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Register a new userOn the role of $L^3$ and $H^{frac{1}{2}}$ norms in hydrodynamics
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Laurent Schoeffel
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In this paper, we extend some results proved in previous references for three-dimensional Navier-Stokes equations. We show that when the norm of the velocity field is small enough in $L^3({I!!R}^3)$, then a global smooth solution of the Navier-Stokes equations is ensured. We show that a similar result holds when the norm of the velocity field is small enough in $H^{frac{1}{2}}({I!!R}^3)$. The scale invariance of these two norms is discussed.
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The past few years have witnessed increased attention to the quest for Majorana-like excitations in the condensed matter community. As a promising candidate in this race, the one-dimensional chiral Majorana edge mode (CMEM) in topological insulator-superconductor heterostructures has gathered renewed interests during recent months after an experimental breakthrough. In this paper, we study the quantum transport of topological insulator-superconductor hybrid devices subject to light-matter interaction or general time-periodic modulation. We report half-integer quantized conductance plateaus at $frac{1}{2}frac{e^2}{h}$ and $frac{3}{2}frac{e^2}{h}$ upon applying the so-called sum rule in the theory of quantum transport in Floquet topological matter. In particular, in a photoinduced topological superconductor sandwiched between two Floquet Chern insulators, it is found that for each Floquet sideband, the CMEM admits equal probability for normal transmission and local Andreev reflection over a wide range of parameter regimes, yielding half-integer quantized plateaus that resist static and time-periodic disorder. The $frac{3}{2}frac{e^2}{h}$ plateau has not yet been computationally or experimentally observed in any other superconducting system, and indicates the possibility to simultaneously create and manipulate multiple pairs of CMEMs by light. The robust half-quantized conductance plateaus, due to CMEMs at quasienergies zero or half the driving frequency, are both fascinating and subtle because they only emerge after a summation over contributions from all Floquet sidebands. Such a distinctive transport signature can thus serve as a hallmark of photoinduced CMEMs in topological insulator-superconductor junctions.
In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system with initial data $(u_0,B_0)in H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ and $sigmain (0,2)$. In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $sigma$ to $0$ with the aid of the new formulation of the Hall-MHD system observed by cite{MR4193644}. Moreover, we establish the global well-posedness in $H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ with $sigmain (0,2)$ for small initial data and get the optimal time-decay rates of solutions. Our results improves some previous works.
In this paper, we study the $frac{1}{H}$-variation of stochastic divergence integrals $X_t = int_0^t u_s {delta}B_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $frac{1}{H}$-variation of $X$ exists in $L^1({Omega})$ and is equal to $e_H int_0^T|u_s|^H ds$, where $e_H = mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $|B_t|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < frac{1}{2}$. Using a multidimensional version of the result on the $frac{1}{H}$-variation of divergence integrals, we prove that if $2dH^2 > 1$, then the divergence integral in the integral representation of the fractional Bessel process has a $frac{1}{H}$-variation equals to a multiple of the Lebesgue measure.
We have systematically investigated the magnetic moments of spin-$frac{1}{2}$ doubly charmed baryons in the framework of the heavy baryon chiral perturbation theory. In this paper, one loop corrections with intermediate spin-$frac{1}{2}$ and spin-$frac{3}{2}$ doubly charmed baryon states are considered. The numerical results are calculated to next-to-leading order: $mu_{Xi^{++}_{cc}}=0.35mu_{N}$, $mu_{Xi^{+}_{cc}}=0.62mu_{N}$, $mu_{Omega^{+}_{cc}}=0.41mu_{N}$. Our results may be useful for future experiment and chiral extrapolation of the lattice QCD.
We consider composition operators $mathscr{C}_varphi$ on the Hardy space of Dirichlet series $mathscr{H}^2$, generated by Dirichlet series symbols $varphi$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $varphi$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $varphi$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of J. H. Shapiro on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that $varphi(s) = c + r2^{-s}$. A number of other examples are given.
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