Abstract: We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ’eddy’ must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice – known to Batchelor (1956) and Wood (1957) - is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.
Abstract: Proving homogenization is a subtle issue for geometric equations due to the discontinuity when the gradient vanishes. To conclude homogenization, the work of Caffarelli-Monneau provides a sufficient condition, namely that perturbed correctors exist. However, some noncoercive equations recently studied do not satisfy this condition. In this talk, we present the homogenization result of geometric equations without using perturbed correctors. For coercive equations, a quantitative result is derived by the fact that they remain coercive under perturbation. We present an example that homogenizes with a rate slower than O(\varepilon) in the last part.
Abstract: We will discuss a recent result concerned with the surface quasi-geostrophic equation (SQG) with low-regularity initial data. Namely, given $s\in (3/2,2)$ and $\varepsilon >0$, we will describe a construction of a compactly supported initial data $\theta_0$ such that $\| \theta_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $\theta$ of the SQG equation that loses Sobolev regularity continuously in time. To be precise, for each $t\in [0,T]$, $ \theta (\cdot ,t ) \in {H^{s/(1+ct)}}$ and $ \theta (\cdot ,t ) \not \in {H^\beta }$ for any $\beta > s/(1+ct)$. Moreover $\theta$ is compactly supported in space, continuous and differentiable on $\R^2\times [0,T]$, and is unique among all solutions with initial condition $\theta_0$ which belong to $C([0,T];H^{1+\alpha })$ for any $\alpha >0$.
Abstract: The study of gravity-capillary water waves has been an important question in mathematical fluid dynamics. In this talk, I will talk about the low regularity well-posedness of two-dimensional deep gravity-capillary water waves. By implementing the cubic modified energy method of Ifrim-Tataru in the context of gravity-capillary waves, the two-dimensional gravity-capillary water wave system is locally well-posed in $H^{s+\frac{1}{2}}\times H^s$ for s>1.
Abstract: The quantitative homogenization of Hamilton-Jacobi equations has garnered significant interest recently, particularly in the periodic and stochastic settings. In this talk, we present an algebraic convergence rate in a quasi-periodic setting. We establish a link between the convergence rate and the regularity of the effective Hamiltonian, using a new quantitative ergodic estimate for bounded quasi-periodic functions with Diophantine frequencies. As an application, we also analyze the convergence rate of the Birkhoff averages for unbounded quasi-periodic functions.
Abstract: In recent years, the behavior of solution fronts of reaction-diffusion equations in the presence of obstacles has attracted attention among many researchers. In this talk, I will consider the case where the obstacle is a wall with many holes and discuss whether the front can pass through the wall and continue to propagate (“propagation”) or is blocked by the wall (“blocking”). The answer depends largely on the size and the geometric configuration of the holes.
This problem has led to a variety of interesting mathematical questions that are far richer than we had originally anticipated. Many questions still remain open. This is joint work with Henri Berestycki and Francois Hamel.
Abstract: In this talk we will discuss recent results concerning stochastic (and deter- ministic) moving boundary problems, particularly arising in fluid-structure interaction (FSI), where the motion of the boundary is not known a priori.Fluid-structure interaction refers to physical systems whose behavior is dictated by the interaction of an elastic body and a fluid mass and it appears in various applications, ranging from aerodynamics to structural engi- neering. Our work is motivated by FSI models arising in biofluidic applications that describe the interactions between a viscous fluid,such as human blood, and an elastic structure, such as a human artery. To account for the unavoidable numerical and physical uncertainties in applications we analyze these PDEs under the influence of external stochastic (random) forces.We will consider nonlinearly coupled fluid-structure interaction (FSI) problems involving a viscous fluid in a 2D/3D domain,where part of the fluid domain boundary consists of an elastic deformable structure, and where the system is perturbed by stochastic effects.The fluid flow is described by the Navier-Stokes equations while the elastodynamics of the thin structure are modeled by shell equations. The fluid and thestructure are coupled via two sets of coupling conditions imposed at the fluid-structure interface. We will consider the case where the structure is allowed to have unrestricted deformations and explore different kinematic coupling conditions (no-slip and Navier slip)imposed at the randomly moving fluid-structure interface, the displacement of which is not known a priori. We will present our results on the existence of (martingale) weak solutions to the(stochastic) FSI models. This is the first body of work that analyzes solutions of stochastic PDEs posed on random and time-dependent domains and a first step in the field toward further research on control problems, singular perturbation problems etc. We will further discuss our findings, which reveal a novel hidden regularity in the structure’s displacement. This result has allowed us to address previously open problems in the 3D (deterministic)case involving large vec- torial deformations of the structure. We will discuss both the cases of compressible and incompressible fluid.
