for the solutions of this equation. 

In this talk I will discuss the development and justification of linearised shock-capturing for aeronautical applications such as flutter, forced response and design optimisation.  At its core is a double-limiting process, reducing both the viscosity and the size of the unsteady or steady perturbation to zero. The design optimisation also requires the consideration of the adjoint equations, but with shock-capturing this is best done at the level of the numerical discretisation, rather than the PDE.

We  report  on the theory of evolution equations that combine a strongly nonlinear parabolic character with the presence of fractional operators representing long-range interaction effects, mainly of fractional Laplacian type. Examples include nonlocal porous media equations and fractional p-Laplacian operators appearing in a number of variants. 

Recent work concerns the time-dependent fractional p-Laplacian equation with parameter p>1 and fractional exponent 0<s<1. It is the gradient flow corresponding to the Gagliardo–Slobodeckii fractional energy. Our main interest is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on a delicate analysis. The superlinear and sublinear ranges involve different analysis and results. 
 

I will present a new existence result for isoperimetric sets of  large volume on manifolds with nonnegative Ricci curvature and  Euclidean volume growth, under an additional assumption on the structure of tangent cones at infinity. After a brief discussion on the sharpness of the additional  assumption, I will show that it is always verified on manifolds with nonnegative sectional curvature. I will finally present the main ingredients of proof emphasizing the key role of nonsmooth techniques tailored for the study of RCD  spaces, a class of metric measure structures satisfying a synthetic notion of Ricci curvature bounded below. This is based on a joint work with G. Antonelli, M. Fogagnolo and M. Pozzetta.

The Alfven waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfven waves are governed by the MHD system. In the talk,  we construct and study the long time behavior of (viscous and non-viscous) Alfven waves.

As applications, (1) We provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects;

(2) We prove 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.

There is a striking relationship between Willmore surfaces of revolution and elastic curves in hyperbolic half-space. Here the term elastic curve refer to a critical point of the energy given by the integral of the curvature squared. In the talk we will discuss this relationship and use it to study long-time existence and asymptotic behavior for the L2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. As in the case of Willmore flow of spheres, we show that if an initial datum has Willmore energy below 8 \pi then the solution of the Willmore flow converges to the Clifford Torus, possibly rescaled and translated. The energy threshold of 8 \pi turns out to be optimal for such a convergence result. 

The lecture is based on joint work with M. Müller (Univ. Freiburg), R. Schätzle (Univ. Tübingen) and A. Spener (Univ. Ulm).

 Given a solution of the Einstein equations, a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution. The latter question, as will be explained in the talk, is intimately tied to the strong cosmic censorship conjecture in general relativity which states in the language of partial differential equations that global uniqueness holds generically for the initial value problem for the Einstein equations. I will then focus in the talk on recent results showing the locally Lipschitz inextendibility of FLRW models with particle horizons and spherically symmetric weak null singularities. The latter in particular apply to the spherically symmetric spacetimes constructed by Luk and Oh, improving their C^2-formulation of strong cosmic censorship to a locally Lipschitz formulation.

We will discuss recent developments of the theory of a-contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows, like the compressible Euler equation.

In the one dimensional configuration, the Bressan theory shows that small BV solutions are stable under small BV perturbations (together with a technical condition known as bounded variations on space-like curve).

The theory of a-contraction allows to extend the Bressan theory to a weak/BV stability result allowing wild perturbations fulfilling only the so-called strong trace property. Especially, it shows that the technical condition of BV on space-like curve is not needed. (joint work with Sam Krupa and Geng Chen). 

We will show several applications of the theory of a-contraction with shifts on the barotropic Navier-Stokes equation. Together with Moon-Jin Kang and Yi wang, we proved the conjecture of Matsumura (first mentioned in 1986). It consists in proving the time asymptotic stability of composite waves of viscous shocks and rarefactions. 

Together with Moon-Jin Kang, we proved also that inviscid shocks of the Euler equation, are stable among the family of inviscid limits of Navier-Stokes equation (Inventiones 2021). This stability result holds in the class of wild perturbations of inviscid limits, without any regularity restriction (not even strong trace property). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem.

This is obtained thanks to a stability result at the level of Navier-Stokes, which is uniform with respect to the viscosity, allowing asymptotically infinitely large perturbations (JEMS 2021).

A first multi D result of stability of contact discontinuities without shear, in the class of inviscid limit of Fourier-Navier-Stokes, shows that the same property is true for some situations even in multi D (joint work with Moon-jin Kang and Yi Wang). 

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. In previous works, we have established stable generic shock formation for the compressible Euler system, showing that at the first singularity the solution has precisely C^{1/3} Holder regularity, a so-called preshock. The focus of this talk is a complete space-time description of the solution after this initial singularity. We show that three surfaces of discontinuity emerge simultaneously and instantaneously from the preshock: the classical shock discontinuity that propagates by the Rankine–Hugoniot conditions, together with two distinct surfaces in space-time, along which C^{3/2} cusp singularities form.

