Holder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Abstract

with some Hölder continuous solution that is constant outside (−3/2, 3/2) × T 3 . We also propose a conjecture related to our main result that would imply Onsager's conjecture that there exist energy dissipating solutions to Euler whose velocity fields have Hölder exponent 1/3 − ǫ.

Local Well-Posedness of the (4 + 1)-Dimensional Maxwell–Klein–Gordon Equation at Energy Regularity

“…In our context, we need to develop a sharp version that works at the critical regularity. Our key tool is an explicit solution operator to the divergence equation [2,3,11] which preserves the compact support property; see Proposition 4.4.…”

“…This solution operator was first used by Bogovskiȋ [2,3]. We remark that a similar solution operator was used in [11] in the context of the incompressible Euler equations.…”

“…Our construction below follows the approach of [11], in which a similar solution operator was constructed for the symmetric divergence equation ∂ j R j = U . We sharpen the estimates for V compared to [11] (where non-sharp estimates sufficed), which turns out to be necessary due to the criticality of our problem. The class of domains we work with is that of star-shaped domains, and unions thereof.…”

High dimensionality and h-principle in PDE

“…A key remark of Isett in [46] compared to [37,38] is that advective derivatives behave much better than simple time derivatives. For instance, since…”

“…Statement (B) has been shown for X = L ∞ (0, 1; C 1/10−ε ) in [38], whereas P. Isett in his PhD thesis [46] was the first to prove statement (A) for X = L ∞ (0, 1; C 1/5−ε ), thereby reaching the current best "uniform" Hölder exponent for part (b) of Onsager's conjecture. Subsequently, T. Buckmaster, the two authors, and P. Isett proved statement (B) for X = L ∞ (0, 1; C 1/5−ε ) in [14].…”