Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows

1. Problem Statement

The authors address the Cauchy problem for a specific class of non-symmetric Keyfitz–Kranzer systems of conservation laws, which model the dynamics of thin-film flows under the influence of soluble surfactants. The system is given by:
$$\begin{aligned} u_t + (u \phi(r))_x &= 0 \\ v_t + (v \phi(r))_x &= 0 \end{aligned}$$
where $r(u, v) = uv$ is the coupling function, $\phi$ is a smooth function with $\phi' > 0$, and the state space is restricted to $U_m = \{(u, v) \in \mathbb{R}^2 \mid u, v \ge m > 0\}$.

The Core Challenge:
While Keyfitz–Kranzer systems are well-studied, this specific class (where the coupling is multiplicative $uv$) presents unique difficulties:

1. Failure of Standard Approximation: The standard vanishing viscosity approach using a linear diffusion matrix $B(U) = I$ (i.e., adding $\epsilon u_{xx}$ and $\epsilon v_{xx}$) fails. Specifically, the maximum principle cannot be applied to the Riemann invariants, and the standard entropy pairs do not satisfy the necessary dissipation inequalities for this linear approximation.
2. Non-Temple Class: Although the shock and rarefaction curves coincide, the system does not belong to the Temple class (where curves are straight lines in phase space). Consequently, classical $L^1$-theory for bounded variation (BV) data cannot be directly applied to guarantee global existence for $L^\infty$ initial data.
3. Goal: Prove the global existence of weak entropy solutions for $L^\infty$ initial data by constructing a tailored approximation that preserves the system's intrinsic entropy structure.

2. Methodology

The authors employ a vanishing diffusion limit strategy, but with a crucial modification: instead of using standard linear diffusion, they design a nonlinear, structure-adapted approximation inspired by lubrication theory.

A. Analysis of the Hyperbolic System

• Hyperbolicity: The system is strictly hyperbolic in $U_m$ with eigenvalues $\lambda_1 = \phi(r)$ and $\lambda_2 = \phi(r) + 2r\phi'(r)$.
• Riemann Invariants: The authors explicitly identify the Riemann invariants:
  • $w_1 = r = uv$ (associated with the linearly degenerate field $\lambda_1$).
  • $w_2 = \xi = u/v$ (associated with the genuinely nonlinear field $\lambda_2$).
• Entropy Pairs: They construct a family of entropy/entropy-flux pairs $(E, Q)$ of the form:
  $$E(u, v) = \Psi(r) + \sqrt{r}\Theta(\xi) + K$$
  where $\Psi$ and $\Theta$ satisfy specific convexity conditions. A specific family of strictly convex entropies is identified:
  $$E_{k,p}(u, v) = \frac{1}{(uv)^k} + \sqrt{uv}\left(\frac{u}{v}\right)^p - C$$
  for $k > 0, p \ge 1/2$.

B. The Novel Approximation Scheme

Recognizing that $B=I$ fails, the authors propose a nonlinear approximation motivated by thin-film lubrication equations (specifically involving anti-surfactants):
$$\begin{aligned} u^\epsilon_t + (u^\epsilon \phi(r^\epsilon))_x &= \epsilon \left( \frac{(u^\epsilon v^\epsilon)_x}{v^\epsilon} \right)_x \\ v^\epsilon_t + (v^\epsilon \phi(r^\epsilon))_x &= \epsilon \left( \frac{(u^\epsilon v^\epsilon)_x}{u^\epsilon} \right)_x \end{aligned}$$
This system corresponds to a diffusion matrix $B(U) = \begin{pmatrix} 1/v & u/v \\ v/u & 1/u \end{pmatrix}$ (in a transformed sense), which is specifically tailored to the system's geometry.

