We study a simplified equation governing turbulent kinetic energy $k$ in a bounded domain, arising from turbulence modeling where the eddy diffusion is given by $\varrho(x) + \varepsilon$, with ...$\varrho$ representing the Prandtl mixing length of the order of the distance to the boundary, and a right-hand side in $L^1$. We obtain estimates of $\sqrt \varrho \g k$ in $L^q$ spaces and we establish the convergence toward the formal limit equation in the sense of the distributions as $\varepsilon$ goes to 0.
We prove that certain families of compact Coxeter polyhedra in 4- and 5-dimensional hyperbolic space constructed by Makarov give rise to infinitely many commensurability classes of reflection groups ...in these dimensions.
Abstract We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). ...This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.
Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in ...dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb{R}^3$ such that each vector field $X\in\mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincaré map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.
In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects ...caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.
We give a practical tool to control the L∞-norm of the Steklov eigenfunctions in a Lipschitz domain in terms of the norm of the BV-trace operator. The norm of this operator has the advantage to be ...characterized by purely geometric quantities. As a consequence, we give a spectral stability result for the Steklov eigenproblem under geometric domain perturbations and several examples where stability occurs. In particular we deal with geometric domains which are not equi-Lipschitz, like vanishing holes, merging sets, approximations of inner peaks.
We consider optimal control problems associated to generally non-well posed Cauchy problems in a general framework. Firstly, we approximate the ill-posed problem with a family of well-posed one and ...show that solutions of the latter one converge to solutions of the former one. Secondly, we investigate the minimization problem associated with the approximated state equation. We prove the existence and uniqueness of minimizers that we characterize with the optimality systems. Finally, we show that minimizers of the approximated problems converge to the minimizers of the optimal control subjected to the ill-posed state equation that we characterize with a singular optimality system. This characterization is obtained as the limit of the optimality systems of the approximated minimization problem. We use the techniques of quasi-reversibility developed by Lattès and Lions in 1969. Our general framework includes classical elliptic second order operators with Dirichlet and Robin conditions, as well as the fractional Laplace operator with the Dirichlet exterior condition.
Abstract
Using a particle-based model, we examine the collective dynamics of skyrmions interacting with a funnel potential under dc driving as the skyrmion density and relative strength of the Magnus ...and damping terms are varied. For driving in the easy direction, we find that increasing the skyrmion density reduces the average skyrmion velocity due to jamming of skyrmions near the funnel opening, while the Magnus force causes skyrmions to accumulate on one side of the funnel array. For driving in the hard direction, there is a critical skyrmion density below which the skyrmions become trapped. Above this critical value, a clogging effect appears with multiple depinning and repinning states where the skyrmions can rearrange into different clogged configurations, while at higher drives, the velocity-force curves become continuous. When skyrmions pile up near the funnel opening, the effective size of the opening is reduced and the passage of other skyrmions is blocked by the repulsive skyrmion–skyrmion interactions. We observe a strong diode effect in which the critical depinning force is higher and the velocity response is smaller for hard direction driving. As the ratio of Magnus force to dissipative term is varied, the skyrmion velocity varies in a non-linear and non-monotonic way due to the pile up of skyrmions on one side of the funnels. At high Magnus forces, the clogging effect for hard direction driving is diminished.
Abstract
Using a particle-based model, we examine the collective dynamics of skyrmions interacting with a funnel potential under dc driving as the skyrmion density and relative strength of the Magnus ...and damping terms are varied. For driving in the easy direction, we find that increasing the skyrmion density reduces the average skyrmion velocity due to jamming of skyrmions near the funnel opening, while the Magnus force causes skyrmions to accumulate on one side of the funnel array. For driving in the hard direction, there is a critical skyrmion density below which the skyrmions become trapped. Above this critical value, a clogging effect appears with multiple depinning and repinning states where the skyrmions can rearrange into different clogged configurations, while at higher drives, the velocity-force curves become continuous. When skyrmions pile up near the funnel opening, the effective size of the opening is reduced and the passage of other skyrmions is blocked by the repulsive skyrmion-skyrmion interactions. We observe a strong diode effect in which the critical depinning force is higher and the velocity response is smaller for hard direction driving. As the ratio of Magnus force to dissipative term is varied, the skyrmion velocity varies in a non-linear and non-monotonic way due to the pile up of skyrmions on one side of the funnels. At high Magnus forces, the clogging effect for hard direction driving is diminished.
PDF
