Schedule for: 25w5330 - Singularities and Asymptotic Patterns in Fluids and Evolutionary PDEs

The solutions of Patlak-Keller-Segel equation are well known to form singularities in two dimensions. We prove that on a large class of two-dimensional regions, a coupling to Darcy's law by buoyancy completely regularizes the solutions, leading to global regularity. The main novel technical tool in the proof is a generalized Nash inequality that uses an anisotropic norm that is natural in the context of the incompressible porous media equation. The talk is based on work joint with Naji Sarsam. 

The two-dimensional Keller-Segel system admits finite time blowup solutions, which is the case if the initial density has a total mass greater than $8\pi$ and a finite second moment. Several constructive examples of such solutions have been obtained, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in a $8\pi$-mass concentration. It was conjectured that singular solutions concentrating simultaneously more than one solitons could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a $16\pi$-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Velazquez. We develop for the first time a robust framework to construct rigorously such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons, which we expect to find applications to other evolution problems. This is joint work with T.-E. Ghoul & N. Masmoudi (New York University in Abu Dhabi) & V. T. Nguyen (National Taiwan University).

In the main part of the talk we will describe some aspects of a study developed in a joint paper with L. Brasco concerning the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary and sign-changing initial datum. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant sign solution of a sublinear Lane-Emden equation, once suitably rescaled. We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one. The last part of the talk will be devoted to some new advances obtained in collaboration with G. Franzina, in the spirit of the ones explained above, for the study of the asymptotics of signed solutions for the Fractional Porous Medium Equation.

I will review the existence and uniqueness of global minimizers for interaction energy functionals. Euler-Lagrange equations in the infinity wasserstein distance will be discussed. Based on linear convexity/concavity arguments, qualitative properties of the global minimizers will also be treated. Anisotropic singular potentials appearing in dislocations will be shown to have rich qualitative properties with loss of dimension and ranges of explicit minimizers. This talk will be based on several works in collaboration with Ruiwen Shu (University of Oxford).

The aim of the talk will be to present some theory on singular limits in thermodynamics of viscous fluids and how to rigorously obtain incompressible models from compressible ones in certain regimes. In particular we consider a general compressible viscous, heat and magnetic conducting fluid described by compressible Navier–Stokes–Fourier system coupled with induction equation. In particular, we do not assume conservative boundary condition for temperature and allow heating or cooling on the surface of the domain We are interested in mathematical analysis when Mach, Froude, and Alvé numbers are small - converging to zero. We give a mathematical justification that i the limit, in case of low stratification, one obtains a modified Oberbeck–Boussinesq–MHD system with nonlocal term or non-local boundary condition for the temperature deviation. Choosing proper form of background magnetic field, gravitational potential and domain between parallel plates one also found that the flow is horizontal. The proof is based on the analysis of weak solutions to primitive system and relative entropy method. This is a recent joint work with Florian Oschmann and Piotr Gwiazda.

In this talk, I will review recent progress on the formation of singularities in incompressible fluid equations.

The purpose of this talk is to describe recent results on the asymptotic stability of nonlinear patterns or solitary waves in 1D Boussinesq type models. These are joint works with Christopher Maulén (Bielefeld)

In this talk we are interested in compactly supported solutions of the steady Euler equations. In 3D the existence of such solutions has been an open problem until the recent result of Gavrilov (2019). In 2D, instead, it is easy to construct solutions via radially symmetric stream functions. Low regularity solutions without radial symmetry have been found in the literature, but even the $C^1$ case was open. In this talk we construct such solutions with regularity $C^k$, for any fixed $k$ given. For the proof, we look for stream functions which are solutions to non-autonomous semilinear elliptic equations. In this framework we look for a local bifurcation around a conveniently constructed 1-parameter family of solutions. The linearized operator turns out to be critically singular, and is defined in anisotropic Banach spaces. This is joint work with A. Enciso (ICMAT, Madrid) and Antonio J. Fernández (UAM, Madrid).

In this talk I will review some results concerning the existence of stationary solutions to the 3D Euler equations that are topologically equivalent, in some sense, to a given smooth divergence-free field. The rule of thumb is that, for “most” fields, there does not exist a $C^1$ topologically equivalent steady Euler flow, but there aways exists a (weakly) topologically equivalent steady Euler flow of class $C^\alpha$. The talk is based on joint work with Javier Peñafiel-Tomás and Daniel Peralta-Salas.

