Generic regularity and Lipschitz metric for a two-component Novikov system

Abstract

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions between peakon-like structures. Our study has two main objectives: first, to analyze the generic regularity of global conservative solutions; and second, to construct a new metric that guarantees the Lipschitz continuity of the flow. Building on the geometric framework developed by Bressan and Chen for quasilinear second-order wave equations, we prove that the solution retains $C^k$ regularity away from a finite number of piecewise $C^{k-1}$ characteristic curves. Furthermore, we provide a description of the solution behavior in the vicinity of these curves. By introducing a Finsler norm on tangent vectors in the space of solutions, expressed in the transformed Bressan-Constantin variables, we introduce a Lipschitz metric representing the minimal energy transportation cost between two solutions.

Figures (4)

• Figure 1: Illustration of the image of $\Gamma^W=\gamma_1^W\cup\gamma_2^W$ and $\Gamma^Z=\gamma_1^Z$ (see \ref{['GammaWZ']} and \ref{['gammaWZ']} with $N_1=2$ and $N_2=1$) under the characteristics map $(t,\xi)\mapsto(t,y(t,\xi))$. Here $C_i^W$ is the image of $\gamma_i^W$, $i=1,2$, and $C_1^Z$ is the image of $\gamma_1^Z$. Notice that the characteristic curves $C_1^W$ and $C_2^W$ have discontinuous derivatives only at the images of the points $(t_i,\xi_i)$ where $W_\xi(t_i,\xi_i)=0$, $i=1,2,3$.
• Figure 2: Eight types of points $(t_i,\xi_i)$, $i=1,\dots,8$ on the characteristic curves $\Gamma^W\cup\Gamma^Z$, see \ref{['GammaWZ']}, discussed in Theorem \ref{['Thmchc']}.
• Figure 3: A diagram of the Bressan-Constantin approach applied to the two-component Novikov equation. Starting from the initial data $\mathbf{u}_0=\left(u_0,v_0,\mu_0;D_{W,0},D_{Z,0}\right)$, the direct transform \ref{['id2']} yields $\mathbf{U}_0=(U_0,V_0,W_0,Z_0,q_0)$, which serves as the initial data for the associated ODE system \ref{['ODE']}. The initial Eulerian data $\mathbf{u}_0$ can be recovered from $\mathbf{U}_0$ via the inverse transform \ref{['uvdef']}--\ref{['mut']} and \ref{['DN-a']} evaluated at $t=0$. Given the unique global solution $U(t)=(U,V,W,Z,q)(t)$ of the ODE system, Theorem \ref{['Thm']} provides a corresponding global conservative solution $\mathbf{u}(t)=(u(t),v(t),\mu_{(t)};D_W(t),D_Z(t))$ of the Novikov system. We emphasize that applying the direct transform \ref{['id2']} to $\mathbf{u}(t)$ produces a solution $\widetilde{\mathbf{U}}(t)$ which, in general, does not coincide with $\mathbf{U}(t)$ (in particular, one has $\tilde{q}\equiv 1$ in $\widetilde{\mathbf{U}}$). Consequently, the Bressan-Constantin and Eulerian variables do not form a bijective correspondence (cf. KR25, BC07, and BC07d).
• Figure 4: Mapping the path $\widehat{\mathbf{U}}^\theta(t)$ in the transformed variables to a path $\tilde{\mathbf{u}}^\theta(t)$ in the Euler variables ($t$ is considered fixed here), and then lifting it back to the Bressan-Constantin variables, produces the path $\widetilde{\mathbf{U}}^\theta_0$ (note that $\tilde{\mathbf{u}}^\theta(t)$ is a piecewise regular path in the sense of BC17A). Because the transformation is not bijective, one generally has $\widetilde{\mathbf{U}}^\theta_0 \neq \widehat{\mathbf{U}}^\theta(t)$ for $t\neq0$ and $\theta\in[0,1]$. Nevertheless, since both $\widehat{\mathbf{U}}^\theta(t)$ and $\widetilde{\mathbf{U}}^\theta_0$ correspond to the same path $\tilde{\mathbf{u}}^\theta(t)$ in the Euler variables, their lengths $\|\cdot\|_{\mathcal{L}}$ coincide (see Definition \ref{['lPR']}).

Theorems & Definitions (63)

• Theorem 1.1: Generic regularity of conservative solutions
• Corollary 1.2
• Remark 1.3: Concentration of energy
• Remark 1.4
• Remark 1.5
• Theorem 1.6: Solution behavior near the characteristic curves
• Corollary 1.7
• Remark 1.8
• Corollary 1.9: Hölder continuity
• Remark 1.10