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In this paper, we give regularity criteria in terms of the magnetic pressure in Lorentz spaces.

In this paper, we consider the conditional regularity for the 3D incompressible Navier-Stokes equations in Vishik spaces. These results will be regarded an improvement of the results given by Huang-Li-Xin, (SIAM J. Math. Anal., 2011) and Jiu-Wang-Ye, (J. Evol. Equ., 2021).

Background We aimed to evaluate dynamic alterations in cerebral total hemoglobin concentration (HbT) in individuals with orthostatic hypotension (OH) and orthostatic intolerance (OI) symptoms using a portable near-infrared spectroscopy (NIRS) system. Methods Participants comprised 238 individuals (mean age, 47.9 years) without a history of cardiovascular, neurodegenerative, or cerebrovascular diseases, including those with unexplained OI symptoms and healthy volunteers. Participants were categorized by the presence of OH based on the supine-to-stand blood pressure (BP) drop and OI symptoms using on OH questionnaires: classic OH (OH-BP), OH symptoms alone (OH-Sx), and control groups. Random case–control matching sets were constructed, resulting in 16 OH-BP and 69 OH-Sx-control sets. The time-derivative of HbT change in the prefrontal cortex during the squat-to-stand maneuver was measured using a portable NIRS system. Results There were no differences in demographics, baseline BP, and heart rate among matched sets. The peak time of maximum slope variation in HbT change, indicating the recovery rate and speed of cerebral blood volume (CBV) change, was significantly longer in OH-Sx and OH-BP groups than in the control group under transition to a standing position after squatting. In the OH-BP subgrouping, the peak time of maximum slope variation in HbT change was significantly longer only in OH-BP with OI symptoms, but did not differ between OH-BP without OI symptoms and controls. Conclusions Our results suggest that OH and OI symptoms are associated with dynamic alterations in cerebral HbT. Regardless of the severity of the postural BP drop, OI symptoms are associated with prolonged CBV recovery.

We show that the energy norm of weak solutions to Vlasov equation coupled with a shear thickening fluid on the whole space has a decay rate the energy norm $E(t) \leq {C}/{(1+t)^{\alpha }}, \forall t \geq 0$ for $\alpha \in (0,3/2)$.

We consider the motion of an incompressible shear-thickening power-law-like non-Newtonian fluid in $R^3$ with a variable power-law index. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the large-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strong solution of the given equations with the initial data $u_0$ and $\tilde{u}$ is the strong solution of the same equations with perturbed initial data $u_0+w_0$. The initial perturbation $w_0$ is not required to be small, but is assumed to satisfy certain decay condition. In particular, we can show that $(1+t)^{-\frac{\gamma}{2}}\lesssim \|u(t)-\tilde{u}(t)\|_2\lesssim(1+t)^{-\frac{\gamma}{2}}$, for sufficiently large $t>0$, where $\gamma\in(2,\frac{5}{2})$. The proof is based on the observation that the solution of the linear heat equation describes the asymptotic behaviour of the solutions of the electrorheological fluids well for sufficiently large time $t>0$, and the generalized Fourier splitting method with an iterative argument. Furthermore, it will also be discussed that the argument used in the present paper can improve the previous results for the generalized Newtonian fluids with a constant power-law index.

We investigate the space-time decay properties for strong solutions to the 3D Navier–Stokes equations with power-law type nonlinear viscous fluid. Based on the temporal decay results for this equation, we find that one can obtain decay estimate of the weight for the velocity field u by direct weighted energy estimate.

We give some regularity criterion (of a weak $$-L^{p}$$ Serrin type) of a weak solution to the 3D Navier–Stokes equations in a bounded domain $$\Omega \subset \mathbb {R}^3$$ with a smooth boundary. In particular, in case of the half space, we give a regularity condition of a weak solution with respect to a tangential component of the velocity flow vector.

