Special Session 77: Singularity and regularity in nonlinear PDEs

Helical vortex structures with compactly supported cross-sectional vorticity for the 3D incompressible Euler equations

Averkios Averkiou University of Bath England Co-Author(s):    Monica Musso

Abstract: In this talk, we first revisit the vortex filament conjecture for three-dimensional incompressible Euler flows with helical symmetry and no swirl. By adapting gluing methods, we obtain the first construction of a smooth helical vortex filament in the whole space $\mathbb{R}^3$ whose cross-sectional vorticity remains compactly supported in $\mathbb{R}^2$ for all times. Building on this construction, we then investigate the leapfrogging phenomenon of helical vortex filaments, in which several interacting vortex helices with a common symmetry axis alternately overtake one another while preserving their coherent structure. This is joint work with Monica Musso.

Weighted Hardy-Rellich inequalities via the Emden-Fowler transform

Elvise Berchio Politecnico di Torino Italy Co-Author(s):    Paolo Caldiroli

Abstract: We exploit a technique based on the Emden-Fowler transform to prove optimal Hardy-Rellich inequalities on cones, including the punctured space and the half space as particular cases. We find optimal constants for classes of test functions vanishing on the boundary of the cone and possibly orthogonal to prescribed eigenspaces of the Laplace Beltrami operator restricted to the spherical projection of the cone. Furthermore, we show that extremals do not exist in the natural function spaces. Depending on the parameters, certain resonance phenomena can occur. For proper cones, this is excluded when considering test functions with compact support. Finally, for suitable subsets of the cones we provide improved Hardy-Rellich inequalities, under different boundary conditions, with optimal remainder terms.

Wasserstein Theory in a Goulash Medium

Iwona Chlebicka University of Warsaw Poland Co-Author(s):

Abstract: When a medium has a nontrivial heterogeneous structure of a goulash, the tools to model diffusion in it is in a short supply. Driven by the idea to describe its evolution as a gradient flow, we face deep structural challenges in a generalized Wasserstein theory.

Fujita`s critical exponent for Fractional Reaction-Diffusion Systems on $\mathbb{R}^{N}$

Soon-Yeong Chung Sogang University, Seoul Korea Co-Author(s):    Soon-Yeong Chung

Abstract: In this talk, we are going to introduce Fujita`s critical exponent to determine whether a fractional reaction-diffusion system (S) admits a global solutions or only blow-up solutions. In fact, the number $(pq)^{*}$ is introduced as the critical exponent to prove i If $(pq)^* < pq$, then (S) has a global solution for some initial data. ii If $(pq)^* \geq pq$, then every nontrivial and non-negative solution to (S) blows up in finite time.

Competing effects in fractional thin film equations

Antonio Esposito University of L'Aquila Italy Co-Author(s):

Abstract: The talk concerns the analysis of fractional thin film equations with linear mobility and an aggregation term. The problem is posed in a bounded convex domain with homogeneous Neumann boundary conditions, in dimension $d \geq 1$. We study the existence of weak solutions by interpreting the problem as a 2-Wasserstein gradient flow of an energy functional presenting two competing effects: the fractional Dirichlet energy and the power-law internal energy. The seminar is based on a joint work with J. A. Carrillo, A. Massucco, and S. Lisini.

On the decay rate of solutions to an anisotropic diffusion equation of porous media type

Filomena Feo University of Naples Parthenope Italy Co-Author(s):

Abstract: In this talk, we will present some recent results concerning on the decay rate of solutions to the following Cauchy-Dirichlet problem \begin{equation}\label{PCD}\tag{$P$} \left\{ \begin{aligned} &u_t=\sum_{i=1}^N(u^{m_i})_{x_ix_i}& \quad &\text{in }\Omega\times (0,T), \ &u(x,0)=u_0(x)& \quad & \text{on }\Omega, \ &u(x,t)=0 &\quad & \text{on }\partial \Omega\times (0,T), \end{aligned} \right. \end{equation} where $\Omega$ is a bounded open set of $\mathbb{R}^N$ with smooth boundary, $N\geq 2$,$00$, for all $i=1, \dots, N$, and $0\leq u_0\in L^{r_0}(\Omega)$ with $r_0\geq 1$. Our approach relies on showing that the solution $u$ satisfies some suitable integral inequalities, which allow us to derive quantitative estimates for the decay rate of solutions as $t \rightarrow+\infty$. We distinguish different regimes depending, in particular, on the relation between 1 and the minimum or the maximum of the exponents $m_i$. Moreover, by an appropriate choice of parameters in these inequalities, we prove that finite-time extinction occurs in certain cases. These results are based on recent joint work with A. G. Grimaldi and M. M. Porzio.

Blow-up and stability for quadratic derivative nonlinear wave equations

Oliver Gough University of Bath England Co-Author(s):    Manuel del Pino, Monica Musso

Abstract: This talk concerns finite-time blow-up in nonlinear wave equations with quadratic derivative nonlinearities. A central question is to understand the blow-up profile and mechanism in such equations, and whether the corresponding singular solutions continue to describe the dynamics under small perturbations. I will discuss recent work on these questions.

Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term

Tatsuki Kawakami Ryukoku University Japan Co-Author(s):    Kazuhiro Ishige, Ryo Takada

Abstract: In this talk, we study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. To this end, we establish decay estimates for the fractional heat semigroup in several uniformly local Zygmund spaces. Furthermore, we apply the real interpolation method in uniformly local Zygmund spaces to obtain sharp integral estimates for the inhomogeneous and nonlinear terms. This enables us to derive sharp sufficient conditions for the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. This talk is based on joint work with Prof. K. Ishige (University of Tokyo) and Prof. R. Takada (University of Tokyo).

Aggregation-diffusion models with nonlinear potentials

Edoardo Mainini University of Genoa Italy Co-Author(s):    Francesco Bozzola

Abstract: We investigate dynamics, stationary states and singular limits for aggregation diffusion models with nonlinear Riesz potentials. Radial stationary states of the dynamics are related with extremals of suitable Hardy-Littlewood-Sobolev inequalities. In the case that aggregation does not dominate over diffusion, radial stationary states also relate to global minimizers of the natural free energy functional. The evolution can be seen as the gradient flow of the free energy, and its analysis requires suitable gradient estimates for the nonlinear Riesz potential.

The Porous Medium Equation in Cones and Half-spaces

Troy Petitt Universidad Carlos III de Madrid Spain Co-Author(s):    Kazuhiro Ishige, Matteo Muratori, Troy Petitt

Abstract: We consider existence, uniqueness, and asymptotics of the porous medium equation in cones and half-spaces with homogeneous Dirichlet boundary conditions. We find that solutions depend quantitatively on the aperture of the cone via the first Dirichlet eigenvalue. Along the way, we prove the existence and uniqueness of dipole solutions in general cones, i.e. solutions that take as an initial datum a Dirac delta measure on the point of the cone.

Propagation and Extinction for a KPP-Type Heat Equation on Regular Metric Trees

Fabio Punzo Politecnico di Milano Italy Co-Author(s):

Abstract: We present some recent joint results with Alberto Tesei on the Cauchy-Neumann problem for a semilinear heat equation with a KPP-type forcing term on a regular metric tree. We investigate the propagation and extinction of solutions, as well as their asymptotic speed of propagation. Particular attention is devoted to the analogies and differences with related results in Euclidean and hyperbolic spaces.

The lifespan estimates of solutions to systems of weighted nonlinear wave equations in 1D

Hiroyuki Takamura Tohoku University Japan Co-Author(s):    Masakazu Kato, Takiko Sasaki

Abstract: In this talk, I will introduce you harp lifespan estimates of solutions to systems of weighted nonlinear wave equations by time-variable in one space dimension. The targeted systems include Nakao type problem among wave and scale-invariant damped wave equations.

Mass concentration comparison for nonlinear diffusion on model manifolds

Bruno Volzone Politecnico di Milano Italy Co-Author(s):    Matteo Muratori

Abstract: We investigate the validity of the mass concentration comparison for a class of nonlinear diffusion equations, commonly known as filtration equations, posed on Riemannian manifolds that are spherically symmetric, that is, model manifolds. Our main result states that, given any continuous bijection $\varphi : [0,+\infty) \rightarrow [0,+\infty)$, the filtration equation $\partial_{t}u=\Delta\varphi(u)$ satisfies the concentration comparison if and only if the underlying model manifold supports the P\`olya-Szeg\H{o} inequality. As a simple corollary, the validity of such a comparison for the heat equation is sufficient to guarantee that the same holds for all filtration equations. Moreover, we prove that if the manifold supports a centered isoperimetric inequality then the P\`olya-Szeg\H{o} inequality holds, allowing us to include important examples such as the hyperbolic space and the sphere. This is a joint work with Matteo Muratori.

Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain

Yuta Wakasugi Hiroshima University Japan Co-Author(s):    Masahiro Ikeda, Motohiro Sobajima, Koichi Taniguchi

Abstract: Consider the initial-boundary value problem for the two-dimensional semilinear damped wave equation with the critical nonlinearity $u_{tt} - \Delta u + u_t = u^2$ in the exterior of the unit ball in $\mathbb{R}^2$ with the Dirichlet boundary condition. We obtain a sharp double-exponential type lifespan estimate $T(\verepsilon) \geq \exp(\exp(C \varepsilon^{-1}))$ under the assumption of radial symmetry. To achieve this result, we introduce a new technique to control an $L^1$-type norm and a new Gagliardo-Nirenberg type estimate with logarithmic weight.

Blow-up rate for the subcritical semilinear heat equation in non-convex domains

Erbol Zhanpeisov Tohoku University, Graduate School of Science Japan Co-Author(s):    Hideyuki Miura, Jin Takahashi

Abstract: We study the blow-up rate for solutions of the subcritical semilinear heat equation. Type I blow-up means that the rate agrees with that of the associated ODE. In the Sobolev subcritical range, type I estimates have been proved for positive solutions in convex or general domains (Giga and Kohn, 1987; Quittner, 2021) and for sign-changing solutions in convex domains (Giga, Matsui and Sasayama, 2004). We extend these results to sign-changing solutions in possibly non-convex domains. The proof uses the Giga-Kohn energy together with a geometric inequality that controls the effect of non-convexity. As a corollary, we obtain blow-up of the scaling critical norm in the subcritical range. Based on joint work with Hideyuki Miura and Jin Takahashi (Institute of Science Tokyo).