We are interested in the density patch problem for the inhomogeneous, incompressible Navier-Stokes equations in $\mathbb{R}^2$ at a critical level of regularity. More precisely, we assume that the initial density of the fluid is the indicator function of a bounded Lipschitz domain and that the initial velocity $u_0$ lies in the Besov space $\dot{B}^0_{2,1}(\mathbb{R}^2)$.

We first introduce a suitable class of solutions and give an overview of the most important a priori estimates. This allows us to prove the global existence and uniqueness of solutions in the critical regularity framework above and to conclude that the Lipschitz regularity of the patch is preserved over time. Compared to previous works related to the density patch problem, the main novelty is an $L^1$-global-in-time Lipschitz estimate on the velocity field. This enables us to fully describe the long-time behavior of the patch, which is the rigid motion of an emerging Lipschitz domain.

This is based on joint work with Alessandro Violini.

Nonlocal traffic flow models have been introduced to macroscopically model the effect of downstream information on driver behavior, allowing them to anticipate traffic conditions ahead. Here, the term "nonlocal" refers to the dependence of the flux function on a convolution between a kernel function and the conserved quantity. In real-world applications, we must additionally account for stochastic influences, which can arise from measurement errors in autonomous vehicles or uncertainty in human driver behavior.

In this talk, we consider nonlocal traffic flow models with random velocity functions. The randomness is introduced via a space-dependent multiplicative noise defined by a random field. For such stochastic nonlocal conservation laws, the existence, uniqueness, and stability of solutions must be reassessed in the presence of random, space-dependent fluxes. Furthermore, new challenges arise when characterizing stochastic moments, such as the expected density. To establish the pathwise well-posedness of weak solutions under space-dependent stochasticity, assumptions regarding the regularity of the random field are crucial. This talk focuses on the specific conditions under which the well-posedness of numerical solutions can be established.

Finally, we present numerical simulations using a Hilliges–Weidlich-type scheme. Suitable realizations of the random field are generated using a Karhunen–Loève expansion with a Matérn covariance kernel, and approximations of the expected density are investigated.

In simulations of rarefied gases, we have to resort to a gas model which goes beyond classical fluid dynamics, as in the presence of shocks the Navier-Stoker-Fourier equations fail to describe the correct behaviour of the thermodynamical quantities. Kinetic methods, which take into account the molecular description of a gas, are a resourceful and accurate alternative. These methods are based on kinetic theory, which states that the thermodynamical quantities of a gas can be described as moments of the molecular velocity distribution $f(v,x,t)$. For example, we can write the density, bulk velocity and energy $\rho, U, E$ as
$$
\rho = \int_{R^3} f \ dv, \ U = \cfrac{1}{\rho} \int_{R^3} v\ f \ dv, \ E = \cfrac{1}{2\rho} \int_{R^3} \vert v -U\vert^2 f \ dv.
$$
The gas dynamics is obtained by following the evolution of $f$ with respect to the binary interaction between the particles (i.e. collisions). This is described by the Boltzmann equation
$$
\partial_t f + v \cdot \nabla_x f = S[f]
$$
where $S[f]$ is called the collision operator. In general, $S(f)$ has a complex integral form and therefore it needs to be modeled in a computationally advantageous way. Common approaches such as Direct Simulation Monte Carlo (DSMC) sample colliding pairs through a Monte Carlo procedure, however this becomes unfeasible when the average number of collisions is high.
In recent years, stochastic approximations of the Boltzmann equation based on simplified collision operators have gained popularity for being a computationally affordable alternative to DSMC. Within these methods, collision operators constructed from the Fokker-Planck equation have been widely studied given their promising potentials.
In fact, the Fokker-Planck equation
$$
\partial_t f + v \cdot \nabla_x f = -\nabla_v \cdot ( A[f] \ f ) + \Delta_{v,v}(D[f] \ f)
$$
gives rise to a drift-diffusion process with drift functional $A[f]$ and diffusion matrix $D[f]$. This leads to a system of Stochastic Differential Equations (SDEs) of Langevin type for the stochastic processes $X_t,V_t$ which describe the position and velocity of the particles:
$$
dX_t = V_t dt, \
dV_t = A[f] dt + \sqrt{2D[f]}dW_t,
$$
where $dW_t$ is the standard Wiener process. This leads to algorithms which are independent of the collision frequency and only depend on the number of particles, which become particularly advantageous in the near-continuum regimes. However, despite this potential, several limitaton in the model still exist. In particular, we want to address the problem of devising a model which honours prescribed moment dynamics along with the entropy constraint. While the former ensures recovery of correct conservation laws, and thus Navier-Stokes-Fourier system in the continuum limit, the latter leads to the model stability for an arbitrary distribution function.
In this talk, we present a Fokker-Planck model which satisfies the aforementioned criteria. In particular, the model includes a novel closure that enforces negative entropy decay proportional to the Fisher information. After having discussed the model, we will present validation of theoretical results. we will conclude by showing the comparison of runtimes against DSMC in a real-world setting with varying rarefaction regimes. Next steps and addressing of remaining open problems will be also outlined.

Active Flux methods are high-order finite volume methods for hyperbolic conservation laws that combine cell averages with point values located on cell interfaces. Various versions of Active Flux methods are currently under development. In this talk, I focus on third-order accurate, fully discrete Active Flux methods with compact stencils in space and time. A crucial step of these methods is the evolution of the point values. To update these degrees of freedom, exact or approximate evolution operators are required. The construction of such operators is therefore a key component of fully discrete Active Flux schemes.

We focus on the construction of evolution operators for the linearised Euler equations. By introducing moving coordinates, the advective part of the system can be removed, reducing the problem to the acoustic equations. This reformulation allows us to employ both exact and approximate truly multidimensional evolution operators originally developed for acoustics. In this way, the construction of evolution operators for the linearised Euler equations is directly linked to their acoustic counterparts, and many of their properties can be transferred to the Euler setting.

The resulting evolution operators form the basis of fully discrete Active Flux methods for the compressible Euler equations. We compare Active Flux methods based on exact and approximate evolution operators and investigate how the choice of evolution operator influences the resulting numerical schemes. Numerical experiments illustrate the performance of these methods for different flow regimes, ranging from discontinuous solution structures with shock waves to vortex structures in the low-Mach-number regime.