Global existence for a system without self-diffusion and different mobilities

1. Problem Statement

The paper investigates the global existence of weak solutions for a one-dimensional cross-diffusion system describing two interacting populations, $m(t,x)$ and $n(t,x)$, on the flat torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. The system is given by:
$$\begin{cases} \partial_t m - \partial_x (m \partial_x (m + n)) = 0, \\ \partial_t n - \nu \partial_x (n \partial_x (m + n)) = 0, \end{cases}$$
where $\nu > 0$ and $\nu \neq 1$ is a parameter representing the ratio of mobilities.

Key Characteristics and Difficulties:

• Rank-One Diffusion: The diffusion matrix has rank one. The total density $\rho = m + n$ diffuses, but the individual species do not have self-diffusion terms. The second mode is transported but not regularized.
• Different Mobilities: The species move with different mobilities (1 and $\nu$). When $\nu = 1$, the system reduces to a porous medium equation for $\rho$, allowing for standard compactness arguments. The case $\nu \neq 1$ is significantly harder.
• Lack of Compactness: The classical entropy estimate controls $\partial_x \rho$ but not $\partial_x m$ and $\partial_x n$ separately. Consequently, while the total density $\rho$ may converge strongly, the individual densities and their fluxes ($m \partial_x \rho$, $n \partial_x \rho$) may only converge weakly. The primary challenge is identifying the limit of the nonlinear fluxes in the presence of oscillations between the species densities and the pressure gradient.

2. Methodology

The author avoids constructing a specific approximation scheme (like viscous regularization) and instead introduces the concept of an admissible approximation sequence. The proof relies on three main pillars:

A. Change of Variables

The system is reformulated using the total density $\rho = m + n$ and an activity variable $a$:
$$a = \frac{m + \nu n}{m + n} = \frac{m + \nu n}{\rho}.$$
This transformation allows the system to be written as:
$$\partial_t \rho - \partial_x (a \rho \partial_x \rho) = 0,$$
where $a$ acts as a mobility for the diffusive variable $\rho$. The individual species are recovered via $m = \frac{\nu - a}{\nu - 1}\rho$ and $n = \frac{a - 1}{\nu - 1}\rho$. The core difficulty shifts to proving that the weak limit of the product $a_k \partial_x \rho_k$ equals $a \partial_x \rho$.

B. Entropy Estimates and Balance Laws

The author derives a family of balance laws (conservation laws) involving the activity variable $a$.

• Standard Entropy: The system satisfies an entropy inequality $H(m,n) = \int (m \log m + \frac{1}{\nu} n \log n) dx$, which provides $L^2$ bounds on $\partial_x \rho$.
• New Balance Laws: By introducing specific test functions $\phi_s(a)$ and $M_s(a)$ (derived from a polynomial $B(a)$ related to the roots of the mobility), the author constructs an infinite family of balance laws of the form:
  $$\partial_t (\rho^s \phi_s(a)) - \partial_x (M_s(a) \rho^s \partial_x \rho) = (1-s)M_s(a)\rho^{s-1}(\partial_x \rho)^2.$$
  These laws are crucial for applying compensated compactness techniques.

C. Young Measures and Div-Curl Lemma

To handle the oscillations of the sequence $(a_k, \xi_k)$ where $\xi_k = \partial_x \rho_k$, the author employs Young measures $\lambda_{t,x}$.

• Admissibility: The approximation sequence is assumed to satisfy specific boundedness and convergence properties (A1–A6), ensuring the balance laws hold in the limit.
• Div-Curl Lemma: The author applies a consequence of the div-curl lemma (Lemma 3.2) to pairs of vector fields derived from the balance laws. This allows the identification of weak limits of nonlinear products.
• Collapse of the Measure: The main technical achievement is proving that the Young measure $\mu_{t,x}$ associated with the activity variable $a$ is a Dirac mass (i.e., no oscillations) on the set where the pressure gradient is non-zero. This is achieved by:
  1. Using the first div-curl identity to relate the correlation $\langle a \xi \rangle$ to $\langle a \rangle \langle \xi \rangle$.
  2. Using a second div-curl identity involving the specific functions $\phi_s$ and $M_s$ to establish a covariance relation.
  3. Applying Chebyshev's integral inequality to show that if the measure were not a Dirac mass, the covariance relation would lead to a contradiction due to the strict monotonicity of the constructed functions.

3. Key Contributions

1. Generalized Existence Result: Proves global existence of weak solutions for arbitrary bounded non-negative initial data for the Busenberg-Travis system with $\nu \neq 1$, a case previously unresolved for general data.
2. Admissible Approximation Framework: Instead of relying on a specific regularization (like viscosity), the paper defines a class of "admissible" sequences. This makes the result robust, implying that any numerical or analytical scheme preserving the system's structural balance laws will converge to a weak solution.
3. Resolution of Oscillations: Successfully identifies the limit of the nonlinear fluxes $m \partial_x \rho$ and $n \partial_x \rho$ by proving the collapse of the Young measure for the activity variable. This demonstrates that while species densities may oscillate, the specific ratio variable $a$ does not oscillate in the presence of a pressure gradient.
4. New Balance Laws: The construction of the infinite family of balance laws involving $\phi_s(a)$ is a novel contribution specifically designed to facilitate the compensated compactness argument for this rank-one system.

4. Main Results

• Theorem 1.3: For any admissible approximation sequence $(m_k, n_k)$, there exists a subsequence converging to a weak solution $(m, n)$ of the system (1.1).
• Weak Solution Definition: The limit satisfies the integral formulation of the PDEs with initial data $(m_0, n_0)$.
• Flux Identification: The paper rigorously proves that:
  $$\lim_{k \to \infty} (m_k \partial_x \rho_k) = m \partial_x \rho \quad \text{and} \quad \lim_{k \to \infty} (n_k \partial_x \rho_k) = n \partial_x \rho$$
  in the sense of weak convergence in $L^1$.

5. Significance

• Theoretical Impact: This work bridges a gap in the theory of cross-diffusion systems. It demonstrates that even without self-diffusion (which usually provides regularity), global existence can be established in 1D if the system possesses a specific algebraic structure (rank-one diffusion) and if one utilizes the correct variables (activity) and tools (Young measures).
• Methodological Advancement: The combination of entropy methods with a tailored family of balance laws and the div-curl lemma provides a powerful template for analyzing other hyperbolic-parabolic systems with incomplete diffusion.
• Applications: The results are directly relevant to modeling population dynamics, cell motion, and crowd behavior where species compete for space and move with different mobilities, phenomena where sharp interfaces and segregation are observed.
• Future Directions: The author notes that extending this result to higher dimensions or systems with more than two species remains an open problem, suggesting that the 1D structure and the specific properties of the activity variable are critical to the current proof.

In summary, Elbar's paper provides a rigorous mathematical foundation for the global existence of solutions to a challenging class of cross-diffusion systems, overcoming the lack of self-diffusion by exploiting the system's hidden structure through advanced compactness arguments.