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

We report a novel strategy for the synthesis of Pt@Au nanorings possessing near-field focusing capabilities at the center through which single-particle surface enhanced Raman scattering could be readily observed. We utilized Pt@Au nanorings as a light-absorber; the absorbed light could be focused at the center with the aid of a Au nanoporous structure. We synthesized the Au nanolens structure through a Galvanic exchange process between Au ions and Ag block at the inner domain of the Pt@Au nanoring. For this step, Ag was selectively pre-deposited at the inner domain of the Pt@Au nanorings through electrochemical potential-tuned growth control and different surface energies with regard to the inner and outer boundaries of the nanoring. Then, the central nanoporous architecture was fabricated through the Galvanic exchange of sacrificial Ag with Au ions leading to the resulting Au nanoring with a Au nanoporous structure at the center. We monitored the shape-transformation by observing their corresponding localized surface plasmon resonance (LSPR) profiles. By varying the rim thickness of the starting Pt@Au nanorings, the inner diameter of the nanolens was accordingly tuned to maximize near-field focusing, which enabled us to obtain the reproducible and light-polarization independent measurements of single-particle SERS. Through theoretical simulation, the near-field electromagnetic field focusing capability was visualized and confirmed through single-particle SERS measurement showing an enhancement factor of 1.9 × 108 to 1.0 × 109., We synthesized a Au nanolens with electromagnetic near-field focusing capability by integrating a Au nanoporous structure at the center of the Pt@Au nanoring via synthetic steps of eccentric growth of Ag and nanoscale Galvanic exchange reaction.

This paper is concerned with time decay rates of the strong solutions of an incompressible the coupled modified Navier-Stokes and Maxwell equations in a half space $ \mathbb{R}^3_+ $. With the use of the spectral decomposition of the Stokes operator and $ L^p-L^q $ estimates developed by Borchers and Miyakawa [2], we study the $ L^2 $-decay rate of strong solutions.

We investigate the following elliptic equations: $$\left\{ {\matrix{ { - M\left( {\int_{{\mathbb{R}^N}} {\phi ({{\left| {\nabla u} \right|}^2}){\rm{d}}x} } \right){\rm{div(}}\phi \prime ({{\left| {\nabla u} \right|}^2})\nabla u{\rm{) + }}{{\left| u \right|}^{\alpha - 2}}u = \lambda h(x,u),} \hfill \cr {u(x) \to 0,\;\;\;\;\;{\rm{as}}\left| x \right| \to \infty ,} \hfill \cr } } \right.\;\;\;\;\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{R}^N},$$ where N ≥ 2, 1 < p < q < N, α < q, 1 < α < p*q†/p† with $${p^ * } = {\textstyle{{Np} \over {N - p}}},\;\;\phi (t)$$ behaves like tq/2 for small t and tp/2 for large t, and p′ and q′ are the conjugate exponents of p and q, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.

We developed a synthetic strategy to form cyclodextrin-based intrananogap particles (CIPs) with a well-defined ∼1 nm interior gap in a high yield (∼97%), and were able to incorporate 10 different Raman dyes inside the gap using the cyclodextrin-based host-guest chemistry, leading to strong, reproducible, and highly multiplexable surface-enhanced Raman scattering (SERS) signals. The average SERS enhancement factor (EF) for CIPs was 3.0 × 109 with a very narrow distribution of the EFs that range from 9.5 × 108 to 9.5 × 109 for ∼95% of the measured particles. Remarkably, 10 different Raman dyes can be loaded within the nanogap of CIPs, and 6 different Raman dye-loaded CIPs with little spectral overlaps were distinctly detected for cancer cell imaging applications with a single excitation source. Our synthetic strategy provides new platforms in precisely forming plasmonic nanogap structures with all key features for widespread use of SERS including strong signal intensity, reliability in quantification of signal and multiplexing capability.

In this Letter, a rational and stepwise method for the solution-phase synthesis of asymmetric Au split nanorings by adopting Au nanoprisms as a template has been demonstrated. The selective chemical etching of Au nanoprism tips activated the surface reactivity of edges and led to the selective deposition of Pt at the periphery of Au nanoplates. By controlling the total amount of Pt on the edges, different degrees of split Au@Pt nanorings were obtained; the subsequent Au coating around the Au@Pt scaffold eventually resulted in asymmetric Au hexagonal split nanorings. Their surface plasmonic features as a function of split degrees were investigated, including straight nanorods, bent nanorods, split nanorings, and full nanorings. The electrical field focusing using single-particle surface-enhanced Raman spectroscopy was evaluated under different polarization angles of the incident light for two different structures with the point gap and line gap between two arms.

Silver double nanorings with circular intra-nanogaps between two nanorings of different diameters were synthesized without a linker molecule to confine an incident electromagnetic field in a single entity. We used on-demand, rational, and systematic multi-stepwise reactions consisting of (1) selective etching of gold, (2) rim-on deposition of platinum, (3) eccentric growth of gold, and (4) concentric growth of silver. The resulting silver double nanorings exhibited a high degree of homogeneity in both shape and size, with strongly coupled circular hot zones (or "hot halos", referring to the circular intra-nanogaps capable of focusing the near electromagnetic field) resulting from strong surface plasmon coupling between the inner and outer nanorings. Remarkably, these silver double nanorings exhibited strong, stable, and reproducible single-particle surface-enhanced Raman scattering signals without blinking. The signals appeared independently of polarization directions, which is a unique feature of a circular hot halo. The estimated enhancement factor was between 2 × 108 and 7 × 108. The measured limit of detection was 10-7 M in bulk concentration, and the signal appeared 570 s after sample exposure.

Herein, plasmonic metal tripod nanoframes with three-fold symmetry were synthesized in a high yield (∼83%), and their electric field distribution and single-particle surface-enhanced Raman scattering (SERS) were studied. We realized such complex frame morphology by synthesizing analogous tripod nanoframes through multiple transformations. The precise control of the Au growth pattern led to uniform tripod nanoframes embedded with circle or line-shaped hot spots. The linear-shaped nanogaps ("Y"-shaped hot-zone) of the frame structures can strongly and efficiently confine the electric field, allowing for strong SERS signals. Coupled with a high synthetic yield of the targeted frame structure, strong and uniform SERS signals were obtained inside the nanoframe gaps. Remarkably, quite reproducible SERS signals were obtained with these structures-the SERS enhancement factors with an average value of 7.9 × 10

The development of a stepwise synthetic strategy for Au ring-in-a-triangle nanoframes with a high degree of structural solidity is essential to the advancement of highly amplified near-field focusing. This strategy leads to the formation of an inscribed nanoring in a triangular metal frame with stability to withstand elevated temperatures and an oxidizing environment, which is critical for successful single-particle surface-enhanced Raman scattering (SERS). The existence of inscribed nanorings plays an important role in enhancing the so-called "lightning rod effect," whereby the electromagnetic near-field enhancement occurs on the highly curved curvature of a metallic interface. We evaluated the corresponding single-particle SERS as a function of the thickness of the rims and then constructed two-dimensional (2D) bulk SERS substrates, wherein an ensemble of hotspots exists. The synergic contribution from both inter- and intrahotspots allowed the outstanding linearity of the calibration curve and the lowest limit of detection, ∼10

The aim of this paper is to examine the existence of at least two distinct nontrivial solutions to a Schrödinger-type problem involving the nonlocal fractional $p(\cdot )$ p ( ⋅ ) -Laplacian with concave–convex nonlinearities when, in general, the nonlinear term does not satisfy the Ambrosetti–Rabinowitz condition. The main tools for obtaining this result are the mountain pass theorem and a modified version of Ekeland’s variational principle for an energy functional with the compactness condition of the Palais–Smale type, namely the Cerami condition. Also we discuss several existence results of a sequence of infinitely many solutions to our problem. To achieve these results, we employ the fountain theorem and the dual fountain theorem as main tools.

A portable near infrared spectroscopy (NIRS) system was used to evaluate alterations in cerebral total hemoglobin concentration (HbT) in individuals with orthostatic hypotension (OH) and orthostatic intolerance (OI) symptoms. We enrolled 238 healthy participants (mean age, 47.9 years) and assessed the presence of OH (orthostatic blood pressure (BP) drop of systolic BP ≥ 20 mmHg or diastolic BP ≥ 10 mmHg within 3 minutes of supine-to-stand) and OI symptoms using the OH questionnaire. The participants were categorized into three groups based on the presence of OH and OI symptoms: the classic OH (OH-BP) group, symptom alone (OH-Sx) group, and control group. Random case-control matching sets (age, sex, hypertension, and diabetes mellitus) were constructed consisting of 16 OH-BP and 69 OH-Sx-control sets. We measured the time-derivative of HbT change in the prefrontal cortex during the squat-to-stand maneuver. There were no differences in demographics, baseline BP, and heart rate among the matched sets. Among the NIRS parameters, the peak-time of maximum slope variation was significantly longer in the OH-Sx and OH-BP groups than in the matched control groups during transition to the standing position after squatting. Our results suggested that OH and OI symptoms are associated with dynamic alteration in cerebral HbT.

In this paper we prove some Liouville-type theorems for the stationary magneto-micropolar fluids under suitable conditions in three space dimensions. We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials. And then we show similar results assuming that the the solutions are contained in L^p(\R^3) with p\in[2,9/2). Finally we show the same result for lower values of p\in[1,9/4) with the further assumption that the solutions vanish at infinity.

