A Lower Bound on Blowup Rates for the 3d Incompressible Euler Equation and a Single Exponential Beale-kato-majda Estimate
نویسنده
چکیده
We prove a Beale-Kato-Majda criterion for the loss of regularity for solutions of the incompressible Euler equations in Hs(R3), for s > 5 2 . Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on ‖u(t)‖Hs involving the dimensionless parameter introduced by P. Constantin in [2]. In particular, we derive lower bounds on the blowup rate of such solutions.
منابع مشابه
Solution Properties of a 3d Stochastic Euler Fluid Equation
We prove local well posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s 2nd Law in every Lagrangian domain.
متن کاملPotentially Singular Solutions of the 3d Incompressible Euler Equations
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in ax...
متن کاملToward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in ax...
متن کاملInterplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem.
Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to 4096^{3}. The results are analyzed in terms of the classical analyticity-strip method and Beale, Kato, and Majda (BKM) theorem. A well-resolved acceleration of the time decay of the width of the analyticity strip δ(t) is observed at the highest resolut...
متن کاملOn the Collapse of Tubes Carried by 3D Incompressible Flows
∇x · u = 0 (x ∈ R , t ≥ 0) u(x, 0) = u(x) (x ∈ R) with u a given, smooth, divergence-free, rapidly decreasing vector field on R. Here, u(x, t) and p(x, t) are the unknown velocity and pressure for an ideal, incompressible fluid flow at zero viscosity. An outstanding open problem is to determine whether a 3D Euler solution can develop a singularity at a finite time T . A classic result of Beale-...
متن کامل