An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions

Abstract

Using the method of paracontrolled distributions, we show the local well-posedness of an additive noise approximation to the fluctuating hydrodynamics of the Keller-Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green's function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

arXiv comment60 pages; added new subsection on maximal time of existence

References in corpus (11)

• A theory of regularity structures
• Langevin Equation for the Density of a System of Interacting Langevin Processes
• KPZ reloaded
• The dynamic ${\Phi }_3^4$ model comes down from infinity
• Langevin dynamic for the 2D Yang-Mills measure
• Singular SPDEs in domains with boundaries
• On Dean-Kawasaki Dynamics with Smooth Drift Potential
• Nonequilibrium polarity-induced mechanism for chemotaxis: emergent Galilean symmetry and exact scaling exponents
• Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions
• An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions
• Corrigendum to "Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions"

Cited by in corpus (2)

• An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions
• Weak Error of Dean-Kawasaki Equation with Smooth Mean-Field Interactions