Asymptotic stability of strong rarefaction waves to a parabolic-hyperbolic system arising from chemotaxis

Technical Summary: Asymptotic Stability of Strong Rarefaction Waves to a Parabolic-Hyperbolic System Arising from Chemotaxis

Problem Statement
This paper investigates the large-time asymptotic behavior of solutions to the Cauchy problem for a parabolic-hyperbolic system derived from a chemotaxis model with logarithmic sensitivity. The system is given by:
$$\begin{cases} n_t - (nq)_x = n_{xx}, \\ q_t - n_x = 0, \end{cases}$$
where $n(x,t)$ represents the density of vascular endothelial cells and $q(x,t)$ is a transformed variable related to the concentration of the signaling molecule VEGF via the Hopf-Cole transformation $q = -(\ln c)_x$. The study focuses on the stability of solutions when the initial data is a small perturbation of a rarefaction wave. Specifically, the authors address the asymptotic stability of:

1. Single strong rarefaction waves: Solutions connecting two constant states $(n_-, q_-)$ and $(n_+, q_+)$ where the states lie on a single rarefaction curve.
2. Composite strong rarefaction waves: Solutions formed by the superposition of two rarefaction waves (a 1-rarefaction and a 2-rarefaction) connecting three states.

A critical feature of this work is that the stability results hold for strong rarefaction waves, meaning the wave strengths (amplitudes $|n_- - n_+| + |q_- - q_+|$) can be arbitrarily large, rather than being restricted to weak waves. Furthermore, the results apply to general perturbations, not restricted to zero-mass perturbations.

Methodology
The authors employ the energy method combined with the construction of smooth approximations for the rarefaction waves. The proof strategy involves several key technical components:

1. Smooth Approximation Construction: Following the framework of Matsumura and Nishihara, the authors construct smooth approximate rarefaction waves $(N, Q)$ by solving initial value problems for the inviscid Burgers equation. These approximations converge to the true rarefaction waves as $t \to \infty$.
2. Decomposition and Perturbation Analysis: The solution $(n, q)$ is decomposed into the sum of the smooth approximation $(N, Q)$ and a perturbation $(\phi, \psi)$ (or $(\Phi, \Psi)$ for the composite case). The perturbation equations are derived, revealing a coupled parabolic-hyperbolic structure.
3. Key Observations for Energy Estimates:
   • Monotonicity: A crucial observation is the monotonicity of the approximate rarefaction waves with respect to both space ($x$) and time ($t$). Specifically, $\partial_x N < 0$, $\partial_x Q < 0$, $\partial_t N < 0$, and $\partial_t Q < 0$. This provides "good" signs in the basic energy estimates, allowing the authors to control certain terms without requiring small wave amplitudes.
   • Derivative Replacement: A significant challenge arises because the hyperbolic component of the perturbation ($\psi$) satisfies a first-order equation ($\psi_t - \phi_x = 0$), making the direct estimation of $\psi_x$ difficult. The authors overcome this by exploiting the specific coupling structure of the system. They replace the spatial derivative $\psi_x$ with terms involving the parabolic component $\phi$ in the $L^2$ and $H^1$ estimates. This allows them to utilize the strong dissipation inherent in the parabolic equation for $\phi$ to close the a priori estimates.
4. Handling Degenerate Viscosity: Unlike previous works on systems with positive definite viscosity matrices (e.g., Yang and Zhao [32]), the system (1.1) exhibits partial viscosity, resulting in a degenerate viscosity coefficient matrix. The authors demonstrate that their specific energy estimates and the monotonicity properties are sufficient to handle this degeneracy without relying on the theories designed for strictly positive definite viscosity.

Key Contributions and Results
The paper establishes two main theorems regarding the global existence and asymptotic stability of solutions:

• Theorem 1.1 (Single Rarefaction Wave): For any fixed state $(n_-, q_-)$ with $n_- > 0$, if the initial data is a sufficiently small perturbation (in the $H^1$ norm) of a smooth approximation of a single rarefaction wave, the system admits a unique global solution. This solution converges asymptotically to the rarefaction wave as $t \to \infty$ in the $L^\infty$ norm.
• Theorem 1.2 (Composite Rarefaction Wave): Similarly, for the superposition of two rarefaction waves, if the initial data is a small perturbation of the composite wave approximation, the system possesses a unique global solution that converges asymptotically to the composite rarefaction wave.

Significance and Claims
The authors claim that their results represent a significant advancement in the understanding of chemotaxis models and parabolic-hyperbolic systems:

• Extension to Strong Waves: While previous stability theories for rarefaction waves in viscous conservation laws (e.g., Matsumura and Nishihara [24, 25]) often required weak wave amplitudes or specific constraints on the adiabatic constant, this work proves stability for strong rarefaction waves with arbitrary amplitudes.
• General Perturbations: The results hold for general perturbations, overcoming the limitation of earlier studies on this specific chemotaxis model that were restricted to zero-mass perturbations (which rely on the anti-derivative approach).
• Overcoming Degeneracy: The paper successfully adapts the energy method to a system with degenerate viscosity, a case where existing general stability theories for strong waves (such as those by Yang and Zhao [32]) are not directly applicable.
• Biological Implication: The authors note that the results imply that if the initial bacterial distribution is close to a rarefaction wave profile, the distribution will asymptotically evolve into a sparse rarefaction wave structure over time.

The proofs rely entirely on rigorous a priori energy estimates, utilizing the monotonicity of the wave profiles and the specific cancellation mechanisms within the perturbation equations to close the estimates without smallness assumptions on the wave strength.