Tag - Applied mathematics

Mohamed Majdoub: The Fujita exponent for a Hardy-Henon equation with a spatial-temporal forcing term

8th March, 2023

The purpose of this work is to analyse the well-posedness and the blow-up of solutions of the higher-order parabolic semilinear equation

ut+(-Δ)du=|x|α|u|p+ζ(t)w(x)   for   (x,t) ∈ ℝN×(0,∞),

where d ∈ (0,1) ∪ ℕ, p > 1, -α ∈ (0, min(2d,N)) or α ≥ 0 and ζ as well as w are suitable given functions. The novelty here regards to previous works lies in considering a forcing term ζ(t)w(x) depending on both time and space variables, a fractional or high-order Laplace operator (-Δ)d, and a weighted (possibly singular) non-linearity |x|α|u|p. We prove small Lebesgue data global existence for p larger than a critical value depending on the behavior of ζ. Furthermore, the global existence fails to hold for p less than the critical value under the additional condition ∫ℝN w(x) dx > 0. As a consequence, we derive the explicit value of the Fujita exponent pF(σ) in the case ζ(t)=tσ, σ > -1.

Alexander Migdal: Vortex lines, anomalous dissipation, and intermittency

22nd February, 2023

We develop a new theory of circulation statistics in strong turbulence (ν → 0 in the Navier-Stokes equation), treated as a degenerate fixed point of a Hopf equation. We use spherical Clebsch variables to parametrize vorticity in the stationary singular Euler flow. This flow has a tangent velocity gap due to the phase gap in the angular Clebsch variable across a discontinuity surface bounded by a stationary loop C in space. We find a circular vortex with a singular core on this loop, regularized as a limit of the Burgers vortex. We compute anomalous contributions to the Euler Hamiltonian, helicity, and the energy flow, staying finite in the vanishing viscosity limit. The normalization constant in the spherical Clebsch variables is determined from the energy balance between incoming flow and anomalous dissipation. The randomness (spontaneous stochastization) comes from the Gaussian fluctuations of a background velocity due to random locations of remote vortex structures. Assuming weak fluctuations of the background velocity field, we compute the probability distribution of velocity circulation Γ, which decays exponentially with pre-exponential factor 1/√Γ in perfect match with numerical simulations of conventional forced Navier-Stokes equations on periodic lattice 8K3. We also compute effective multifractal indexes for the tails of velocity circulation probability density as a function of conditional probability below that tail. The anomalous dimensions are independent of this probability and decrease as inverse powers of the logarithm of the size of the loop.

Jonas Sauer: Time-Periodic Weighted Lp-Estimates

20th February, 2023

In between elliptic PDEs, which do not depend on time (think of the steady-state Stokes equations), and honest parabolic PDEs, which do depend on time and are started at a given initial value (think of the instationary Stokes equations), there are time-periodic parabolic PDEs: On the one hand, time-independent solutions to the elliptic PDE are also trivially time-periodic, which gives periodic problems an elliptic touch, on the other hand solutions to the initial value problem which are not constant in time might very well be periodic.

I want to advocate for time-periodic problems not being the little sister of either elliptic or parabolic problems, but being a connector between the two and a class of its own right. This is highlighted by a direct method for showing a priori Lp estimates for time-periodic, linear, partial differential equations. The method is generic and can be applied to a wide range of problems, for example the Stokes equations and boundary value problems of Agmon-Douglas-Nirenberg type. In the talk, I will present these ideas and show how they can be extended to the setting of weighted Lp estimates, which is advantageous for extrapolation techniques and rougher boundary data.

Diogo Gomes: Hessian Riemannian flows in mean-field games

15th February, 2023

Hessian Riemannian flows are a powerful tool for the construction of numerical schemes for monotone mean-field games that have their origin in constrained optimization problems. In this talk, we discuss the general construction of these flows for monotone mean-field games, their existence and regularity properties, and their asymptotic convergence.

Riccardo Montalto: Quasi-periodic solutions and inviscid limit for Euler and Navier Stokes equations via KAM methods

15th February, 2023

In this talk I will discuss some recent results on Euler and Navier-Stokes equations concerning the construction of quasi-periodic solutions and the problem of the invscid limit for the Navier-Stokes equation. I will discuss the following two results:

1) Construction of quasi-periodic solutions for the Euler equation with a time quasi-periodic external force, bifurcating from a constant, diophantine velocity field;

2) I shall discuss the inviscid limit problem from the perspective of KAM theory, namely I shall prove the existence of quasi-periodic solutions of the Navier Stokes equation converging to the one of the Euler equation constructed in 1).

The main difficulty is that this is a singular limit problem. We overcome this difficulty by implementing a normal form methods which allow to prove sharp estimates (global in time) with respect to the viscosity parameter.

Satyvir Singh: High-fidelity simulations on the development of shock-induced hydrodynamic instabilities

25th January, 2023

The hydrodynamic instability research in fluid mechanics establishes whether a flow is stable or unstable, and if so, how these instabilities produce turbulent mixing. The Richtmyer–Meshkov (RM) instability is a shock-driven hydrodynamic instability that occurs in a combination with the Kelvin-Helmholtz instability when an initially perturbed surface separating by distinct fluid properties is driven by an incident shock wave. The RM instability can be considered as the impulsive limit of Rayleigh-Taylor instability where primary perturbations expand across the surface and ultimately emerge into a turbulent fluid mixing as the uniform gravitational acceleration increases. The studies on development of shock-induced instability are essential for the investigation of difficult issues related to shock propagation through arbitrarily inhomogeneous materials because of its wide range of applications, such as inertial confinement fusion, supersonic combustion, and supernova explosions.

In this talk, high-fidelity simulations on the development of shock-induced hydrodynamic instabilities for light/heavy bubbles of various shapes are presented. The focus is placed on presenting more intuitive details of the flow-fields visualizations, wave patterns, bubble deformation, vorticity production, and enstrophy evolution. For these simulations, two-dimensional compressible Euler/Navier-Fourier equations are simulated with a high-order mixed-type modal discontinuous Galerkin method. Additionally, a thorough investigation is made into the impact of shock strength, Atwood number, aspect ratios, and bulk viscosity in diatomic and polyatomic gases on the flow morphologies of shock-induced hydrodynamic instabilities.

Gigliola Staffilani: The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together

25th January, 2023

Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. In the first lecture, which will be colloquium style, I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic wave interactions provided also the tools to prove a famous conjecture in number theory, or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples. In the second and third lecture I will open a small window into the concept of weak wave turbulence. I will start with the deterministic approach of Bourgain, involving the study of long time asymptotic of higher Sobolev norms of solutions of dispersive equations, and I will end with the rigorous derivation of a 3-wave kinetic equation.