Anthony Suen

Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms

We study non-negative solutions to the chemotaxis system under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Ω × [0, ∞)2 with values in [0, ∞) and ℝ2×2, respectively, which are assumed to satisfy certain regularity assumptions and growth restrictions. Systems of type (⋆), in the special case reducing to a version of the standard Keller–Segel system with signal consumption, have recently been proposed as a model for swimming bacteria near a surface, with the sensitivity tensor then given by , reflecting rotational chemotactic motion. It is shown that for any choice of suitably regular initial data (u0, v0) fulfilling a smallness condition on the norm of v0 in L∞(Ω), the corresponding initial-boundary value problem associated with (⋆) possesses a globally defined classical solution which is bounded. This result is achieved through the derivation of a series of a priori estimates involving an interpolation inequality of Gagliardo–Nirenberg type which appears to be new in this context. It is next proved that all corresponding solutions approach a spatially homogeneous steady state of the form (u, v) ≡ (μ, κ) in the large time limit, with μ := fΩu0 and some κ ≥ 0. A mild additional assumption on the positivity of f is shown to guarantee that κ = 0. Finally, numerical solutions are presented which suggest the occurrence of wave-like solution behavior.

Hölder Continuity of Solutions to the Kinematic Dynamo Equations

We study the propagation of regularity of solutions to a three dimensional system of linear parabolic PDE known as the kinematic dynamo equations. The divergence free drift velocity is assumed to be at the critical regularity level with respect to the natural scaling of the equations.

Existence and uniqueness of small energy weak solution to multi-dimensional compressible Navier-Stokes equations with large external potential force

We study the 3-D compressible Navier-Stokes equations with an external potential force and a general non-decreasing pressure. We prove the global-in-time existence of weak solutions with small-energy initial data and with densities being non-negative and essentially bounded. A solution may have large oscillations and contain vacuum states. No smallness assumption is made on the external force nor the initial perturbation in L∞ for density. Initial velocity u0 is taken to be bounded in Lq for some q > 6 and no further regularity assumption is imposed on u0. Finally, we discuss the uniqueness of weak solutions.