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Small scale formation for the 2D Boussinesq equation
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Small scale formation for the 2D Boussinesq equation
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We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying $\sup_{\tau\in[0,t]} \|\nabla \rho(\tau)\|_{L^2}\gtrsim t^\alpha$ for some $\alpha>0$. For the inviscid equation in the strip, we construct examples satisfying $\|\omega(t)\|_{L^\infty}\gtrsim t^3$ and $\sup_{\tau\in[0,t]} \|\nabla \rho(\tau)\|_{L^\infty} \gtrsim t^2$ during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth.
Forward citations
Cited by 3 Pith papers
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A unified Boussinesq--Euler formulation and finite-time blow-up for a Hou--Luo type boundary-jet system
The authors unify the Boussinesq and axisymmetric Euler systems into a parameterized boundary-jet model and prove finite-time blow-up for its closed truncation using a Riccati argument.
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2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)
A coefficient-based unification of two fluid equations yields exact (1+1)D reductions whose apex dynamics blow up in finite time under stated conditional stability assumptions.
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2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)
A unified polar subsystem of 2D Boussinesq and 3D axisymmetric Euler yields exact (1+1)D axis reductions whose apex obeys a CLM-type ODE and blows up in finite time.
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