Segregated solutions of a degenerate cross-diffusion system with drifts

Technical Summary: Segregated Solutions of a Degenerate Cross-Diffusion System with Drifts

Problem Statement
This paper investigates the global existence of segregated weak solutions for a one-dimensional degenerate cross-diffusion system with independent drifts. The system is given by:
$$\begin{cases} \partial_t \varrho = \partial_x \left( \varrho \partial_x (f'(\varrho + \mu) + V) \right), \\ \partial_t \mu = \partial_x \left( \mu \partial_x (f'(\varrho + \mu) + W) \right), \end{cases}$$
where $\varrho$ and $\mu$ represent the densities of two competing species, $V$ and $W$ are given Lipschitz drift potentials, and $f$ is a convex function governing the interaction based on the total mass $S = \varrho + \mu$. The primary objective is to prove the existence of solutions where the species remain segregated for all time, meaning their supports are disjoint ($\varrho(x)\mu(x) = 0$ almost everywhere), starting from segregated initial data.

The system is endowed with a Wasserstein gradient flow structure associated with the energy functional:
$$E[\varrho, \mu] = \int_{\mathbb{R}} f(\varrho + \mu) \, dx + \int_{\mathbb{R}} (\varrho V + \mu W) \, dx.$$
The authors address the challenge that standard parabolic theory fails due to the lack of maximum principles and the formation of sharp interfaces (segregation boundaries). Furthermore, existing literature on this specific system with drifts has relied on restrictive assumptions regarding the ordering of potentials or initial configurations, or has focused on "totally mixed" solutions.

Methodology
The core strategy involves a Lagrangian reformulation of the problem. Instead of analyzing the Eulerian densities $\varrho$ and $\mu$ directly, the authors utilize the cumulative mass function of the total density, $F_t(x) = \int_{-\infty}^x (\varrho + \mu) \, dy$, and its (pseudo-)inverse $u_t = F_t^{-1}$.

1. Lagrangian Reformulation: By assuming a specific segregated structure where the supports of $\varrho$ and $\mu$ are determined by a fixed measurable set $A \subset [0, 2]$ in the Lagrangian coordinate space (via $F_t$), the system collapses into a single scalar equation for the inverse function $u_t$. This equation takes the form of a parabolic PDE:
   $$\partial_t u = \partial_y (\tilde{f}'(u')) - b(y, u),$$
   where $\tilde{f}(s) = s f(1/s)$ and $b$ is a drift term derived from $V$ and $W$. For $f(s) \sim s^\alpha$, this resembles a $p$-Laplace equation with $p = 1-\alpha \in (-\infty, 1)$.

2. Variational Approach: The authors solve this Lagrangian equation using a Minimising Movement Scheme (implicit Euler scheme) in the setting of $L^2 \cap BV_{loc}$. They define an energy functional $\mathcal{E}$ on the space of functions with non-decreasing derivatives and construct a sequence of minimizers $\{v_k\}$ for discrete time steps $\tau$.

3. Analytical Tools:

   • Monotonicity Methods: The proof relies heavily on Minty–Browder monotonicity arguments to establish strong convergence of the discrete scheme as the time step $\tau \to 0$.
   • BV and $L^2$ Estimates: The authors derive uniform bounds in $L^2$ and $BV_{loc}$ to ensure compactness. A crucial step involves proving that the absolutely continuous part of the derivative, $u'$, remains strictly positive ($u' > 0$), which prevents the collapse of the Lagrangian map and ensures the validity of the inverse transformation.
   • Handling Singularities: The analysis carefully treats the singular part of the derivative measure $(\partial_y u)_s$, showing that the term $\tilde{f}'(u')$ vanishes on the support of the singular part, effectively decoupling the singular and regular components in the limit.

Key Contributions and Results
The paper establishes the following main results:

• Global Existence of Segregated Solutions: The authors prove the existence of a global weak solution $(\varrho_t, \mu_t)$ to the cross-diffusion system (1.1) that preserves the segregation property from the initial data. This holds for arbitrary segregated initial configurations and general Lipschitz drifts, without requiring the drifts to favor segregation or the masses to be strictly ordered (e.g., $\varrho$ entirely to the left of $\mu$).
• Broad Class of Diffusions: The results apply to a wide range of diffusion behaviors, including:
  • Porous medium type ($f(s) = s^\alpha, \alpha > 1$).
  • Log-entropy ($f(s) = s \log s - s$).
  • Fast diffusion ($f(s) = -s^\alpha$) for indices $\alpha \in (1/3, 1)$.
• Lagrangian Well-Posedness: The paper provides a rigorous existence theory for the associated Lagrangian problem (Theorem 3.6), characterizing the solution $u$ in the space $L^\infty_{loc}([0, \infty); X) \cap H^1_{loc}([0, \infty); L^2)$, where $X = L^2 \cap BV_{loc}$.
• Novelty in Gradient Flow Structure: The work demonstrates that the Lagrangian reformulation yields a parabolic problem with a $p$-Laplace-like structure where $p < 1$, solved via a gradient flow in $L^2 \cap BV_{loc}$. This approach is distinct from previous works that utilized ratio-based formulations or compensated compactness.

Significance and Claims
The authors position this work as a significant advancement in the theory of cross-diffusion systems, specifically regarding the preservation of segregation.

• Comparison with Prior Work: The paper notes that while previous results (e.g., [31, 46, 47, 51]) established global existence for various initial data and diffusion types, they often focused on "totally mixed" solutions or required restrictive assumptions on the initial ordering of species. The work of [42] specifically studied segregated solutions but imposed very restrictive conditions on the potentials and initial data.
• Removal of Restrictions: This paper claims to remove these restrictions, allowing for arbitrary segregated initial data and general drifts. The authors state that their result is "much stronger" than [42] in this regard.
• Methodological Contribution: The authors emphasize that their contribution is not merely to obtain the most general existence result possible, but to study the specific segregation dynamics using variational techniques. The Lagrangian formulation and the use of monotonicity methods for the resulting $p$-Laplace-like equation ($p < 1$) are presented as novel technical achievements that complement existing literature.

In summary, the paper provides a robust existence theory for segregated solutions in a degenerate cross-diffusion system by transforming the problem into a Lagrangian framework, solving it via a minimizing movement scheme, and rigorously passing to the limit using monotonicity and compactness arguments in low-regularity spaces.