C. Key Analytical Steps

1. Entropy Dissipation: They prove that the proposed approximation dissipates the identified entropy pairs $E_{k,p}$. This yields an a-priori bound on the spatial derivative of the product $r^\epsilon = u^\epsilon v^\epsilon$.
2. Invariant Region & Maximum Principle: By transforming the system into the Riemann invariants $(r^\epsilon, \xi^\epsilon)$, they derive a scalar parabolic equation for $\theta^\epsilon = \sqrt{r^\epsilon}$ and a transport equation for $\xi^\epsilon$.
   • The parabolic equation for $\theta^\epsilon$ allows the application of the parabolic maximum principle, establishing that if initial data lies in $[m, M]^2$, the solution remains in this bounded invariant region for all time.
   • This provides uniform $L^\infty$ bounds for $u^\epsilon$ and $v^\epsilon$.
3. Compactness (Vanishing Diffusion Limit):
   • Using the entropy dissipation and the $L^\infty$ bounds, they apply compensated compactness theory.
   • They show that the sequence of Riemann invariants $(r^\epsilon, \xi^\epsilon)$ converges strongly (pointwise a.e.) to a limit $(r, \xi)$.
   • The strong convergence of the invariants implies the strong convergence of the original variables $(u^\epsilon, v^\epsilon) \to (u, v)$.
4. Limit Passage: They rigorously pass to the limit $\epsilon \to 0$ in the weak formulation, proving that the limit function satisfies the entropy inequality for the original hyperbolic system.

3. Key Contributions

1. Identification of Entropy Structure: The paper explicitly constructs a family of entropy/entropy-flux pairs for a non-symmetric Keyfitz–Kranzer system with multiplicative coupling, which were previously unknown or difficult to characterize.
2. Tailored Approximation: The authors demonstrate that the standard linear viscosity ($B=I$) is insufficient for this class of systems. They introduce a nonlinear diffusion operator derived from physical thin-film models that respects the system's entropy structure and allows for the application of the maximum principle.
3. Global Existence for $L^\infty$ Data: Unlike many results that require initial data of Bounded Variation (BV), this work establishes global existence for $L^\infty$ initial data (with bounded variation in the ratio $u/v$), significantly broadening the applicability of the theory.
4. Resolution of Non-Temple Behavior: The paper provides a rigorous framework for handling Keyfitz–Kranzer systems that are not in the Temple class, overcoming the failure of the maximum principle on the original variables by working with transformed Riemann invariants.

4. Main Results

• Theorem 4.1 (Main Result): For initial data $(u_0, v_0) \in [m, M]^2$ with $u_0, v_0 \in L^\infty$ and $\int |(\frac{u_0}{v_0})_x| dx < \infty$, there exists a global weak entropy solution to the Cauchy problem for the system (1.4).
• Convergence: The sequence of solutions to the tailored approximate system (1.9) converges strongly (pointwise a.e.) to the unique weak entropy solution of the hyperbolic system as the diffusion parameter $\epsilon \to 0$.
• Regularity: The approximate system admits global classical solutions, and the limit solution satisfies the entropy inequality in the distributional sense.

5. Significance

• Physical Relevance: The results directly apply to thin-film flow models involving soluble surfactants, providing a rigorous mathematical foundation for the existence of solutions in regimes where standard hyperbolic theory fails.
• Methodological Advancement: The work highlights the necessity of system-adapted regularization in hyperbolic conservation laws. It shows that for systems with delicate entropy structures, the choice of the diffusion matrix is not arbitrary but must be "designed" to preserve the entropy dissipation mechanism.
• Future Directions: The authors suggest that the invariant region structure and the analytical framework developed here could be extended to multi-layer thin-film flows (systems with $n > 2$ equations) and could inform the development of entropy-stable numerical schemes that preserve positivity and physical bounds at the discrete level.

In summary, this paper resolves a long-standing open problem regarding the global existence of solutions for a specific, physically motivated class of Keyfitz–Kranzer systems by bridging the gap between physical modeling (lubrication theory) and rigorous mathematical analysis (entropy methods and compensated compactness).