We prove the regularity of weak 1/2-harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2-harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.
First we introduce the Bailleul-Hoshino's result [4], which links the theory of regularity structures and the paracontrolled calculus. As an application of their result, we give another algebraic proof of the multicomponent commutator estimate [3], which is a generalized version of the Gubinelli-Imkeller-Perkowski's commutator estimate [11, Lemma 2.4].
We present a general approach to obtain a weak Harnack inequality for rough hypoellipitic equations, e.g. kinetic equations. The proof is constructive and does not study the commutator structure but rather compares the rough solution with a smooth problem for which the estimates are assumed.
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.
Another look at regularity in transport-commutator estimates
Carnegie Mellon University（卡内基梅隆大学）；Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL)（索邦大学、法国国家科学研究中心、巴黎大学、雅克-路易·利翁斯实验室）；Institut Universitaire de France（法国高等研究院）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
If the smooth vector fields $X_1,\ldots,X_m$ and their commutators span the tangent space at every point in $Ω\subseteq \mathbb{R}^N$ for any fixed $m\leq N$, then we establish the full interior regularity theory of quasi-linear equations $\sum_{i=1}^m X_i^*A_i(X_1u, \ldots,X_mu)= 0$ with $p$-Laplacian type growth condition. In other words, we show that a weak solution of the equation is locally $C^{1,α}$.
We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion's directions. The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide an alternative proof to that of Theorem 1.5 of [9] for the optimal spatial regularity.
This paper establishes the global regularity of classical solution to the 2D MHD system with only horizontal dissipation and horizontal magnetic diffusion in a strip domain $\mathbb{T}\times\mathbb{R}$ when the initial data is suitable small. To prove this, we combine the Littlewood-Paley decomposition with anisotropic inequalities to establish a crucial commutator estimate. We also analysis the asymptotic behavior of the solution. In addition, the global existence and uniqueness of classical solution is obtained for the 2D simplified tropical climate model with only horizontal dissipations.
本文证明了从Rn的子集到球面的n/2-调和映射的Hoelder连续性。这推广了F. Da Lio和T.里维耶尔任意尺寸。证明依赖于补偿效果，我们量化适应的方法温特不等式L。鞑靼，而不是贝索夫空间的参数，其中使用的一维情况下。此外，Hodge分解和高阶Poincare不等式的分数类似物以及类似于分数Laplacian的非局部算子的几个局部化效应也得到了发展和应用.
We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions that the solutions belong to $W^{n/2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivière--Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman--Rochberg--Weiss commutator theorem.
In this paper, we continue to study the fractional harmonic gradient flow on $S^{n-1}$ taking values in a general closed manifold $N \subset \mathbb{R}^n$, addressing global existence and uniqueness of solutions of energy class with sufficiently small energy, adding to the existing body of knowledge pertaining to the half-harmonic gradient flow and expanding upon our previous work in [34]. We extend the techniques by Struwe in [30] and Rivière in [22] to the non-local framework analogous to [34] to derive uniqueness, employ commutator estimates as in [8] for regularity and follow [30] for a ge…
We prove an $ε$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets. Instead, we follow the spirit of the last-named author's recent work on free boundary harmonic maps …
In this paper we study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model defines a family of dispersive equations, that can be seen as a dispersive interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and Benjamin-Ono equation, satisfy the following property: if the initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. I…
In this work we investigate some regularization properties of the incompressible Euler equations and of the fractional Navier-Stokes equations where the dissipative term is given by $(-Δ)^α$, for a suitable power $α\in (0,\frac{1}{2})$ (the only meaningful range for this result). Assuming that the solution $u \in L^\infty _t(C^θ_x)$ for some $θ\in (0,1)$ we prove that $u \in C^θ_{t,x}$, the pressure $p\in C^{2θ-}_{t,x}$ and the kinetic energy $e \in C^{\frac{2θ}{1-θ}}_t$. This result was obtained for the Euler equations in [Is13] with completely different arguments and we believe that our proo…
Commutator Estimates for Low-Temperature Fermi Gases
Beijing Institute of Technology（北京理工大学）；École Normale Supérieure de Lyon（里昂高等师范学院）；Kyung Hee University（庆熙大学）；Universität Freiburg（弗赖堡大学）；Universität Basel（巴塞尔大学）
We prove existence and regularity of minimisers for the Canham-Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the $2$-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to prove the main results we establish the lower semicontinuity of the Canham-Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.
We consider critical points $u:Ω\to N$ of the bi-energy \[ \int_Ω|Δu|^2\,d x, \] where $Ω\subset\mathbb{R}^m$ is a bounded smooth domain of dimension $m\ge 5$ and $N\subset\mathbb{R}^L$ a compact submanifold without boundary. More precisely, we consider variationally biharmonic maps $u\in W^{2,2}(Ω,N)$, which are defined as critical points of the bi-energy that satisfy a certain stationarity condition up to the boundary. For weakly convergent sequences of variationally biharmonic maps, we demonstrate that the only obstruction that can prevent the strong compactness up to the boundary is the pr…
In my previous paper I have contrived a Ginzburg-Landau heat flow with a time-dependent parameter and by using it, I constructed a harmonic heat flow into spheres with a monotonical inequality and a reverse Poincaré inequality. This paper establishes these two energy inequalities near the boundary and then by making the best of them, we discuss a partial boundary regularity. In addition to it, we demonstrate a whole domain's regularity under "the one-sided condition." This has been proposed by S.Hildebrandt and K.-O.Widman.
We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_{x}^α\partial_{x}u+u\partial_{x}u=0, & x,t\in\mathbb{R},\,0<α<1, \\ u(x,0)=u_{0}(x).& \\ \end{array} \right. \end{equation} It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This pro…
We consider transport equations with an incompressible transporting vector field. Whereas smooth solutions of such equations conserve every $L^p$ norm simply by the chain rule, the question arises how regular a weak solution needs to be to guarantee this conservation property. Whereas the classical DiPerna-Lions theory gives sufficient conditions in terms of the regularity of the coefficients, with no regularity requirement on the transported scalar, we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar. This is motivated by the case of act…
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as $ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $ with zero boundary data, have unexpected degenerate nature.
We establish global existence, uniqueness, regularity and long-time asymptotics of strong solutions to the half-harmonic heat flow and dissipative Landau-Lifshitz equation, valid for initial data that is small in the homogeneous Sobolev norm $\dot{H}^{\frac n 2}$ for space dimensions $n \le 3$.
