Lie symmetry analysis of a 2+1-dimensional flux limited Keller Segel system

Technical Summary: Lie Symmetry Analysis of a (2+1)-Dimensional Flux-Limited Keller–Segel System

Problem Statement
The paper investigates the two-dimensional flux-limited Keller–Segel (FLKS) system, a mathematical model describing chemotactic aggregation where cell velocities are bounded by a nonlinear flux-limiting mechanism. The system is governed by the coupled partial differential equations:
$$u_t = \nabla \cdot \left( \nabla u - \frac{\chi u \nabla v}{\sqrt{1 + |\nabla v|^2}} \right), \quad v_t = \Delta v + u - v,$$
where $u$ and $v$ represent cell and chemoattractant concentrations, respectively, and $\chi > 0$ is the chemotactic sensitivity. While the classical Keller–Segel model is known to exhibit finite-time blow-up and unbounded accumulation, the FLKS model introduces a flux limiter $\Phi(\nabla v) = \nabla v / \sqrt{1 + |\nabla v|^2}$ to ensure bounded transport speeds and more realistic biological behavior. However, this modification introduces a complex non-polynomial nonlinear structure that complicates the construction of analytical solutions. The primary objective of this work is to perform a complete Lie point symmetry analysis to determine the continuous transformation groups admitted by the system and to derive invariant solutions and similarity reductions.

Methodology
The authors employ classical Lie group analysis. The methodology proceeds in three main stages:

1. Symmetry Determination: An infinitesimal generator $X$ is introduced, and its second prolongation is applied to the system. The invariance conditions are solved by splitting the resulting determining equations with respect to the independent derivatives of $u$ and $v$.
2. Algebraic Classification: The admitted Lie algebra is identified, and an optimal system of one-dimensional subalgebras is constructed using the adjoint representation. This involves calculating the commutator table and applying adjoint actions to classify inequivalent subalgebras up to scaling and conjugation.
3. Similarity Reductions and Solution Construction: For each representative of the optimal system, invariant surface conditions are solved to reduce the original (2+1)-dimensional system to lower-dimensional systems (PDEs or ODEs). The reduced equations are then analyzed for exact solutions, integral representations, and special cases (e.g., zero chemotactic sensitivity).

Key Contributions and Results

• Lie Symmetry Algebra: The analysis reveals that the flux-limiting mechanism fundamentally alters the symmetry structure compared to the classical Keller–Segel model. The admitted Lie algebra is four-dimensional, spanned by:

  • $X_1 = \partial_t$ (Temporal translation)
  • $X_2 = \partial_x$ (Spatial translation in $x$)
  • $X_3 = \partial_y$ (Spatial translation in $y$)
  • $X_4 = y\partial_x - x\partial_y$ (Planar rotation)
    Crucially, the non-polynomial structure of the flux limiter breaks the classical scaling symmetries and eliminates any $u$ or $v$-dependent transformations. The algebra consists solely of space-time translations and rotations.

• Optimal System of Subalgebras: The authors construct a complete optimal system of five inequivalent one-dimensional subalgebras:

  1. $\langle \partial_t \rangle$: Leads to stationary (steady-state) reductions.
  2. $\langle \partial_x \rangle$: Leads to translationally invariant reductions (1+1D system in $y, t$).
  3. $\langle y\partial_x - x\partial_y \rangle$: Leads to radially symmetric reductions.
  4. $\langle \partial_t + c\partial_x \rangle$: Leads to travelling-wave reductions.
  5. $\langle \partial_t + \mu(y\partial_x - x\partial_y) \rangle$: Leads to rotating-wave reductions (spiral patterns).

• Reduced Equations and Solutions:

  • Stationary and Radial Solutions: The stationary reduction yields an elliptic system. The authors identify a spatially homogeneous equilibrium solution ($u=v=U_0$). For radial stationary solutions satisfying a zero-flux condition at the origin, an integral representation is derived: $U(r) = K \exp(\chi \text{arsinh}(V_r(r)))$, reducing the problem to a single ODE for $V(r)$.
  • Travelling Waves: For the travelling-wave reduction, the system reduces to a pair of coupled ODEs. In the absence of chemotaxis ($\chi=0$), exact travelling-wave solutions are found in terms of exponential functions. For the general case ($\chi \neq 0$), an integral representation for the cell density $U(\xi)$ is derived in terms of the chemoattractant gradient $V'(\xi)$, assuming zero flux at infinity.
  • Rotating Waves: The rotating-wave reduction yields a system describing spiral patterns. While the homogeneous equilibrium is an exact solution, the authors note that constructing nontrivial rotating patterns with angular dependence remains an open problem likely requiring numerical methods.

Significance and Claims
The paper claims that the flux-limiting mechanism significantly restricts the class of admissible invariant solutions by eliminating the scaling symmetries present in the standard Keller–Segel model. The work establishes that the remaining symmetries (translations and rotations) reflect the homogeneous and isotropic nature of the underlying biological mechanism.

The authors present the derived similarity reductions as a foundation for further analytical investigation, providing lower-dimensional models that facilitate the study of chemotactic dynamics and pattern formation. The results include the determination of exact spatially homogeneous equilibria and the derivation of integral representations for stationary radial and travelling-wave profiles. The paper concludes that while explicit closed-form solutions for the general nonlinear case are difficult to obtain, the symmetry analysis provides a systematic framework for identifying solution structures and suggests directions for future work, including numerical analysis of the reduced systems and the investigation of nonlocal symmetries and conservation laws.