Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations

1. Problem Statement

The authors investigate the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations on the real line, given by:
$$u_t - u_{xxt} + (f(u))_x - (f(u))_{xxx} + \left(g(u) + \frac{f''(u)}{2}u_x^2\right)_x + \lambda(u - u_{xx}) = 0$$
with initial data $u(0, x) = \bar{u}(x)$.

Key Components:

• Dissipation: The term $\lambda(u - u_{xx})$ with $\lambda > 0$ represents weak dissipation.
• Generality: This equation generalizes several classical models:
  • Weakly dissipative Camassa-Holm (CH) equation: When $f(u) = \frac{1}{2}u^2$, $g(u) = 2ku + u^2$.
  • Weakly dissipative hyperelastic rod wave equation: When $f(u) = \frac{\gamma}{2}u^2$, $g(u) = \frac{3-\gamma}{2}u^2$.
• Objective: To establish the global existence of solutions despite the potential for finite-time blow-up (wave-breaking) known to occur in the non-dissipative case ($\lambda=0$) when initial derivatives are sufficiently negative. The paper specifically targets two types of solutions:
  1. Global Conservative Weak Solutions: For initial data $\bar{u} \in H^1(\mathbb{R})$.
  2. Global Strong Solutions: For higher regularity data $\bar{u} \in H^s(\mathbb{R})$ ($s > 3/2$), under specific conditions on initial data size or sign.

2. Methodology

The paper employs distinct strategies for weak and strong solutions, leveraging the specific structure of the dissipation term.

A. Transformation to a Semilinear System (For Weak Solutions)

To handle the low regularity of $H^1$ data and the formation of singularities (wave-breaking), the authors adapt the Bressan-Constantin approach used for the CH equation.

1. Variable Transformation: They introduce a new time-weighted variable $k(t, x) = e^{\lambda t} u(t, x)$. This transforms the original equation into an equivalent form where the linear dissipation induces an exponentially time-weighted energy conservation law:
   $$\frac{d}{dt} \left( e^{2\lambda t} \|u(t)\|_{H^1}^2 \right) = 0 \implies e^{\lambda t} \|u(t)\|_{H^1} = \|\bar{u}\|_{H^1}$$
2. Characteristic Coordinates: They introduce a non-decreasing map $\xi \mapsto \bar{y}(\xi)$ and define characteristic curves $y(t, \xi)$.
3. New Variables: They define a semilinear system in terms of $(k, v, q)$:
   • $k$: The transformed solution.
   • $v = 2 \arctan(k_x)$: A variable to handle the slope $u_x$ (which may become infinite).
   • $q = (1 + k_x^2) \frac{\partial y}{\partial \xi}$: A variable related to the Jacobian of the coordinate transformation.
4. Global Existence for Semilinear System: They prove the global existence of solutions to this semilinear system in a Banach space $X = H^1 \times (L^2 \cap L^\infty) \times L^\infty$ by establishing uniform bounds on the energy and utilizing the conservation of a specific integral quantity derived from the system.
5. Reconstruction: The global conservative weak solution for the original equation is recovered by mapping the solution of the semilinear system back to the original coordinates $(t, x)$.

B. Energy Estimates and Blow-up Criteria (For Strong Solutions)

For strong solutions ($s > 3/2$), the authors use two different mechanisms depending on the initial data:

1. Small Initial Data:

   • Method: A bootstrap argument using $H^s$-energy estimates.
   • Mechanism: They show that if the initial data is sufficiently small ($\|\bar{u}\|_{H^s} \leq \epsilon$) and the dissipation coefficient $\lambda$ is large enough ($\lambda \geq M$), the linear dissipation dominates the nonlinear terms. This allows them to close the a priori estimates and extend the local solution globally.

2. Sign-Changing Initial Data:

   • Method: Analysis of the momentum density $m = u - u_{xx}$ along characteristics.
   • Conditions: Requires specific structural conditions on $f$ and $g$ (specifically $f''' \equiv 0$ and $g'(u) = 2f''(u)u$).
   • Mechanism: They prove a sign-preservation property for the momentum $m$ along characteristics. Combined with the exponentially weighted energy conservation, this yields a uniform lower bound for the slope $u_x$.
   • Result: Since the blow-up criterion for these equations relies on $u_x \to -\infty$, establishing a uniform lower bound for $u_x$ prevents wave-breaking, ensuring global existence even for large initial data, provided the sign condition is met.

3. Key Contributions and Results

Main Theorems

1. Theorem 1.2 (Global Conservative Weak Solutions):

   • For any initial data $\bar{u} \in H^1(\mathbb{R})$, there exists a global conservative weak solution.
   • The solution is Hölder continuous in space and Lipschitz continuous in time (into $L^2$).
   • Conservation Law: The quantity $e^{2\lambda t} \int_{\mathbb{R}} (u^2 + u_x^2) dx$ is conserved for almost every $t$.
   • Continuous Dependence: The solution depends continuously on the initial data in the $H^1$ norm.

2. Theorem 1.3 (Global Strong Solutions for Small Data):

   • If $\bar{u} \in H^s(\mathbb{R})$ ($s > 3/2$) is sufficiently small and the dissipation $\lambda$ is sufficiently large, a unique global strong solution exists.
   • This extends previous results by relaxing assumptions on the specific form of the equation while requiring strong dissipation.

3. Theorem 1.5 (Global Strong Solutions for Sign-Changing Data):

   • If $\bar{u} \in H^s(\mathbb{R})$ satisfies a specific sign condition on the initial momentum $\bar{m}$ (changing sign exactly once) and the structural conditions on $f, g$ hold, the solution exists globally.
   • The solution decays to 0 in the $H^s$ norm as $t \to \infty$.
   • This generalizes results for the weakly dissipative CH equation to a broader class of equations.

Technical Innovations

• Time-Weighted Energy Space: The identification of the invariant quantity $e^{\lambda t} \|u\|_{H^1}$ is crucial. It allows the authors to control the growth of the solution in the presence of dissipation, which is essential for constructing weak solutions where standard energy methods fail due to singularity formation.
• Generalized Semilinear System: The construction of the semilinear system $(k, v, q)$ is adapted to handle the general functions $f(u)$ and $g(u)$, not just the quadratic nonlinearity of the standard CH equation.
• Sign-Preservation Analysis: The proof for sign-changing data provides a robust mechanism to rule out blow-up without requiring smallness of the data, relying instead on the topological structure of the initial momentum.

4. Significance

• Unification: The paper unifies the study of several important physical models (CH, hyperelastic rod) under a single general framework with weak dissipation.
• Resolution of Singularity: It rigorously addresses the "wave-breaking" phenomenon. While solutions may develop singularities in finite time for the non-dissipative case, the inclusion of weak dissipation (even with $\lambda > 0$) combined with specific data conditions ensures global existence.
• Extension of Theory:
  • It extends the theory of conservative weak solutions (previously established for CH) to a much broader class of nonlinear dispersive equations.
  • It provides new global existence criteria for strong solutions that do not rely solely on smallness assumptions but also on the sign structure of the initial data.
• Physical Relevance: By including dissipation, the model becomes more physically realistic for fluid dynamics and elastic rod deformation, where energy loss is inevitable. The results confirm that global solutions exist even in these dissipative regimes, bridging the gap between idealized conservative models and real-world dissipative systems.

In summary, this work provides a comprehensive mathematical foundation for the long-time behavior of a wide class of weakly dissipative dispersive waves, establishing both the existence of global weak solutions (handling singularities) and strong solutions (under specific data conditions).