The liquid drop model was originally introduce by Gamow in 1928 to model atomic nuclei. The model describes the competition between surface tension (which keeps the nuclei together) and Coulomb force (which corresponds to repulsion among the protons). Equilibrium shapes correspond to sets in the 3-dimensional Euclidean space which satisfies an equation that links the mean curvature of the boundary of the set to the Newtonian potential of the set. In this talk I will present the construction of toroidal surfaces, modelled on a family of Delaunay surfaces, with large volume which provide new equilibrium shapes for the liquid drop model. This work is in collaboration with M. del Pino and A. Zuniga.

The liquid drop model with a positive background density describes matter in the crust of neutron stars and is believed to feature transitions between patterns of different dimensionality, sometimes referred to as nuclear pasta phases. We present some results in the dilute limit, which are in accordance with this conjecture in dimensions 1, 2 and 3. The talk is based on joint works with Elliott Lieb, with Lukas Emmert and Tobias König and with Mathieu Lewin and Robert Seiringer.

We find solutions to the 2D incompressible Euler equations that, in a precise sense, are close to a superposition of traveling vortices as time tends to infinity. Our approach is constructive: we glue together classical traveling waves, specifically two vortex-antivortex pairs that travel at leading order with constant speed in opposite directions.

I will discuss some ongoing work involving using relative-entropy methods to investigate weak-strong uniqueness and the low Mach low Froude number limit for the gravitational compressible Navier-Stokes-Poisson system with degenerate, density-dependent viscosity in three space dimensions.

In this talk, we will analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani and Kim-Sakajo-Sohn. Such solutions are piecewise constant vorticity distributions, subject to the Gauss constraint and rotating uniformly around the vertical axis. The proof is based on the bifurcation from zonal solutions given by spherical caps. This is a collaboration with Z. Hassainia and E. Roulley.

The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. We shall present a construction of the Sadovskii vortex patch by solving the energy maximization problem under the exact impulse condition.

We consider the Cauchy and Cauchy-Dirichlet problems for a semilinear heat equation with a superlinear nonlinearity and we study the properties of threshold or subthreshold solutions lying on or below the boundary between blow-up and global existence, respectively. Among other things, we find sufficient conditions for the boundedness of all subthreshold solutions: This boundedness implies, in particular, that all global unbounded solutions - if they exist - are threshold solutions. Our results improve known results even in the special case of power nonlinearities. This is a joint work with Philippe Souplet.

The space and time dependence of free boundary extinction will be addressed by formal asymptotic methods in a number of examples, with an emphasis on non-radially symmetric cases.

Mean curvature flow is the gradient flow of the area functional where an embedded hypersurface evolves in direction of its mean curvature vector. This constitutes a natural geometric heat equation for hypersurfaces, which ideally will evolve the embedding into a nicer shape. But due to the nonlinear nature of the equation singularities are guaranteed to form. Nevertheless, a key observation in geometry and physics is that generic solutions, obtained by small perturbations, can exhibit simpler singularities. In this direction, a conjecture of Huisken posits that a generic mean curvature flow encounters only the simplest singularities. We will discuss work together with Chodosh, Choi and Mantoulidis which together with recent work of Bamler-Kleiner establishes this conjecture for embedded hypersurfaces in R^3.

We will discuss the asymptotic behavior of compact solutions to Mean Curvature flow around the peanut solution which is known to develop a degenerate neck pinch. In particular we will show that by choosing appropriate perturbations of the peanut solution the flow will develop either spherical or cylindrical singularities and describe the exact behavior of the flow. This is joint work with Sigurd Angenent and Natasa Sesum.

We study the behavior of the scaling critical Lebesgue norm for blow-up solutions to the nonlinear heat equation (the Fujita equation). For the energy supercritical nonlinearity, we give estimates of the blow-up rate for the critical norm, without any assumptions such as radial symmetry or sign conditions. In certain situations, the rates we obtain are optimal. This is based on joint work with Hideyuki Miura (Institute of Science Tokyo) and Jin Takahashi (Institute of Science Tokyo).

