Get started
Work

Sharp anisotropic Gevrey regularization and directional microlocal smoothing for dissipative–dispersive Fourier evolutions

Year: 2026
Type: article
Abstract: We establish a sharp Fourier–Gevrey regularization theory for a broad class of nonautonomous dissipative–dispersive evolution equations on $$\mathbb {R}^n$$ R n . The dissipative symbol is allowed to have different fractional orders in different coordinate directions, while the dispersive symbol is only required to be real. We prove an exact moving-radius estimate in exponentially weighted Fourier spaces: each anisotropic regularity radius increases by precisely the time integral of the corresponding dissipative coefficient. For the model symbol $$\sum _j c_j(t)|\xi _j|^{\kappa _j}$$ ∑ j c j ( t ) | ξ j | κ j , this threshold is optimal in every coordinate, in the strong sense that the evolution operator is bounded between two Fourier–Gevrey spaces if and only if the target radius does not exceed the initial radius plus the accumulated dissipation. Explicit factorial derivative estimates identify the sharp anisotropic Gevrey indices $$1/\kappa _j$$ 1 / κ j , including analytic and ultra-analytic regimes. We also treat inhomogeneous problems and complex lower-order Fourier perturbations. In the microlocal part we develop a nonquasianalytic anisotropic formulation for mixed exponents $$0<\kappa _j<1$$ 0 < κ j < 1 . Writing $$\kappa _*=\max _j\kappa _j$$ κ ∗ = max j κ j , $$q_j=\kappa _*/\kappa _j$$ q j = κ ∗ / κ j , and $$\sigma =1/\kappa _*$$ σ = 1 / κ ∗ , we use $$\varvec{q}$$ q -conic neighborhoods and the anisotropic Gevrey scale $$\mathcal {G}^{\sigma \varvec{q}}$$ G σ q , in the spirit of Zanghirati’s anisotropic microlocal theory. If the integrated dissipation is elliptic only in a
DOI: 10.1007/s11565-026-00757-7
Language: English
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7214085010
FWCI:
Cites: 12
Cited by:
Related to: