Blow-up of Solutions to the Generalized Inviscid Proudman–Johnson Equation

Abstract

For arbitrary values of a parameter λ ∈ R, finite-time blowup of solutions to the generalized, inviscid Proudman-Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem. Mathematics Subject Classification (2010). 35B44, 35B10, 35B65, 35Q35.

Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

“…If we are solving this equation on S 1 , we should impose that S 1 ω = 0 (which we may assume on the initial data). Now we recall from [49] that this system satisfies…”

On the global well-posedness of the inviscid generalized Proudman–Johnson equation using flow map arguments

“…In the appendix, we also gave a physical derivation of the GPJ equation from the compressible, two-dimensional Euler equations. While the finite-time existence results in, e.g., [12,13] also specified the nature of the singularity, such as blow-up in the L p -norm of u x , Theorem 2.3 only guarantees that the solution cannot be extended, but does not give further information on the nature of the singularity. Does the solution blow up or does the solution lose its (Lipschitz) regularity, i.e., is there blow-up in its derivatives?…”

“…In [12], the authors showed, for a = 1, that there exist smooth initial data such that the solutions exist globally. Since their results were obtained with quite different techniques, it would be interesting to compare the singularity and global existences results in Theorem 2.3 for a ≥ 1 and a > −1 respectively to the case of a = 1 in [12] Also, it would be interesting to extend the flow map approach laid out in the present paper to also give information on how a solution becomes singular. This will, however, also require information on the second derivative of t → η(t), which has to be derived from new estimates.…”

Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations

“…Regularity criteria for non-smooth initial velocities, including piecewise-linear functions and maps with "cusps" and/or "kinks" on their graphs, can be studied via an argument similar to that used in the proof of Theorem 3.2 (see, e.g., [23,21]). …”

“…(8) is known as the inviscid Proudman-Johnson equation [20]. In [23], a general solution formula for solutions of (8), along with blowup and global-in-time criteria, were established (see [3,4,24,19,21] for additional regularity results). Eq.…”