Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces M_{q, σ}^s (Rⁿ). Though the case of the derivative index s = 0 has been treated in our previous work, the case s ≠ 0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of M_{q, σ}^s (Rⁿ) for the existence of local and global solutions for initial data u₀ ∈ M_{q, σ}^s (Rⁿ).