An Overview on Some Results Concerning the Transport Equation and its Applications to Conservation Laws

We provide an informal overview on the theory of transport equations with non smooth velocity fields, and on some applications of this theory to the well-posedness of hyperbolic systems of conservation laws.Comment: 12 page

On smooth approximations of rough vector fields and the selection of flows

In this work we deal with the selection problem of flows of an irregular vector field. We first summarize an example from \cite{CCS} of a vector field $b$ and a smooth approximation $b_\epsilon$ for which the sequence $X^\epsilon$ of flows of $b_\epsilon$ has subsequences converging to different flows of the limit vector field $b$. Furthermore, we give some heuristic ideas on the selection of a subclass of flows in our specific case.Comment: Proceeding of the "XVII International Conference on Hyperbolic Problems: Theory, Numerics, Applications.

Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy

We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in $L^p$ with $1\leq p\leq \infty$. Moreover, if $p\geq 3/2$ all weak solutions are conservative. In this work we prove that solutions obtained via the vortex method are Lagrangian, and that they are conservative if $p>1$.Comment: 28 page

Polynomial mixing under a certain stationary Euler flow

We study the mixing properties of a scalar $\rho$ advected by a certain incompressible velocity field $u$ on the two dimensional unit ball, which is a stationary radial solution of the Euler equation. The scalar $\rho$ solves the continuity equation with velocity field $u$ and we can measure the degree of mixedness of~$\rho$ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted for the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a large class of initial data. This extends previous results for the functional mixing scale, based on the explicit expression for the solution in Fourier variable, results that are also partially recovered by our approach.Comment: 21 pages, 6 figure

Lagrangian solutions to the 2D euler system with L^1 vorticity and infinite energy

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under only the assumption of L^1 weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler system follows for arbitrary L^1 vorticity. Relations with previously known notions of solutions are established

Renormalized solutions of the 2d Euler equations

In this paper we prove that solutions of the 2D Euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized

On the singular local limit for conservation laws with nonlocal fluxes

We give an answer to a question posed in [P. Amorim, R. Colombo, and A. Teixeira, ESAIM Math. Model. Numerics. Anal. 2015], which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity depends on the solution via the convolution by a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law.Comment: 26 page

DLR Contribution to the First High Lift Prediction Workshop

DLR’s contribution to the first AIAA High Lift Prediction Workshop (HiLiftPW-1) covers computations of all three scheduled test cases for the NASA trapezoidal wing in high lift configuration. The DLR finite volume code TAU has been employed as the flow solver. In a standard set-up the one-equation turbulence model of Spalart and Allmaras in the original formulation is used to model effects of turbulence. For selected grids and flow conditions, the k-ω SST model of Menter and a differential Reynolds stress model (SSG/LLR-ω ) developed by DLR have been considered. DLR contributed with two hybrid unstructured grid families to the workshop. The grids have been generated with the grid generation packages Centaur and Solar. A grid family with three Solar grids has been generated and provided to the workshop featuring grids of 12·10^6 , 37·10^6 , and 111·10^6 points for test case 1. In addition, a Solar grid of 37·10^6 points has been provided for test case 2, and a grid of 40·10^6 for the configuration including the slat and flap brackets (test case 3). DLR didn’t succeed in generating a fine-grid with the Centaur package. In order to complete a Centaur grid family with three grid levels an extra-coarse grid has been provided. Thus, the three levels of the Centaur grid family are realized by grids of 13·10^6 , 16·10^6 , and 32·10^6 points. In general a go o d agreement between the experimental evidence and the polar computations on the Solar and Centaur grids is found in terms of forces, moments and wing pressure distributions. The wing tip area with the rearward part of the main wing and the flap represents the most challenging part of the configuration, especially at angles of attack around maximum lift. The deviations between the TAU solutions and the experimental data in this area are only weakly influenced by the different grid topologies or turbulence models used. The influence of the grid resolution of both grid families is comparable, taking into account the different absolute resolution levels of both grid families. Including the slat and flap brackets leads to the expected lift decrease. Concerning the convergence properties, a strong dependence on the numerical start-up procedure has been detected in many of the computations at higher angles of attack

Exponential self-similar mixing and loss of regularity for continuity equations

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.Comment: 8 pages, 3 figures, statement of Theorem 11 slightly revise