arXiv subjects
Publications and source records attributed to Tristan Buckmaster.
Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by recent work of Glatt-Holtz and Vicol.
We prove that given any $β<1/3$, a time interval $[0,T]$, and given any smooth energy profile $e \colon [0,T] \to (0,\infty)$, there exists a weak solution $v$ of the three-dimensional Euler equations such that $v \in C^β([0,T]\times \mathbb{T}^3)$, with $e(t) = \int_{\mathbb{T}^3} |v(x,t)|^2 dx$ for all $t\in [0,T]$. Moreover, we show that a suitable $h$-principle holds in the regularity class $C^β_{t,x}$, for any $β<1/3$. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.
We consider the nonlinear Schrödinger (NLS) equation posed on the box $[0,L]^d$ with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when $L$ is large and the characteristic size $ε$ of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in theory of statistical physics of dispersive waves, that goes by the name of "wave turbulence". Our main result is deriving a new equation, the continuous resonant (CR) equation, that describes the effective dynamics for large $L$ and small $ε$ over very large time-scales. Such time-scales are well beyond the (a) nonlinear time-scale of the equation, and (b) the Euclidean time-scale at which the effective dynamics are given by (NLS) on $\mathbb R^d$. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.
We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Székelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.
In recent work by Isett (arXiv:1211.4065), and later by Buckmaster, De Lellis, Isett and Székelyhidi Jr. (arXiv:1302.2815), iterative schemes where presented for constructing solutions belonging to the Hölder class $C^{1/5-ε}$ of the 3D incompressible Euler equations which do not conserve energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of $C^{1/3-ε}$ solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial Hölder continuous solutions which for almost every time belong to the critical Onsager Hölder regularity $C^{1/3-ε}$ and have compact temporal support.
For any $ε>0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{\frac{1}{3}-ε})$, namely $x\mapsto v (x,t)$ is $(\frac{1}{3}-ε)$-Hölder continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{\frac{1}{3}-ε} dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{\frac{1}{3}-ε})$.
Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper P. Isett (arXiv:1211.4065) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent - albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett's final result, adhering more to the original scheme and introducing some new devices. More precisely we show that for any positive $ε$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class $C^{\frac{1}{5}-ε}$.
In this paper we will prove the existence of weak solutions to the Korteweg-de Vries initial value problem on the real line with H^{-1} initial data; moreover, we will study the problem of orbital and asymptotic H^{s} stability of solitons for integers s\ge -1; finally, we will also prove new a priori H^{-1} bounds for solutions to the Korteweg-de Vries equation. The paper will utilise the Miura transformation to link the Korteweg-de Vries equation to the modified Korteweg-de Vries equation.