Instantaneous continuous loss of regularity for the SQG equation

2024/09/27 by Córdoba, Diego, Martínez-Zoroa, Luis, Ożański, Wojciech S. · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.18900

Abstract

Given s∈ (3/2,2) and ε >0, we construct a compactly supported initial data θ₀ such that ‖ θ₀ ‖_H^s≤ ε and there exist T>0, c>0 and a local-in-time solution θ of the SQG equation that is compactly supported in space, continuous and differentiable in t and in x on ℝ²× [0,T], and, for each t∈ [0,T], θ(⋅ ,t ) ∈ H^s/(1+ct) and θ(⋅ ,t ) \not ∈ H^β for any β> s/(1+ct). Moreover, θ is unique among all solutions with initial condition θ₀ which belong to C([0,T];H^1+α) for any α>0 and is continuous and differentiable in t and in x on ℝ²× [0,T].

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