Critical Well‐Posedness and Decay for an Active Scalar Equation With Mixed Dissipation and Hardy–Littlewood–Sobolev Drift

ABSTRACT The present work is concerned with the global well‐posedness of small mild solutions for an active scalar equation driven by a mixed local–nonlocal diffusion operator and a nonlocal drift: Here , , and is a Hardy–Littlewood–Sobolev‐type drift operator of order . The mixed dissipative operator is nonhomogeneous and prevents exact scaling invariance; nevertheless, an effective scaling analysis shows that the critical Lebesgue exponent is . In a suitable time‐weighted critical space, we establish global existence and uniqueness of solutions for sufficiently small initial data in . The proof relies on semigroup estimates for the mixed diffusion operator, a bilinear estimate for the Duhamel term based on Hardy–Littlewood–Sobolev‐type bounds, and a contraction argument. Moreover, the resulting global solutions satisfy scale‐consistent decay and smoothing estimates, together with continuity and stability properties.