On self-similar singular solutions to a vorticity stretching equation

2025/12/17 by Du, Dapeng, Li, Jingyu, Shi, Xinyue
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #Navier-Stokes equation solutions

paper · doi:10.48550/arxiv.2512.15017

Abstract

We consider the following model equation: ω_t = Z₁₁ω ω, where Z₁₁ = ∂₁₁Δ^-1 is a Calderon-Zygmond operator. We get the existence of self-similar singular solutions with a special form. The main difficulty is the degeneracy of the operator Z₁₁ that is overcome by the spectral uncertainty principle. We also show that the solution to this model blows up in finite time if the initial datum is compactly supported and has a positive integral.

Citations

• Finite-time singularity formation for scalar stretching equations
• Singularity formation in the incompressible Euler equation in finite and infinite time
• On some model equations of Euler and Navier-Stokes equations
• Small scale creation for solutions of the incompressible two dimensional\n Euler equation
• Remarks on Oldroyd-B and Related Complex Fluids Models

Related

• Finite-time singularity formation for scalar stretching equations
• On the Effects of Advection and Vortex Stretching
• The role of self-similarity in singularities of PDE's
• Spectral theory and self-similar blowup in wave equations
• Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations