Nonlocalised damping estimates for hyperbolic relaxation systems in one space dimensions

1. Problem Statement

The paper addresses the stability theory of shock profiles for general nonlinear hyperbolic relaxation systems in one space dimension. The system is modeled by the equation:
$$U_t + A(U)U_x = q(U)$$
where $U \in \mathbb{R}^N$, $A(U)$ is the flux Jacobian, and $q(U)$ is a source term.

The specific goal is to establish damping estimates for perturbations around a stationary shock profile $\bar{U}(x)$. These estimates are crucial for closing nonlinear stability arguments, particularly for nonlocalised perturbations (perturbations that do not necessarily decay rapidly at infinity) and in the $L^\infty$ setting (supremum norms), extending previous work that was limited to $L^2$ or symmetric settings.

Key Assumptions:

1. Existence of a Profile: A smooth stationary solution $\bar{U}(x)$ exists, connecting two end-states $U^\pm$ at exponential rates.
2. Strict Hyperbolicity & Non-characteristicity: The matrix $A(U)$ has distinct real eigenvalues $\lambda_j$ bounded away from zero along the profile.
3. High-Frequency Stability: At the end-states, the linearized operator satisfies a spectral gap condition (Shizuta-Kawashima condition) ensuring dissipation at high frequencies, without requiring symmetry assumptions on the source term Jacobian.

2. Methodology

The author employs a combination of the method of characteristics and a commutator argument inspired by Fourier analysis. The approach is structured in three main stages:

A. Linearization and Diagonalization

The perturbation $U(t,x) = \tilde{U}(t, x+\delta(t)) - \bar{U}(x)$ is introduced, where $\delta(t)$ is a dynamic phase shift (shock location). The equations are linearized and transformed using left and right eigenvectors ($L(U)$ and $R(U)$) of the flux Jacobian $A(U)$.

• This diagonalizes the principal part of the operator, converting the system into characteristic variables $\Phi$ (for $U$) and $\Psi$ (for $U_x$).
• The source term matrix $Q(U)$ is decomposed into a diagonal dissipative part $E(U)$ and an off-diagonal part $F(U)$.

B. The Method of Characteristics (for $L^\infty$ estimates)

For the $C^k_b$ (bounded $C^k$) estimates, the author integrates the characteristic equations along paths $X_j(t)$.

• Damping Mechanism: The diagonal entries of the source matrix $E_{jj}$ are negative (dissipative) near the end-states. However, near the shock core, damping may be weak.
• Key Insight: Due to the non-characteristic assumption ($|\lambda_j| \geq c > 0$), characteristics traverse the "non-damping" region (the shock core) in finite time. They spend the majority of their time in the regions where the profile decays to end-states, where strong damping exists. This allows for an overall exponential decay estimate despite local non-damping.

C. Commutator Argument and Variable Transformation

A major technical hurdle is the coupling between characteristic variables caused by the off-diagonal terms ($F(U)$) in the source matrix.

• The Transformation: To decouple the system, the author introduces a new variable $\tilde{\Psi} = \Psi + \Theta \Phi$, where $\Theta$ is a matrix constructed to solve a commutator equation:
  $$[\Theta, \Lambda] = \tilde{F}$$
  Here, $\Lambda$ is the diagonal matrix of eigenvalues, and $\tilde{F}$ represents the off-diagonal forcing.
• Effect: This transformation eliminates the linear off-diagonal terms in the equation for $\tilde{\Psi}$, effectively "slaving" the derivative variable $\Psi$ to the primary variable $\Phi$ and the phase shift $\dot{\delta}$. This allows the derivation of a closed damping estimate for $\Psi$ that depends only on initial data and lower-order terms, not on $\Psi$ itself.

D. Extension to Higher Derivatives and $L^2$

• Higher Derivatives: The method is iterated for higher derivatives ($U_{xx}, \dots$). The structure of the equations ensures that higher derivatives do not introduce new nonlinearities that break the damping estimates.
• $L^2$ Estimates: The paper recovers $L^2$ damping estimates (recovering results from [7]) without symmetry assumptions. This is achieved using weighted $L^2$ inner products with carefully chosen weights $\alpha_j(x)$ that decay exponentially, balancing the transport and dissipation terms.

3. Key Contributions

1. Extension to $L^\infty$ and Non-Symmetric Settings: The paper generalizes previous damping estimates from the $L^2$ case (and symmetric systems) to the $L^\infty$ case and non-symmetric source terms. This is a significant step toward a general stability theory.
2. Non-Localised Perturbations: The estimates hold for perturbations that are not necessarily localized, provided they are small in the $C^1_b$ norm. This is essential for physical applications where perturbations may have non-decaying tails.
3. Dynamic Phase Shift Handling: The analysis rigorously incorporates a time-dependent phase shift $\delta(t)$, which is necessary for defining orbital stability (convergence to a shifted profile rather than the original profile).
4. Commutator-Based Decoupling: The explicit construction of the variable transformation $\Theta$ to eliminate off-diagonal terms provides a robust algebraic tool for handling general hyperbolic relaxation systems without relying on Fourier multipliers in the physical space derivation.

4. Main Results

The central result is Theorem 1, which establishes the following damping estimate for the perturbation $U$ in the $C^k_b$ norm:

$$\|U(t)\|_{C^k_b} \leq C e^{-\theta t} \|U(0)\|_{C^k_b} + C \int_0^t e^{-\theta(t-s)} \left( \|U(s)\|_{C^0_b} + |\dot{\delta}(s)| \right) ds$$

Implications:

• Exponential Decay: The perturbation decays exponentially in time, modulo the lower-order terms and the phase shift speed.
• Orbital Stability: Combined with the existence of a phase shift $\delta(t)$, this implies that the solution converges to a dynamically shifted shock profile $\bar{U}(x - \delta(t))$.
• Closure of Nonlinear Arguments: These estimates provide the necessary "a priori" bounds to close nonlinear stability arguments for shock profiles in 1D, as referenced in the context of detonations and subshocks.

5. Significance

This work bridges a critical gap in the stability theory of hyperbolic relaxation systems. Previous results often relied on:

• Symmetry assumptions (which are not physically general).
• $L^2$ frameworks (which are insufficient for certain nonlinear stability arguments requiring $L^\infty$ control).
• Localized perturbations.

By removing the symmetry requirement and extending the framework to $L^\infty$ and non-localized perturbations, Bärlin lays the groundwork for a general stability theory for shock profiles in 1D. This is particularly relevant for complex physical systems like viscous conservation laws, gas dynamics, and combustion models (detonations), where the source terms are inherently non-symmetric and perturbations may be broad. The methodology of using commutators to decouple characteristic variables in physical space offers a powerful alternative to purely spectral methods.