Joint Meeting AMS-UMI Palermo 2024

Special Session A22 – Recent Developments on Certain Evolution Partial Differential Equations

July 23-24, 2024

Organizers:

• Sergio Polidoro (Università degli Studi di Modena e Reggio Emilia, Italy)
• Gigliola Staffilani (Massachusetts Institute of Technology, USA)

Description:

The goal of this section is to bring together researchers whose interest is in the study of a variety of evolution equations that represent fundamental physical phenomena. The study of these equations involves harmonic and Fourier analysis, geometry and stochastic calculus. We expect our speakers to give colloquium style talks involving the analysis of the Kolmogorov’s operator, of many body systems of particles and wave packets, of invariant measures, and of singularity formations. Many of the problems discussed could be collected under the wide umbrella of kinetic theory and wave turbulence theory, two subjects that in recent years have seen an incredible number of fundamental advances.

Confirmed Speakers:

• Francesca Anceschi (Università Politecnica delle Marche, Italy)
• Yuri Cacchiò (GSSI, L’Aquila, Italy)
• Nicola Garofalo (Università degli Studi di Padova, Italy)
• Javier Gomez-Serrano (Brown University, Providence, USA)
• Ricardo Grande (SISSA, Trieste, Italy)
• Felipe Hernandez (Massachusetts Institute of Technology, USA)
• Alessia Kogoj (Università “Carlo Bo” di Urbino, Italy)
• Amélie Loher (Cambridge University, UK)
• Nataša Pavlović (University of Texas Austin, USA)
• Mirco Piccinini (Università di Pisa, Italy)
• Annalaura Rebucci (Max Planck Institute, Leipzig, Germany)
• Jia Shi (Massachusetts Institute of Technology, USA)
• Richard Sowers (Illinois University, Urbana-Champaign, USA)
• Nicola Visciglia (Università di Pisa, Italy)

Titles and abstracts of the talks:

