Speaker
Description
This talk concerns the Cauchy problem for the following two-component Novikov system:
$\begin{align}
&m_t+(uvm)_x+u_xvm=0,
&& m=m(t,x),\,u=u(t,x),\,v=v(t,x),\\
&n_t+(uvn)_x+uv_xn=0,
&& n=n(t,x),\\
&m=u-u_{xx},\,\,n=v-v_{xx},
&& u,v\in\mathbb{R},\quad
t,x\in\mathbb{R},
\end{align}$
It is well-known that this system admits the wave-breaking phenomena: the slope of solution corresponding to the smooth initial data can blow-up in finite time.
The goal is to establish the generic regularity for the two-component Novikov system. More in detail, we show that there exists an open dense subset of initial data, such that the corresponding global weak solutions (which exist beyond blow-up) retain certain regularity properties in the $t,x$ plane. Applying the Bressan and Chen's approach [1], originally developed for the nonlinear variational wave equation, we prove the following result [2].
Theorem. Consider the regular initial data $(u_0,v_0)$ in the metric space $\Upsilon^k\subset\left(C^k\right)^2\cap\left(H^1\right)^2$, $k\geq5$, see [2] for details. Then for any $T>0$ there exists an open in $\Upsilon^k$ and dense in $\Upsilon^{k-1}\cap C^k_{\mathrm{loc}}$ subset $\mathcal{M}_T\subset\Upsilon^k$, such that for any initial data $(u_0,v_0)\in\mathcal{M}_T$ the corresponding global weak solution $(u,v)(t)$ satisfies the following regularity properties. There exist two sets of piecewise $C^{k-1}$ curves, say $\left\{\mathbf{C}_i^W\right\}_{i=1}^{N_1}, \left\{\mathbf{C}_j^Z\right\}_{j=1}^{N_2}\subset[-T,T]\times\mathbb{R}$, such that
- both components $u$ and $v$ are $C^k$ regular away from the characteristic sets $\mathbf{C}_i^W$ and $\mathbf{C}_j^Z$
- $u$ is of class $C^1$ along each $\mathbf{C}_j^Z$ and $v$ is of class $C^1$ along each $\mathbf{C}_i^W$.
[1] A. Bressan, G. Chen. Generic regularity of conservative solutions to a nonlinear wave equation. Ann. I. H. Poincaré 34 (2017) 335--354.
[2] K.H. Karlsen, Ya. Rybalko. Generic regularity and Lipschitz metric for a two-component Novikov system. arXiv :2512.13305v2 (2026).
