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Global semigroup of conservative weak solutions of the two-component Novikov equation

Kenneth H. Karlsen, Yan Rybalko

TL;DR

This work addresses the global evolution of the two-component Novikov equation, where wave breaking and energy concentration pose major challenges. By transforming the problem along characteristics into a semilinear, nonlocal ODE system for variables $(U,V,W,Z,q)$ and introducing a Radon energy measure $\mu_{(t)}$, the authors construct a global conservative weak solution in the energy space $\Sigma$ and establish a corresponding data-to-solution continuity in the uniform norm. They prove global well-posedness of the ODE system, reconstruct the physical fields $(u,v)$ from the transformed variables, and define measure-valued energy transport equations that describe energy concentration during wave breaking. These ingredients are assembled into a global semigroup $\Psi_t$ acting on an augmented data space $\mathcal{D}$, yielding unique continuation of solutions past collisions and preserving key invariants such as $E_{u_0}, E_{v_0}, G_0, H_0$. The results provide a rigorous, energy-centered framework for the global dynamics of this nonlocal, two-component system with potential two-way peakon interactions and energy concentration phenomena.

Abstract

We study the Cauchy problem for the two-component Novikov system with initial data $u_0, v_0$ in $H^1(\mathbb{R})$ such that the product $(\partial_x u_0)\partial_x v_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of $(\partial_x u)^2dx$, $(\partial_x v)^2dx$, and $\left((\partial_x u)^2(\partial_x v)^2\right)dx$, which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm.

Global semigroup of conservative weak solutions of the two-component Novikov equation

TL;DR

This work addresses the global evolution of the two-component Novikov equation, where wave breaking and energy concentration pose major challenges. By transforming the problem along characteristics into a semilinear, nonlocal ODE system for variables and introducing a Radon energy measure , the authors construct a global conservative weak solution in the energy space and establish a corresponding data-to-solution continuity in the uniform norm. They prove global well-posedness of the ODE system, reconstruct the physical fields from the transformed variables, and define measure-valued energy transport equations that describe energy concentration during wave breaking. These ingredients are assembled into a global semigroup acting on an augmented data space , yielding unique continuation of solutions past collisions and preserving key invariants such as . The results provide a rigorous, energy-centered framework for the global dynamics of this nonlocal, two-component system with potential two-way peakon interactions and energy concentration phenomena.

Abstract

We study the Cauchy problem for the two-component Novikov system with initial data in such that the product belongs to . We construct a global semigroup of conservative weak solutions. We also discuss the potential concentration phenomena of , , and , which contribute to wave-breaking and may occur for a set of time with nonzero measure. Finally, we establish the continuity of the data-to-solution map in the uniform norm.
Paper Structure (15 sections, 18 theorems, 216 equations)

This paper contains 15 sections, 18 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Change of variables and a semilinear system
  4. Conservation laws
  5. New variables
  6. A semilinear system
  7. Local existence and uniqueness of ODE system
  8. Global well-posedness of ODE system
  9. Global solutions of two-component system
  10. Reconstruction of $(u,v)$ in terms of $(U,V,y)$
  11. Global weak solution
  12. A semigroup of global solutions
  13. Appendix A
  14. Appendix B
  15. Appendix C

Key Result

Theorem 2.5

Consider $(u_0,v_0,\mu_0)\in\mathcal{D}$. Then there exists a flow map $\Psi_t:\mathbb{R}\times\mathcal{D} \to\mathcal{D}$ whose trajectories $\Psi_t(u_0,v_0,\mu_0)=(u(t),v(t),\mu_{(t)})$ satisfy the following properties:

Theorems & Definitions (47)

  • Definition 2.1: Global weak solution of \ref{['t-c-N-n']}--\ref{['iid']}
  • Definition 2.2: Global conservative weak solution of \ref{['t-c-N-n']}--\ref{['iid']}
  • Remark 2.3: Sets $N_W$ and $N_Z$
  • Remark 2.4: Measure-valued solution
  • Theorem 2.5: Global semigroup of conservative solutions
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Proposition 3.1: HQ21HLQ23L19
  • ...and 37 more