	Francesca Anceschi (Università Politecnica delle Marche, Italy)
Title:	Harnack inequalities for kinetic integral equations
Abstract:	In this talk, based on a joint project with M. Piccinini and G. Palatucci, I will present some recent results for weak solutions to the kinetic integral equation $$ \big[ \partial_t + v\cdot\nabla_x\big] f = \mathcal{L}_v f, $$ where the diffusion term $\mathcal{L}_v$ is an integro-differential operator, whose nonnegative kernel is of fractional order $s\in(0,1)$ and has measurable coefficients. In particular, I will focus on the proof of a strong Harnack inequality for nonnegative weak solutions $f$ of the type $$ \sup_{Q^-} f \ \leq \ c\inf_{Q^+} f, $$ where $Q^{\pm}$ are suitable slanted cylinders defined in accordance with the non-Euclidean geometry underlying the operator and no a priori boundedness for the solution is assumed, as it is usually done in literature.
	Yuri Cacchiò (GSSI, L’Aquila, Italy)
Title:	On the effect of the Coriolis force on the enstrophy cascade
Abstract:	Geophysical fluid dynamics refers to the fluid dynamics of naturally occurring flows, such as oceans and planetary atmospheres on Earth and other planets. These flows are primarily characterized by two elements: stratification and rotation. In this article we investigate the effects of rotation on the dynamics, by neglecting stratification, in a 2D model where we incorporate the effects of the planetary rotation by adopting the $\beta$-plane approximation, which is a simple device used to represent the latitudinal variation in the vertical component of the Coriolis force. We consider the well-known 2D $\beta$-plane Navier-Stokes equations in the statistically forced case. Our problem addresses energy-related phenomena associated with the solution of the equations. To maintain the fluid in a turbulent state, we introduce energy into the system through a stochastic force. In the 2D case, a scaling analysis argument indicates a direct cascade of enstrophy and an inverse cascade of energy. We compare the behaviour of the direct enstrophy cascade with the 2D model lacking the Coriolis force, observing that at small scales, the enstrophy flux from larger to smaller scales remains unaffected by the planetary rotation, confirming experimental and numerical observations. In fact, this is the first mathematically rigorous study of the above equations. In particular, we provide sufficient conditions inspired by [1, 2] to prove that at small scales, in the presence of the Coriolis force, the so-called third-order structure function’s asymptotics follows the third-order universal law of 2D turbulence without the Coriolis force. We also prove well-posedness and certain regularity properties necessary to obtain the mentioned results. This is a Joint work in collaboration with Amirali Hannani (Leuven) and Gigliola Staffilani (MIT). References [1] Bedrossian, J. and Coti Zelati, M. and Punshon-Smith, S. and Weber, F., Sufficient conditions for dual cascade flux laws in the stochastic 2d Navier-Stokes equations. Arch Rational Mech Anal, 237 : 103-145, (2020). [1] Xie, J. and Bühler, O., Exact third-order structure functions for two-dimensional turbulence. Journal of Fluid Mechanics, 851 : 672-686, (2018).
	Felipe Hernandez (Massachusetts Institute of Technology, USA)
Title:	Quantum Diffusion and Approximate Semigroups
Abstract:	The random Schr¨odinger equation is a toy model for waves in random media and for the motion of an electron in an imperfect metal. The evolution of smooth observables for this equation are described by a linear Boltzmann equation, which gives a classical picture for the motion of electrons in terms of a Markov process with discontinuous jumps in the momentum. In this talk I will introduce both of these equations and discuss some of the ideas needed to rigorously the classical description from the quantum mechanical system.
	Amélie Loher (Cambridge University, UK)
Title:	Local behaviour of non-local hypoelliptic equations
Abstract:	We discuss local regularity properties of solutions to a class of non-local kinetic equations in divergence form. In particular, we explain why it is not clear if the Strong Harnack inequality holds for solutions of non-local equations, and how we manage to derive it. The key is to obtain a non-local-to-local bound on the tail terms that naturally affect the function values inside any local domain.
	Nataša Pavlović (University of Texas Austin, USA)
Title:	On the effective dynamics of Bose-Fermi mixtures
Abstract:	Investigating degenerate mixtures of bosons and fermions is an extremely active area of research in experimental physics for constructing and understanding novel quantum bound states such as those in superconductors, superfluids, and supersolids. These ultra-cold Bose-Fermi mixtures are fundamentally different from degenerate gases with only bosons or fermions. They not only show an enriched phase diagram, but also a fundamental instability due to energetic considerations coming from the Pauli exclusion principle. Inspired by this activity in the physics community, recently we started exploring the mathematical theory of Bose-Fermi mixtures. One of the main challenges is understanding the physical scales of the system that allow for suitable analysis. We will describe how we overcame this challenge by identifying a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical. In this regime, the bosons are much lighter and more numerous than the fermions. This is a Joint work in collaboration with Esteban Cárdenas (University of Texas Austin, USA) and Joseph Miller (University of Texas Austin, USA). References [1] E. Cárdenas, J.K. Miller, N. Pavlović, On the effective dynamics of Bose-Fermi mixtures, arXiv:2309.04638, Submitted for publication.
	Mirco Piccinini (Università di Pisa, Italy)
Title:	Interpolative boundedness estimates for kinetic integral equations
Abstract:	We investigate local properties of weak solutions to a wide class of kinetic equations where the diffusion term in velocity is an integro-differential operator, having nonnegative kernel of fractional order $s \in (0, 1)$ with merely measurable coefficients. We provide an explicit local boundedness estimate by combining together a suitable gain of integrability and a kinetic Caccioppoli-type inequality, by giving also a precise control of the long-range interactions arising from the nonlocal behaviour of the involved diffusion operator. Moreover, we will show how to deduce a similar result in the case when a p-growth conditions is assumed on the diffusion operator. This is based on a joint project in collaboration with F. Anceschi and G. Palatucci.
