| Abstract: |
| In this talk, I will present recent results on the one-dimensional pressureless Euler alignment system with a strongly singular interaction kernel and initial data allowing vacuum
\begin{align*}
\rho_t + (\rho u)_x &= 0 , \
u_t + u u_x &= - \Lambda^\alpha (\rho u) + u\,\Lambda^\alpha \rho.
\end{align*}
Building on a previously developed framework for the case $\alpha \in (0,1)$, we extend the analysis to the full range $\alpha \in (0,2)$. We work under the structural condition imposed on the initial data,
\begin{align*}
0 \leqslant G_0(x) \leqslant a\,\rho_0(x) \quad \text{where} \quad G_0 = \partial_x u_0 - \Lambda^\alpha \rho_0,
\end{align*}
which determines an admissible class of initial velocity profiles and is propagated by the dynamics.
I will discuss the construction of global weak solutions corresponding to compactly supported initial data and show that the support remains compact for all times. Moreover, I will explain how finite speed of propagation can be established by adapting barrier arguments and contact analysis techniques from the theory of nonlocal porous medium equations. |
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