Renjun Duan

Global Solutions to the Coupled Chemotaxis-Fluid Equations

In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier–Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.

OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH POTENTIAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions with zero velocity. By combining the Lp - Lq estimates for the linearized equations and an elaborate energy method, the convergence rates are obtained in various norms for the solution to the stationary profile in the whole space when the initial perturbation of the stationary solution and the potential force are small in some Sobolev norms. More precisely, the optimal convergence rates of the solution and its first order derivatives in L2-norm are obtained when the L1-norm of the perturbation is bounded.

Optimal large‐time behavior of the Vlasov‐Maxwell‐Boltzmann system in the whole space

In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space \input amssym

Optimal Time Decay of the Vlasov–Poisson–Boltzmann System in \({\mathbb R^3}\)

{\Bbb R}^3

THE VLASOV–POISSON–BOLTZMANN SYSTEM FOR SOFT POTENTIALS

. The existence of global-in-time nearby Maxwellian solutions is known from Strain in 2006. However, the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t−3/2 + 3/(2r)) in the L (L)-norm for any 2 ≤ r ≤ ∞ if initial perturbation is smooth enough and decays in space velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to 0 are also provided.

Stability of the One-Species Vlasov–Poisson–Boltzmann System

The Vlasov–Poisson–Boltzmann System governs the time evolution of the distribution function for dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over

The Cauchy Problem on the Compressible Two-fluids Euler–Maxwell Equations

{\mathbb R^3}

Global existence to Boltzmann equation with external force in infinite vacuum

. It is shown that the electric field, which is indeed responsible for the lowest-order part in the energy space, reduces the speed of convergence, hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces; the exact L2-rate for the former is (1 + t)−1/4 while it is (1 + t)−3/4 for the latter. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.