We study the minimization of functionals of the form $$ u  \mapsto \int_\Omega  f(\nabla u) \, dx $$

with a convex integrand $f$ of linear growth (such as the area integrand), among all functions in the Sobolev space W$^{1,1}$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and might in fact fail, as it is well-known already for the non-parametric minimal surface problem. In such cases, the functional is extended suitably to the space BV of functions of bounded variation via relaxation, and for the relaxed functional one can in turn guarantee the existence of minimizers. However, in contrast to the original minimization problem, these BV minimizers might in principle have interior jump discontinuities or not attain the prescribed boundary values.

After a short introduction to the problem I want to focus on the question of regularity of BV minimizers. In past years, Sobolev regularity was established provided that the lack of ellipticity -- which is always inherent for such linear growth integrands -- is mild, while, in general, only some structure results seems to be within reach. In this regard, I will review several results which were obtained in cooperation with Miroslav Bulíček (Prague), Franz Gmeineder (Bonn), Erika Maringová (Vienna), and Thomas Schmidt (Hamburg).

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. This flow is also related to the classical continuous Heisenberg model in ferromagnetism and to the 1-D cubic Schrödinger equation. In this lecture I will first talk about the state of the art of the binormal flow conjecture, as well as about mathematical methods and results for the binormal flow. Then I will introduce a class of solutions at the critical level of regularity that generate singularities in finite time and describe some of their properties. These results are joint work with Luis Vega.

One of the main questions in the theory of the linear transport equation is whether uniqueness of solutions to the Cauchy problem holds in the case the given vector field is not smooth. We will show that even for incompressible, Sobolev (thus quite “well-behaved”) vector fields, uniqueness of solutions can drastically fail. This result can be seen as a counterpart to DiPerna and Lions’ well-posedness theorem, and, more generally, it can be interpreted as an instance of the “flexibility vs. rigidity” duality, which is a common feature of PDEs appearing in completely different fields, such as differential geometry and fluid dynamics (joint with G. Sattig and L. Székelyhidi). 

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

 In this talk I will discuss possible extensions of the euclidean quantitative isoperimetric inequality to compact Riemannian manifolds. 
This is joint work with O. Chodosh (Stanford) and M. Engelstein (University of Minnesota).

We propose a modulated free energy which combines of the method previously developed by the speaker together with 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 more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with 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 Patlak-Keller-Segel system in the subcritical regimes, is obtained.

Hardt-Simon proved that every area-minimizing hypercone $C$ having only an isolated singularity fits into a foliation of $R^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $C$. We prove that if a minimal hypersurface $M$ in the unit ball $B_1 \subset R^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), then $M \cap B_{1/2}$ is a $C^{1,\alpha}$ perturbation of either the cone itself, or some leaf of its associated foliation. In particular, we show that singularities modeled on these cones determine the local structure not only of $M$, but of any nearby minimal surface. Our result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces in $R^{n+1}$ asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation.  This is joint work with Luca Spolaor.

Unlike the classical Cauchy problem in general relativity, which has been well-understood since the pioneering work of Y. Choquet-Bruhat (1952), the initial boundary value problem for the Einstein equations still lacks a comprehensive treatment. In particular, there is no geometric description of the boundary data yet known, which makes the problem well-posed for general timelike boundaries. Various gauge-dependent results have been established. Timelike boundaries naturally arise in the study of massive bodies, numerics, AdS spacetimes. I will give an overview of the problem and then present recent joint work with Jacques Smulevici that treates the special case of a totally geodesic boundary.

In the talk, we will discuss the connection between quantitative hypoelliptic PDE methods and the long-time dynamics of stochastic differential equations (SDEs). In a recent joint work with Alex Blumenthal and Sam Punshon-Smith, we put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a degenerate Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a  quantitative version of Hörmander's hypoelliptic regularity theory in an L1 framework which estimates this (degenerate) Fisher information from below by a W^{s,1} Sobolev norm using the associated Kolmogorov equation for the stationary density. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE and we prove that this class includes the classical Lorenz 96 model in any dimension greater than 6, provided the additive stochastic driving is applied to any consecutive pair of modes. This is the first mathematically rigorous proof of chaos (in the sense of positive Lyapunov exponents) for Lorenz 96 and, more recently, for finite dimensional truncations of the shell models GOY and SABRA (stochastically driven or otherwise), despite the overwhelming numerical evidence. If time permits, I will also discuss joint work with Kyle Liss, in which we obtain sharp, quantitative estimates on the spectral gap of the Markov semigroups. In both of these works, obtaining various kinds of quantitative hypoelliptic regularity estimates that are uniform in certain parameters plays a pivotal role.