OxPDE Lunchtime Seminar

 

We first consider stochastic Cucker--Smale particle systems driven by truncated multiplicative noise. A key difficulty is to control particle positions uniformly in time, since the truncated noise destroys the conservation of the mean velocity. By working in a comoving frame and adapting arguments from deterministic flocking theory, we obtain stochastic flocking together with uniform-in-time $L^\infty$ bounds on particle positions. We also derive quantitative stability estimates in the $\infty$-Wasserstein distance, which allow us to pass to the mean-field limit and obtain corresponding flocking results for the associated stochastic kinetic equation.

 

We then study an infinite-particle Cucker--Smale system with sublinear, non-Lipschitz velocity coupling under directed sender networks. While classical energy methods only yield asymptotic alignment, a componentwise diameter approach combined with Dini derivative estimates leads to finite-time flocking for both fixed and switching sender networks. The resulting flocking-time bounds are uniform in the number of agents and apply to both finite and infinite systems.

For a genuinely nonlinear $2 \times 2$ hyperbolic system of conservation laws, assuming that the initial data have small $\bf L^\infty$ norm but possibly unbounded total variation, the existence of global solutions was proved in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like $t^{-1}$. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with faster decay rate: $\hbox{Tot.Var.}\bigl\{u(t,\cdot)\bigr\}\leq C t^{\alpha-1}$. For these solutions, a uniqueness theorem is proved. Indeed, as the initial data range over a domain of functions with $\|\bar u\|_{{\bf L}^\infty} \leq\varepsilon_1$ small enough, solutions with fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to 1 by further shrinking the value of $\varepsilon_1>0$. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.

This is a joint work with A. Bressan and G. Vaidya.