# Waves on a nutshell

**Time: 11.11.2019, 9am-2pm**

**Room: 03.323 – Besprechungsraum** Chair in Applied Analysis (Alexander von Humboldt-Professorschip)

**09:00-09.30**

**Large time asymptotics for partially dissipative hyperbolic systems
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**Enrique Zuazua (FAU & AvH)**

In this talk we report on the joint work developed in collaboration with Karine Beauchard (2011) in which we analyse the large time dynamics for hyperbolic systems in several space variables in the presence of weak damping. By linearization and Fourier transform we exhibit the role that control theoretical tools, such as the classical Kalman rank condition, play in the long-time analysis of these models. We shall present some challenging open problems too.

**09:30-10:00**

**The limits of stabilizability for networks of strings
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**Martin Gugat (FAU)**

Bastin and Coron pointed out that boundary-feedback stabilization of certain 1-d hyperbolic systems is impossible if the space interval is too long. We show that similar phenomena also occur for star-shaped networks of strings that are governed by the wave equation with a certain source term. If one of the strings is sufficiently long or the number of strings is sufficiently large, boundary-feedback stabilization can be impossible. The influence of the source term is essential for the behavior of the system.

**10:00-10:30**

**“Traffic flow modeled by nonlocal balance laws”
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**Lukas Pflug (FAU)**

Nonlocal balance laws – a class of nonlinear partial integro-differential equations – play a major role in the modeling of real world phenomena. A prominent example therefore is the macroscopic modeling of vehicular traffic flow.

We will discuss theoretical results obtained in the recent years covering the whole range from existence, uniqueness and regularity results to a non-dissipative numerical solution scheme. Additionally, the convergence of the solution of nonlocal balance laws to the entropy solution of their corresponding (local) balance laws will be shown. Further, the effect of time-delay and a model for multi-lane traffic flow will be discussed.

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**10:30-10:45 Coffee Break**

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**10:45-11:15**

**Exact Boundary Controllability for Coupled Wave Equations with Dynamical Boundary Conditions.
**

**Yue Wang (FAU)**

For wave equation on one-dimensional bounded region, in order to characterize the dynamical behavior of tip-masses on the boundary, one should focus on the dynamical boundary conditions with higher-order derivatives. Different from general types, dynamical boundary conditions will cause unique mathematical difficulties when we consider the boundary controllability problem for this kind of system.

In this talk, the following treatment methods are proposed: the dynamical boundary conditions will be transformed into integral non-local boundary conditions equivalently, and then the semi-global classical solutions to the mixed initial boundary value problems will be established, finally the local exact boundary controllability will be obtained by a modular constructive method.

Under the framework of C^{2} solutions, we establishes the corresponding exact boundary controllability for a single wave equation, a coupled system of wave equations, vibration strings in space and in a planar tree-like network, respectively. The mathematical results are combined with a considerable number of application models to help us understanding the physical meaning in different situations.

This is a joint work with Prof. Leugering and Prof.Li.

11:15-11:45

**Fluid-structure coupling of linear elastic model with compressible flow models with multilevel time stepping
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**Aleksey Sikstel (**

**RWTH Aachen University)**

Cavitation erosion is caused in solids exposed to strong pressure waves developing in an adjacent fluid field. The knowledge of the transient distribution of stresses in the solid is important to understand the cause of damaging by comparisons with breaking points of the material. The modeling of this problem requires the coupling of the models for the fluid and the solid. One common approach is to iterate the coupling condition in each time step solving alternatingly the fluid and solid model.

Alternatively, a strategy based on the solution of coupled Riemann problems that has been developed by M. Herty and collaborators. This concept is exemplified for the coupling of a linear elastic structure with an ideal gas. The coupling procedure relies on the solution of a nonlinear equation. Existence and uniqueness of the solution is proven.

The wave speeds in the solid are in general significantly higher than in the fluid imposing a smaller time step for the solid solver. All computations were performed using a synchronized time-stepping for the fluid and solid solver. Thus, the CFL-number in the fluid was much smaller than in the solid. In order to avoid an unnecessary small time-step in the fluid solver, we develop a multilevel local timestep algorithm in the adaptive RKDG solver. By numerical simulations we will verify the improvement in the computational time without spoiling the accuracy.

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**11:45-12:15**

**Feedback Control for Linearized Hyperbolic Balance Laws
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**Stephan Gerster (RWTH Aachen University)**

** **Physical systems such as gas, water and electricity networks are usually operated in a state of equilibrium and one is interested in stable systems, where small perturbations are damped over time. We will consider gas flow on a network with feedback boundary conditions. We focus on linearized isothermal Euler equations that are diagonalizable with Riemann invariants and analyze the stability of a steady state. Explicit conditions are presented yielding an exponential Lyapunov stability. We will focus both on a Lyapunov function with respect to the L^{2}– and H^{2}-norm. Furthermore, not only the convergence to a steady state of the analytical solution, but also of the numerical approximation is guaranteed. Numerical results illustrate our analysis.

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**12:15-13:00 ****Lunch**

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**13:00-13:30**

**Recent results on multi-dimensional nonlocal balance laws
**

**Michele Spinola**

**(FAU)**

In recent years, nonlocal conservation and balance laws have drawn significantly more attention. These are conservation laws with a velocity function depending on the integral of the solution. In application, those models are used in supply chains, traffic flow, opinion formation and especially in the case where multiple spatial variables are considered in crowd dynamics and chemical engineering. While in many publications the nonlocal term will be considered over the entire space domain, in this talk we present an existence and uniqueness result which allows rather general integration area for the nonlocal term. Here, we can get rid of the Entropy condition which is usually postulated to guarantee a unique weak solution of (local) balance laws and by Banach’s fixed-point theorem, we obtain a unique weak solution on every finite time horizon.

**13:30-14:00 **

**Discussion and closing**