Modified compensating functions for the incompressible Euler--Vlasov--Fokker--Planck system: Global classical solutions and pointwise-in-space decay

We consider the Cauchy problem for the incompressible Euler-Vlasov-Fokker-Planck (Euler-VFP) system in the whole space ${\mathbb{R}}^3$ near the global Maxwellian equilibrium. The Fokker-Planck operator and the particle-fluid drag dissipate the relative momentum but do not separately control the common particle-fluid momentum; in Fourier variables, this degeneracy occurs in the transverse momentum components. To recover the missing coercivity, we augment the classical four-moment compensator with a finite-rank skew-adjoint correction constructed from second-order Hermite modes. Combined with the cancellation between the kinetic and fluid drag terms and the incompressibility constraint, the resulting compensated Fourier energy yields a unique global classical solution for sufficiently small initial data $(u_0,f_0)\in H^N\times L_v^2(H^N)$, with $N\ge 4$. The high-order energy argument involves only spatial derivatives of the kinetic perturbation and requires no mixed $x$-$v$ derivative estimates. We further construct a positive-order Lyapunov functional and establish the decay rate $(1+t{)}^{-1\mathrm{/}2}$ for all positive-order spatial derivatives in the $L^2$-norm and for the corresponding pointwise-in-space norms, without any additional $L^1$ integrability or low-frequency assumption on the initial data. Although no uniform algebraic decay rate is asserted for the zero-order energy of $(u,f)$, the directly dissipative variables $u-J(f)$ and $\{\mathbf{I}-{\mathbf{P}}_0\}f$ decay in the $L^2$-norm at the same rate, where $J(f)={\int }_{{\mathbb{R}}^3}v\sqrt{M}f\mathrm{d}v$ denotes the particle momentum and ${\mathbf{P}}_0$ is the orthogonal projection onto $\mathrm{span}\{\sqrt{M},v_1\sqrt{M},v_2\sqrt{M},v_3\sqrt{M}\}$. To the best of our knowledge, these positive-order and zero-order decay estimates have not previously been established for the incompressible Euler-VFP system.