2026/07/17 by Jinkai Ni
#math.AP
We consider the Cauchy problem for the incompressible Euler-Vlasov-Fokker-Planck (Euler-VFP) system in the whole space \(\mathbb R3\) 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 (u0,f0)∈ HN× Lv2(HN), with N≥ 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 \((1+t)-1/2\) for all positive-order spatial derivatives in the \(L2\)-norm and for the corresponding pointwise-in-space norms, without any additional \(L1\) 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 \(\\mathbf I-\mathbf P0\f\) decay in the \(L2\)-norm at the same rate, where J(f)=∫\mathbb R3v√ M f \rm dv denotes the particle momentum and \(\mathbf P0\) is the orthogonal projection onto \(span\√ M,v1√ M,v2√ M,v3√ 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.