Abstract: The characterization of global solutions to the obstacle problem, or equivalently of null quadrature domains, is connected to the famous Shell Theorem of Netwon and has been studied for more than 90 years. In this talk I will discuss a recent result with Eberle and Weiss, where we give a conclusive answer to this question.
Abstract: A qualitative stability estimate for the classical Sobolev inequality is known since the work of Bianchi and Egnell in 1991. Obtaining constructive stability estimates, i.e., giving an estimate of the stability constant, has remained an open question for a very long time. Several results of stability in strong norms have been obtained in recent years not only for the Sobolev inequality but also for the logarithmic Sobolev inequality as well as some families of Gagliardo-Nirenberg interpolation inequalities. The aim of this lecture is to provide an overview of various methods, results and open questions.
Abstract: In this seminar I will present a work in collaboration with Ariela Briani and Hitoshi Ishii that extends the well known result on thin domains of Hale and Raugel. The test function approach of C. Evans is very powerful and gives new results even in the case of the Laplacian.
Abstract: When we imagine a gas at the microscopic scale, we envision a huge number of individual particles flying around and colliding with one another. The sheer number of particles makes this mathematically intractable. The Boltzmann equation provides a way to bypass this complexity through a kinetic partial differential equation. Due to its nonlocal and nonlinear structure, however, it is notoriously difficult to analyze.
In this talk, I will give an overview of a new approach to analyzing the Boltzmann equation by drawing on and generalizing ideas from parabolic equations (which govern systems like heat flow). This approach leads to a novel blow-up criterion; that is, a condition that guarantees the continued existence of a solution. We then use this to significantly expand the class of initial data for which solutions exist and for which we understand the long-time trend to equilibrium.
Abstract: Chemotactic singularity formation in the context of the Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong -- this effect is conjectured to hold for more general classes of nonlinear PDEs. In this talk, I will introduce Keller-Segel equation coupled with several incompressible flows forced by buoyancy, including a flow characterized by Darcy's law for incompressible porous media and a Navier-Stokes equation. We prove that, in contrast to previous results driven by passive mixing flows, such active fluid coupling is capable of suppressing singularity formation via an interesting dynamical, nonlinear mixing mechanism which is manifested in a delicate kinetic-potential energy balance. This talk is based on a recent work by the author and an earlier joint work with Alexander Kiselev and Yao Yao.
Abstract: This talk presents the global existence of entropy weak solutions for scalar balance laws with nonlocal singular sources, along with a partial uniqueness result. A detailed analysis of the solution structure is provided for a general class of initial data, particularly in neighborhoods where two shocks interact. Additionally, some open questions will be discussed.
Abstract: In the first part of this talk, we will introduce an approximation to the Euclidean isoperimetric problem using the Allen-Cahn energy with a novel volume constraint. This approximating model shares several rigidity properties with the isoperimetric model: uniqueness and stability of minimizers; and rigidity of critical points (in the sense of Alexandrov's theorem). In the second part of the talk, we will consider the gradient flow associated to this model, regarding it as an approximation to the Volume Preserving Mean Curvature Flow, and we will show that this flow converges (under very general assumptions) to a single (diffused) ball. This talk is based on joint work with Francesco Maggi and Matteo Bonforte.
Abstract: The numerical discretisation of elliptic equations in nondivergence form is notoriously challenging, due to the lack of a notion of weak solutions based on variational principles. In many cases, there still is a well-posed variational formulation for such equations, which has the particularity of being posed in 𝐻², and therefore leads to a strong solution. Galerkin discretizations based on this formulation have been studied in the literature. Since 𝐻² conforming finite elements tend to be considered impractical, most of these discretizations are of discontinuous Galerkin type. On the other hand, it has been observed that the virtual element method provides a practical way to build 𝐻² conforming discretizations of variational problems. In this talk, I will describe a virtual element discretization of equations in nondivergence form. I will mainly present results we obtained in the setting of a simple linear model problem. Especially, I will show how the 𝐻² conformity of the method allows for a particularly simple well-posedness and error analysis. I will then briefly discuss the extension to equations with lower-order terms and with Hamilton-Jacobi-Bellman type nonlinearities, and present some numerical results.
This is joint work with Andrea Cangiani, Andreas Dedner, and Ricardo Nochetto.
Abstract: This talk is concerned with the Cauchy problem for the Navier-Stokes equations in dimension three. At the critical regularity, it is known that small data leads to global well-posedness, the strongest result being due to Koch and Tataru for velocity data in BMO^{-1}. We will present recent work with M. Coiculescu in which we construct initial data in the same space leading to two distinct smooth solutions, showing the smallness assumption is necessary. Then we discuss evidence in the dyadic setting which suggests a program for constructing Leray-Hopf solutions.
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