具有局部平均Brinkman力的可压缩动理学-流体系统的有限能量弱解与松弛
Finite-energy weak solutions and relaxation for a compressible kinetic--fluid system with locally averaged Brinkman force
AI总结：
本文研究带局部平均Brinkman力的可压缩动理学-流体系统，证明其正则解的条件指数松弛估计，及低维下带密度依赖粘度的Bresch-Desjardins熵弱解的全局有限能量性与指数松弛性质。
AI中文摘要：
我们研究一种动理学-流体系统，其中Vlasov或Vlasov-Fokker-Planck方程与具有密度依赖粘度的可压缩Navier-Stokes方程通过局部平均Brinkman力耦合。该平均以守恒形式选取，使得耦合系统保持总动量并满足自然的能量耗散平衡，同时避免对可能粗糙的流体速度进行逐点评估。首个主要结果是针对足够正则解的条件指数松弛估计，该估计在正、负密度矩界以及密度幂次的Muckenhoupt $\boldsymbol{\frak A}_2$条件下成立，替代了一致逐点上下界。证明结合了调制能量、hypocoercivity分析以及密度涨落的补偿泛函，关键要素是相关椭圆校正子的加权Calderón-Zygmund估计，其允许我们在$\boldsymbol{\frak A}_2$条件下控制粘性贡献。第二个主要结果涉及可容许低维区域中的全局有限能量Bresch-Desjardins熵弱解，利用与密度依赖粘度相关的额外熵结构，我们验证了条件松弛定理所需的密度假设，并得到弱解的指数松弛：无扩散情形下粒子分布趋向单动力学态，有扩散情形下趋向麦克斯韦平衡态。
英文摘要：
We study a kinetic--fluid system in which a Vlasov or Vlasov--Fokker--Planck equation is coupled to the compressible Navier--Stokes equations with density-dependent viscosities through a locally averaged Brinkman force. The averaging is chosen in a conservative form so that the coupled system preserves the total momentum and satisfies a natural energy-dissipation balance, while avoiding the pointwise evaluation of a possibly rough fluid velocity. The first main result is a conditional exponential relaxation estimate for sufficiently regular solutions. The estimate is proved under positive and negative density moment bounds and a Muckenhoupt $\calA_2$ condition on a power of the density, which replace uniform pointwise upper and lower bounds. The proof combines a modulated energy and hypocoercivity analysis with a compensating functional for the density fluctuation. A key ingredient is a weighted Calderón--Zygmund estimate for the associated elliptic corrector, which allows us to control the viscous contribution under the $\calA_2$ condition. The second main result concerns global finite-energy Bresch--Desjardins entropy weak solutions in admissible low-dimensional regimes. Using the additional entropy structure associated with the density-dependent viscosities, we verify the density assumptions required by the conditional relaxation theorem and obtain exponential relaxation for the weak solutions. In the diffusionless case, the particle distribution aligns toward a mono-kinetic state, while in the diffusive case the relaxation is towards a Maxwellian equilibrium.
