Talks will take place in Hodges Hall (Second Floor). View the Full Schedule for presentation times and locations.

In this talk, we consider the well-posedness of the surface quasi-geostrophic (SQG) front equation in low regularity Sobolev spaces. By observing a null structure, we obtain access to a normal form transformation for the equation. Applying this normal form in the context of a paradifferential analysis with modified energies, we are able to prove balanced cubic energy estimates and thus local well-posedness at just half a derivative above the scaling-critical regularity threshold. This is joint work with Ovidiu-Neculai Avadanei.

In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the whole blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of poroelastic equations is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the rest of the circulatory system. We discuss wellposedness analysis for this multiscale model, as well as solution methods focused on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions.

In this talk I shall review some old and new results about uniqueness of solutions to hyperbolic conservation laws. In particular: for any n x n strictly hyperbolic system, any weak solution which takes values inside the domain of the semigroup of vanishing viscosity limits, and whose shocks satisfy the Liu admissibility conditions, actually coincides with a semigroup trajectory. Implications of his result toward a posteriori error estimates will be discussed.

We consider a multiscale interface coupling between a 3D partial differential equations system modeling fluid flow in deformable porous media and a system of ordinary differential equations modeling a lumped hydraulic circuit that accounts for the global features of the problem. The main application of interest is tissue perfusion. We provide recent results related to existence, uniqueness and well-posedness owing to a fixed point solution in the above mentioned system.

In this talk, I will discuss the new progress on the supersonic expanding wave for compressible Euler equations in multiple space dimension with radial symmetry. We define the rarefaction and compression wave for radially symmetric solution, and show that on one hand, supersonic expanding waves with rarefactive initial data won't form a shock, and on the other hand, a strong initial compression wave will guarantee a gradient blowup. We will also discuss the ongoing project on the construction of example including an expanding shock wave.

In this talk, we investigate a system of equations modeling a non-isothermal magnetoviscoelastic fluid. We demonstrate that this model is thermodynamically consistent and that the critical points of the entropy functional with constant energy constraint correspond to the system’s equilibria. Our analysis utilizes the maximal $L_p$- regularity theory of quasilinear parabolic systems, proving the model to be locally well-posed. Additionally, we will discuss global existence and the asymptotic behavior of the entire dynamics. This is joint work with Yuanzhen Shao and Gieri Simonett.

In this talk, we discuss two-dimensional Riemann problems in the framework of potential flow equation and isentropic Euler system. We first review recent results on existence, regularity and properties of global self-similar solutions involving transonic shocks for several 2D Riemann problems in the framework of potential flow equation. Examples include regular shock reflection, Prandtl reflection, and four-shocks Riemann problem. The approach is to reduce the problem to a free boundary problem for a nonlinear elliptic equation in self-similar coordinates. A well-known open problem is to extend these results to compressible Euler system, i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions have low regularity, specifically velocity and density do not belong to the Sobolev space $H^1$ in self-similar coordinates. We further discuss well-posedness of the transport equation for vorticity in the resulting low regularity setting.

We will present a local well-posed result for piecewise regular solutions with a single shock of scalar balance laws, with singular integral of convolution type kernels. In a neighborhood of the shock curve, a description of the solution is provided for a general class of initial data.

Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in $\mathbb{R}_{+} \times \Omega$ with an equation on the boundary satisfied by the trace, where $\Omega$ is either $2 D$ torus or a bounded convex domain in $\mathbb{R}^2$. In this talk, we will show the existence and some regularity results of global in time weak solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The main difficulty is handling the degeneration of the background density profile at the boundary.

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, while a relativistic star is modeled by the Einstein-Euler system. In the talk, I will review some recent progress on the local and global dynamics of Newtonian star solutions, and discuss mathematical construction of self-similar gravitational collapse of Newtonian stars including Larson-Penston solution for the isothermal stars, Yahil solution for polytropic stars, which show the existence of smooth initial data that lead to finite time collapse, characterized by the blow-up of the star density. If time permits, I will also discuss the relativistic analogue of Larson-Penston solutions and formation of naked singularities for the Einstein-Euler system.

The interface problem for the two immiscible incompressible viscous fluids in magnetohydrodynamics
will be considered. Recent results on the solutions to this problem with surface
tension in a three-dimensional bounded domain will be presented and discussed.

The Guderley problem is a well-known hydrodynamic problem of a strong shock propagating radially in an ideal gas medium. The shock originates at infinity and collapses to a central point or an axis. In this talk, I will briefly discuss the recent work on the rigorous proof of the construction of the Guderley solution for $\gamma\in (1,3]$. These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.

A rigorous derivation of point vortex systems from kinetic equations has been a challenging open problem, due to singular layers in the inviscid limit, giving a large velocity gradient in the Boltzmann equations. In this paper, we derive the Helmholtz-Kirchhoff point-vortex system from the hydrodynamic limits of the Boltzmann equations. We construct Boltzmann solutions by the Hilbert-type expansion associated to the point vortices solutions of the 2D Navier-Stokes equations. We give a precise pointwise estimate for the solution of the Boltzmann equations with small Strouhal number and Knudsen number.

Abstract: We prove error estimates for a certain class numerical schemes

approximating nonlocal conservation laws, modeling traffic flow and crowd dynamics.
Numerical simulations are presented to illustrate this result.

The compressible Euler equation can lead to the emergence of shocks-discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question. In this presentation, we will present the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent a-contraction with shifts method. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation.

In this talk, we’ll present some recent work on traveling waves in water that carry vortices in their bulk. We show that for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family of solitary waves with a submerged point vortex in equilibrium. This family bifurcates from an irrotational laminar flow, and, at least for large Froude numbers, it extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. Through a separate numerical study, we find strong evidence that many of the waves do indeed have an overturned air-water interfaces. Finally, we prove that generically one can perform a desingularization procedure to obtain a solitary wave with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air in their bulk. This is joint work with Ming Chen, Kristoffer Varholm, and Miles Wheeler.

We shall consider the hyperbolic and mixed-type problems arising in gas dynamics and geometry. In particular, the transonic flows past obstacles and in nozzles, and the isometric embedding in geometry will be discussed.

When we reduce the Ericksen-Leslie system to the semi-linear symmetric form with the initial data at r=0 with function value =0 and r=infinity with function value =pi. Under the energy condition near the static solution case, since the part of the 2nd equation is a O(3) -sigma model, the blow-up happens at the finite time.

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space-time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.

North Carolina State University

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The Pennsylvania State University

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University of Kansas

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University of Wisconsin-Madison

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University of Southern California

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The University of Texas at Austin

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University of Missouri

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University of Pittsburgh

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