Abstract We consider the ordinary or fractional Laplacian plus a homogeneous, scaling-critical drift term. This operator is non-symmetric but homogeneous, and generates scales of L p -Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. Unlike in previous studies concerning Hardy operators, i.e. ordinary or fractional Laplacians plus scaling-critical scalar perturbations, handling the drift term requires an additional, possibly technical, restriction on the range of comparable Sobolev spaces, which is related to the unavailability of gradient bounds for the associated semigroup.

Abstract In this paper, we are concerned with the following noncommutative Painlevé II (PII) equation D 2 β 1 = 4 s β 1 + 4 β 1 s + 8 β 1 3 , where β 1 = β 1 ( s → ) is an n × n matrix-valued function of s → = ( s 1 , … , s n ) , s = diag ⁡ ( s 1 , … , s n ) and D = ∑ j = 1 n ∂ ∂ s j . If n = 1, it reduces to the classical PII equation up to a scaling. Given an arbitrary n × n constant matrix C = ( c j k ) j , k = 1 n , a remarkable result due to Bertola and Cafasso asserts that there exists a unique solution β 1 ( s → ) = β 1 ( s → ; C ) of the noncommutative PII equation such that its ( k , l )-th entry behaves like − c k l Ai ( s k + s l ) as S = 1 n ∑ i = 1 n s j → + ∞ , where Ai stands for the standard Airy function. For a class of structured matrices C , we establish asymptotics of the associated solutions as S → − ∞ , which particularly include the so-called connection formulas. In the present setting, it comes out that the solution exhibits a hybrid behaviour in the sense that each entry corresponds to either an extension of the Hastings–McLeod solution or an extension of the Ablowitz–Segur solution for the PII equation. It is worthwhile to emphasize the asymptotics of the ( k , l )-th entry as S → − ∞ cannot be deduced solely from its behaviour as S → + ∞ in general, which actually also depends on the positive infinity asymptotics of the ( l , k )-th entry. This new and intriguing phenomenon disappears in the scalar case.

Abstract Since the Ginzburg–Landau theory is concerned with macroscopic phenomena, and gravity affects how objects interact at the macroscopic level. It becomes relevant to study the Ginzburg–Landau theory in curved space, that is, in the presence of gravity. In this paper, some existence theorems are established for the vortex solutions of the magnetic Ginzburg–Landau theory coupled to the Einstein equations. First, when the coupling constant λ = 1, we get a self-dual structure from the Ginzburg–Landau theory, then a partial differential equation with a gravitational term that has power-type singularities is deduced from the coupled system. To overcome the difficulty arising from the orders of singularities at the vortices, a constraint minimisation method and a monotone iteration method are employed. We also show that the quantised flux and total curvature are determined by the number of vortices. Second, when the coupling constant λ > 0, we use a suitable ansatz to get the radially symmetric case for the magnetic Ginzburg–Landau theory in curved space. The existence of the symmetric vortex solutions are obtained through combining a two-step iterative shooting argument and a fixed-point theorem approach. Some fundamental properties of the solutions are established via applying a series of analysis techniques.

Abstract We derive quantitative propagation of chaos in the sense of relative entropy for the 2D viscous vortex model with general circulations, approximating the vorticity formulation of the 2D Navier–Stokes equation on the whole Euclidean space. Our results work on the general setting that the vortices are positioned on the whole space R 2 and that the circulations are allowed to be in different magnitudes and orientations, which can be adapted to general unconfined realistic fluids with vorticity that may change sign. We provide explicit convergence rates which are optimal in N and optimal in t among existing literature. The key technical tools, which are our major novelty, are the sharp logarithmic growth estimates and a new ODE hierarchy and iterated integral estimates.

Abstract We characterize the Leray–Hopf solutions of the 2D inhomogeneous Navier–Stokes system that become strong for positive times. This characterization relies on the strong energy inequality and the regularity properties of the pressure. As an application, we establish a weak-strong uniqueness result and provide a unified framework for several recent advances in the field.

Abstract We prove the existence and some moment estimates for an invariant measure µ for the two-dimensional (2D) deterministic Euler equations on the unbounded domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> and with highly regular initial data. The result is achieved by first showing the existence of Markov (marginally) stationary processes which solve the hyperviscous 2D Navier–Stokes equations with kinematic viscosity ν &gt; 0 and an additive stochastic noise scaling as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msqrt> <mml:mi>ν</mml:mi> </mml:msqrt> </mml:mrow> </mml:math> . We then study the inviscid limit and prove that, as ν tends to 0, these processes converge, in an appropriate trajectory space, to a pathwise solution to the Euler equations. Its law is the sought invariant measure µ .

Abstract In this paper we consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin ). The return dynamics is described by a map T of the annulus A = T × ( 0 , π ) . We prove the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary ∂ A ; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of ∂ A through which there passes no invariant essential curve; and the only coin billiard for which the phase space A is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, we describe the results of some numerical experiments on the elliptical coin billiard.

Abstract We investigate the strong convergence of weak solutions to the two-dimensional quasi-geostrophic shallow-water (QGSW) equation as the inverse Rossby radius tends to zero. In this limit, we recover the Yudovich solution of the incompressible Euler equations. We prove that the vorticity convergence holds in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>t</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msubsup> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , for any finite integrability exponent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> . This extends to the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> provided that the initial vorticities are continuous and converge uniformly. We also discuss the sharpness of this limit by demonstrating that the continuity assumption on the initial data is necessary for the endpoint convergence in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> . The proof of the strong convergence relies on the Extrapolation Compactness method, recently introduced by Arsénio and the first author to address similar stability questions for the Euler equations. The approach begins with establishing the convergence in a lower regularity space, at first. Then, in a later step, the convergence to Yudovich’s vorticity of Euler equations in Lebesgue spaces comes as a consequence of a careful analysis of the evanescence of specific high Fourier modes of the QGSW vorticity. A central challenge arises from the absence of a velocity formulation for QGSW, which we overcome by employing advanced tools from Littlewood Paley theory in endpoint settings. The sharpness of the convergence in the endpoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> case is obtained in the context of vortex patches, drawing insights from key findings on uniformly rotating and stationary solutions of active scalar equations.

Abstract We prove that a Mañé generic real-analytic D -Hamiltonian subjected to a totally non-holonomic real-analytic distribution D , has no non-trivial normal D -singular orbits of minimal rank. If D has co-rank 1, this implies that a Mañé generic D -Hamiltonian does not admit non-trivial normal D -singular orbits.

Abstract In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results in a new hyperregular control Lagrangian and, thus, a new control Hamiltonian whose equations of motion provide necessary optimality conditions. We compare this approach to the weak Pontryagin’s maximum principle (PMP) in this setting, providing geometric insight into their relation. This leads us to define an extended Tulczyjew’s triple with controls. Moreover, we study the relationship between Noether symmetries of this new formulation and those of the PMP.