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A Nash stratification inequality and global regularity for a chemotaxis-fluid system on general 2D domains

Alexander Kiselev, Naji A. Sarsam

Abstract

Incompressible fluid advection has been shown to facilitate singularity suppression in various differential equations, often by mixing-enhanced diffusion or by dimension-reduction effects. To aid with the study of such scenarios, we prove a refinement of the classical Nash inequality, $$ \|f-f_M\|_{L^2}^2 \lesssim \|f-f_M\|_{L^1}^{8/7}\|\nabla f\|_{L^2}^{6/7} + \|\partial_1 f\|_{\dot{H}_0^{-1}}^{2\vartheta} \|f-f_M\|_{L^1}^{1-\vartheta}\|\nabla f\|_{L^2}^{1-\vartheta}, $$ for $f \in H^1$ with mean $f_M$ over a smooth bounded planar domain under the main constraint of having connected horizontal cross-sections. The first term on the right-hand side follows the classical Nash scaling for a formal dimension of $3/2$. The second term introduces a mixing norm that measures how far $f$ is from being stratified. The proof provides an explicit exponent $0 < \vartheta \ll 1$. As an application, we study the 2D parabolic-elliptic Patlak--Keller--Segel (PKS) chemotaxis model over the aforementioned large class of bounded domains. This aggregation-diffusion equation is well-known to produce finite-time singularity formation for large-mass data in finite domains. Using the above Nash stratification inequality, we prove that the 2D PKS equation becomes globally regular when actively coupled via buoyancy to a fluid obeying Darcy's law for incompressible porous media flows. This result holds for arbitrarily large $C^\infty$ initial data, far from perturbative regimes, and for arbitrarily weak coupling strength. Moreover, the spatial domain can have "bottle neck" regions and boundary segments of large curvature. The argument conceptualizes and generalizes recent work by Hu, Yao, and the first author on the analogous result for the periodic channel.

A Nash stratification inequality and global regularity for a chemotaxis-fluid system on general 2D domains

Abstract

Incompressible fluid advection has been shown to facilitate singularity suppression in various differential equations, often by mixing-enhanced diffusion or by dimension-reduction effects. To aid with the study of such scenarios, we prove a refinement of the classical Nash inequality, for with mean over a smooth bounded planar domain under the main constraint of having connected horizontal cross-sections. The first term on the right-hand side follows the classical Nash scaling for a formal dimension of . The second term introduces a mixing norm that measures how far is from being stratified. The proof provides an explicit exponent . As an application, we study the 2D parabolic-elliptic Patlak--Keller--Segel (PKS) chemotaxis model over the aforementioned large class of bounded domains. This aggregation-diffusion equation is well-known to produce finite-time singularity formation for large-mass data in finite domains. Using the above Nash stratification inequality, we prove that the 2D PKS equation becomes globally regular when actively coupled via buoyancy to a fluid obeying Darcy's law for incompressible porous media flows. This result holds for arbitrarily large initial data, far from perturbative regimes, and for arbitrarily weak coupling strength. Moreover, the spatial domain can have "bottle neck" regions and boundary segments of large curvature. The argument conceptualizes and generalizes recent work by Hu, Yao, and the first author on the analogous result for the periodic channel.
Paper Structure (26 sections, 47 theorems, 330 equations, 7 figures)

This paper contains 26 sections, 47 theorems, 330 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Previous literature on the PKS equation
  3. Previous literature on PKS-fluid systems
  4. The PKS-IPM system
  5. Main results and organization
  6. Setting and preliminaries
  7. Domain assumptions and normalizations
  8. Function decomposition into stratified and unstratified components
  9. Function spaces and notation
  10. Anisotropic Nash inequalities that quantify stratification
  11. Nash inequality on the stratified component
  12. Proof of theorem and remarks
  13. Proofs of auxiliary propositions
  14. Nash inequality on the unstratified component
  15. Proof of theorem and remarks
  16. ...and 11 more sections

Key Result

Theorem 1.1

Fix a domain $\Omega \subseteq \mathbb{R}^2$ satisfying as:secSett_1as:secSett_2as:secSett_3, as stated in subsec:Domain, and fix $f \in C^\infty(\overline{\Omega})$. Then, where $C_0 = C_0(\Omega) > 0$ is a universal constant. It immediately follows that by the classical Nash inequality (eq:secIntro_Nash), where $C_1 = C_1(\Omega) > 0$.

Figures (7)

  • Figure 1: Cartoon of how regularization of the solution to (\ref{['eq:secIntro_PKSIPM']}) occurs on the periodic channel $\mathbb{T} \times (0, \pi)$ whenever the mass begins to concentrate in the form of an approximate two-dimensional Dirac delta.
  • Figure 2: Potential scenarios of chemotactic singularity formation in a domain with non-flat boundary geometry.
  • Figure 3: Cartoon depiction of images of $I_{x_2'} \times \{x_2'\}$ under $\Phi_t$, with the arrows representing $\partial_t \Phi_t$. This construction of $\Phi_t$ preserves the length of $I_{x_2'} \times \{x_2'\}$.
  • Figure 4: Cartoon depiction of images of $I_{x_2'} \times \{x_2'\}$ under $\Phi_t$, with the arrows representing $\partial_t \Phi_t$. This construction of $\Phi_t$ stretches and shrinks the length of $I_{x_2'} \times \{x_2'\}$.
  • Figure 5: Cartoon depiction of $\partial_1 \Phi$.
  • ...and 2 more figures

Theorems & Definitions (110)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 100 more