	Annalaura Rebucci (Max Planck Institute, Leipzig, Germany)
Title:	On the De Giorgi-Nash-Moser regularity theory for hypoelliptic operators
Abstract:	We present new results that extend the De Giorgi-Nash-Moser theory to hypoelliptic equations of Kolmogorov (kinetic) type with rough coefficients. A key ingredient to prove these results is a Poincaré inequality, which we derive from the construction of suitable trajectories (see [1]). The trajectories we rely on are quite flexible and allow us to consider equations with an arbitrary number of commutators and whose diffusive part is either local (second-order) or nonlocal (fractional order). Following [2], we later combine the Poincaré inequality with a $L^2-L^\infty$ estimate, a Log-transformation and a classical covering argument (Ink-Spots Theorem) to deduce Harnack inequalities and Hölder regularity along the line of De Giorgi method. References [1] F. Anceschi, H. Dietert, J. Guerand, A. Loher, C. Mouhot, A. Rebucci, Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators, preprint arXiv (2024). [2] F. Anceschi, A. Rebucci, A note on the weak regularity theory for degenerate Kolmogorov equations, J.Diff. Eq. 341, 538–88 (2022).
	Jia Shi (Massachusetts Institute of Technology, USA)
Title:	Non-radial implosion for compressible Euler and Navier-Stokes in $\mathbb{T}^3$ and $\mathbb{R}^3$
Abstract:	We will discuss the smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. The construction is motivated by the radial imploding solutions from Merle-Raphaël-Rodnianski-Szeftel (see [1],[2],[3]), and CaoLabora-Buckmaster-Gómez-Serrano (see [4]) but is flexible enough to handle both periodic and non-radial initial data. This is a joint work with Gonzalo Cao-Labora, Javier Gómez-Serrano, and Gigliola Staffilani [5]. References [1] F. Merle, P. Raphaël, I. Rodnianski, and J. Szeftel, On blow up for the energy super critical defocusing nonlinear Schrödinger equations, Invent. Math. 227.1 (2022), pp. 247-413 [2] F. Merle, P. Raphaël, I. Rodnianski, and J. Szeftel, On the implosion of a compressible fluid I: smooth self-similar inviscid profiles, Ann. of Math. (2) 196.2 (2022), pp. 567-778. [3] F. Merle, P. Raphaël, I. Rodnianski, and J. Szeftel, On the implosion of a compressible fluid II: singularity formation, Ann. of Math. (2) 196.2 (2022), pp. 779-889 [4] T. Buckmaster, G. Cao-Labora, and J. Gómez-Serrano, Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445 (2022) [5] G. Cao-Labora, J. Gómez-Serrano , J. Shi and G. Staffilani, Non-radial implosion for compressible Euler and Navier-Stokes in $\mathbb{T}^3$ and $\mathbb{R}^3$, Arxiv preprint arxiv:2310.05325 (2023)
	Richard Sowers (Illinois University, Urbana-Champaign, USA)
Title:	Lateral boundary conditions for a Kolmogorov-type PDE
Abstract:	We consider a hypoelliptic Kolmogorov-type PDE motivated by a particle under the influence of a white noise force. We are interested in imposing Dirichlet conditions at a side (lateral) boundary. Specifically, we consider $$\begin{aligned} \frac{\partial u}{\partial t}(t, x, y) &= \frac1 2 \frac{\partial^2u}{\partial y^2} (t, x, y) + b(y)\frac{\partial u}{\partial x} (t, x, y) \quad t > 0, 0 < x < \infty, -\infty < y < \infty \\ u(0, x, y) &= u_\circ(x, y) \quad 0 < x < \infty, -\infty < y < \infty \\ u(t, 0, y) &= u_\partial (t, y) \quad t > 0, -\infty < y < \infty \end{aligned}$$ where $u_\circ$ is an initial condition and $u_\partial$ is a lateral boundary condition. We assume that this PDE is hypoelliptic and that the lateral boundary is accessible. Namely, we consider functions $b$ similar to $$b(y) = -2 + \tanh(y);$$ these are bounded, negative, and strictly increasing in $y$ at all points. We construct a simple Gaussian heat kernel $K$ inside the domain, and investigate a boundary-layer kernel $K^\partial$. We show that this boundary layer heat kernel has a novel jump condition (stemming from a careful Laplace-type asymptotic analysis) which captures the structure of the above PDE $u$ near the lateral boundary. We outline a polynomial refinement of the heat kernels, and then construct a Volterra equation stemming from the variation of parameters ansatz $$ \begin{aligned} u(t, x, y) = \int_{s=0}^t & \int_{x_\circ >0, – \infty<y_\circ<\infty} K_{x_\circ, y_\circ}(t – s, x, y)\psi(s, x_\circ, y_\circ)\, dx_\circ dy_\circ ds \\ + &\int_{s=0}^t \int_{-\infty<y_\circ<\infty} K^\partial_{y_\circ}(t – s, x, y) \psi^\partial (s, y_\circ) \,ds dy_\circ \\ + &\int_{x_\circ>0,- \infty<y_\circ<\infty} K_{x_\circ,y_\circ}(t, x, y) u_\circ(x_\circ, y_\circ) \,dx_\circ dy_\circ \end{aligned} $$ where $\psi$ and $\psi^\partial$ are to be found. The boundary condition corresponding to $\psi^\partial$ reflects the novel jump condition. We construct a Volterra equation and show (somewhat nonstandard) convergence of the Volterra equation for $\psi$ and $\psi^\partial$.
	Nicola Visciglia (Università di Pisa, Italy)
Title:	Global existence for the generalized derivative NLS
Abstract:	In this talk, based on a joint work with M. Hayashi (Kyoto U.) and T. Ozawa (Waseda U.), we shall discuss a blow-up criterion for $H^2$ solutions to the generalized DNLS $$ \begin{cases} iu_t + u_{xx} + i|u|^{2 \sigma}u_x = 0, \: \sigma > 1, (t, x) \in \textbf{R} \times \textbf{T} \\ u(0) = \varphi \in H^2(\textbf{T}) \end{cases} $$ For $\sigma > 1$ the equation is no more completely integrable and hence the globalization argument is not trivial. The existence and uniquness of a local solution has been constructed in [1]. Our main result is the following blow-up alternative which improves the one described in [1]. Theorem 1. Let $\varphi \in H^2(\textbf{T})$. For the solution $u \in C([0, T_{\textrm{max}}),H^2(\textbf{T}))$ we have the following alternative: (1) $T_{\textrm{max}} = \infty$, (2) $T_{\textrm{max}}< \infty$ implies $\limsup_{t \uparrow T_{\textrm{max}}} ||u(t)||_{H^1(\textbf{T})} =\infty$. As a consequence of the result above and the Hamiltonian structure of the equation, we get the following global existence result for small data, which provides a positive answer to a question raised in [1]. Corollary 2. There exists $\delta > 0$ such that if $\varphi \in H^2(\textbf{T})$ satisfies $||\varphi||_{H^1(\textbf{T})}< \delta$, then there exists a unique global solution $u \in C(\textbf{R},H^2(\textbf{T}))$. References [1] D. M. Ambrose and G. Simpson, Local existence theory for derivative nonlinear Schrödinger equations with noninteger power nonlinearities, SIAM J. Math. Anal. 47 (2015), 2241–2264.