In this talk we discuss two new results about vortex filament evolution for incompressible Navier-Stokes and Euler equations. For Navier-Stokes, we prove global-in-time regularity for initial helical vortex filament. For Euler, we give existence of weak dissipative solutions with initial vorticity concentrated in a circle.

In this talk, we consider the dynamics of a two-dimensional incompressible viscous fluid evolving through a porous medium or a Hele-Shaw cell, driven by gravity and surface tension. The fluid will be confined within a vessel with vertical walls and below a dry region. Consequently, the dynamics of the contact points between the vessel, the fluid and the dry region are inherently coupled with the surface evolution. We present global-in-time a priori estimates for solutions initially close to equilibrium. Taking advantage of the Neumann problem solved by the velocity potential, the analysis is carried out in non-weighted L^2-based Sobolev spaces and without imposing restrictions on the contact angles.

We study regularity properties of solutions to nonlinear and nonlocal evolution problems driven by the so-called 0-order fractional $p$−Laplacian type operators: $$\partial_t u(x,t) = \mathcal{L}_p u(x,t) := \int_{\mathbb{R}^n} J(x-y) |u(y,t)-u(x,t)|^{p-2} (u(y,t)-u(x,t)) dy,$$ where $n\geq 1, p > 1, J:\mathbb{R}^n\to\mathbb{R}$ is a bounded nonnegative function with compact support, $J(0) > 0$ and normalized such that $\|J\|_{L^1(\mathbb{R}^n)}=1$, but not necessarily smooth. We deal with Cauchy problems on the whole space, and with Dirichlet and Neumann problems on bounded domains. Beside complementing the existing results about existence and uniqueness theory, we focus on sharp regularity results in the whole range $p\in(1,\infty)$. When $p>2$, we find an unexpected $L^q-L^\infty$ regularization: the surprise comes from the fact that this result is false in the linear case $p=2$. We show next that bounded solutions automatically gain higher time regularity, more precisely that $u(x,\cdot)\in C_t^p$. We finally show that solutions preserve the regularity of the initial datum up to certain order, that we conjecture to be optimal (at most $p$-derivatives in space). When $p>1$ is integer we can reach $C^\infty$ regularity (gained in time, preserved in space) and even analyticity in time. The regularity estimates that we obtain are quantita- tive and constructive (all computable constants), and have a local character, allowing us to show further properties of the solutions: for instance, initial singularities do not move with time. We also study the asymptotic behavior for large times of solutions to Dirichlet and Neumann problems. Our results are new also in the linear case and are sharp when $p$ is integer. We expect them to be optimal for all $p > 1$, supporting this claim with some numerical simulations.

V-states are uniformly rotating vortex patches of the 2D Euler equation. The only known explicit examples are circles and ellipses: the rest of positive existence results use local or global bifurcation arguments and don’t give any quantitative information of the solutions. In this talk I will prove the existence of solutions far from the perturbative regime, being able to extract nontrivial features of them and a precise quantitative description. The proof uses a combination of analysis and computer-assisted techniques.

In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model of a self-gravitating Newtonian star is given by the gravitational Euler-Poisson system. In the talk, I will review some recent progress on the local and global dynamics of Newtonian star solutions and discuss self-similar Newtonian gravitational collapse and stability of the Larson-Penston solution for isothermal stars.

The Porous Media Equation, $u_t=\Delta u^m$, is a much studied equation in the class of parabolic equations representing nonlinear diffusion processes. In this talk, we want to contribute to the study of anisotropy in the form of a variant of the model in which the exponents of the nonlinearity are different in each spatial coordinate direction, $u_t=\sum_i \partial^2_{ii}(u^{m_i})$. After settling the questions of existence and uniqueness, and functional properties, we aim at covering selfsimilarity and asymptotic behaviour. Thus, for selected ranges of exponents we construct finite-mass self-similar solutions (source type solutions). Asymptotic theorems allow us to demonstrate that such solutions are attractive for the wide class of finite mass solutions. The coexistence of slow directions and fast directions is an important issue. The ranges of exponents in which such facts are true are indicated. (joint work with F. Feo and B. Volzone)