Direct photoluminescence (PL) from metal nanoparticles (NPs) without chemical dyes is promising for sensing and imaging applications since this offers a highly tunable platform for controlling and enhancing the signals in various conditions and does not suffer from photobleaching or photoblinking. It is, however, difficult to synthesize metal NPs with a high quantum yield (QY), particularly in the near-infrared (NIR) region where deep penetration and reduced light scattering are advantageous for bioimaging. Herein, we designed and synthesized Au-Ag long-body nanosnowman structures (LNSs), facilitated by polysorbate 20 (Tween 20). The DNA-engineered conductive junction between the head and body parts results in a charge transfer plasmon (CTP) mode in the NIR region. The junction morphology can be controlled by the DNA sequence on the Au core, and polythymine and polyadenine induced thick and thin junctions, respectively. We found that the LNSs with a thicker conductive junction generates the stronger CTP peak and PL signal than the LNSs with a thinner junction. The Au-Ag LNSs showed much higher intensities in both PL and QY than widely studied Au nanorods with similar localized surface plasmon resonance wavelengths, and notably, the LNSs displayed high photostability and robust, sustainable PL signals under continuous laser exposure for >15 h. Moreover, the PL emission from Au-Ag LNSs could be imaged in a deeper scattering medium than fluorescent silica NPs. Finally, highly robust PL-based cell images can be obtained using Au-Ag LNSs without significant signal change while repetitively imaging cells. The results offer the insights in plasmonic NIR probe design, and show that chemical dye-free LNSs can be a very promising candidate with a high QY and a robust, reliable NIR PL signal for NIR sensing and imaging applications.

We prove local regularity condition for a suitable weak solution to 3D MHD equations. Precisely, if a solution satisfies $u, b \in L^{\infty}(-(\frac{4}{3})^2, 0;L^{3, q}(B_{\frac{3}{4}}))$, $q\in (3, \infty)$ in Lorentz space, then $(u, b)$ is Hölder continuous in the closure of the set $Q_{\frac{1}{2}}$.

In this note, we give a new proof to the energy conservation for the weak solutions of the incompressible 3D MHD equations. Moreover, we give the lower bounds for possible singular solutions to the incompressible 3D MHD equations.

In this paper, we consider the conditional regularity for 3D inhomogeneous incompressible magnetohydrodynamic equations in Vishik spaces and give a regularity criterion of weak solutions in terms of a gradient velocity vector.

We designed complex Au nanorings with intertwined triple rings (ANITs) in a single entity to amplify the efficacy of near-field focusing. Such a complex and unprecedented morphology at the nanoscale was realized through on-demand multistepwise reactions. Triangular nanoprisms were first sculpted into circular nanorings, followed by a series of chemical etching and deposition reactions eventually leading to ANITs wherein thin metal bridges hold the structure together without any linker molecules. In the multistepwise reaction, the well-faceted growth pattern of Au, which induces the growth of two distinctive flat facets in a lateral direction, is important to evolve the morphology from single to multiple nanorings. Although our synthesis proceeds through multiple steps in one batch without purification steps, it shows a remarkably high yield (>∼90%) at the final stage. The obtained high degree of homogeneity (in both shape and size) of the resulting ANITs allowed us to systematically investigate the corresponding localized surface plasmon resonance (LSPR) coupling with varying nanoring arrangements and observe their single-particle surface enhanced Raman scattering (SERS). Surprisingly, individual ANITs exhibited an enormously large enhancement factor (∼109), which confirms their superior near-field focusing relative to other reported nanoparticles.

Inadequate cerebral perfusion is a risk factor for cerebral ischemia in patients with large artery steno-occlusion. We investigated whether prefrontal oxyhemoglobin oscillation (ΔHbO

We show some regularity criteria (Prodi–Serrin type regularity) to weak solutions of the 3D generalized Navier–Stokes equations in viewpoint of the velocity vector u or the vorticity vector $$\omega :=\nabla \times u$$ in Lorentz space. Moreover, we briefly mention some results for coupled equations with Navier–Stokes equation (see Remark 1.5 and 1.8).

Herein, we report the novel strategy for the synthesis of complex 3-dimensional (3D) nanostructures, mimicking the linker molecule-free 3D arrangement of six Au nanospheres at the vertices of octahedrons. We utilized 3D PtAu skeleton for the structural rigidity and deposited Au around the PtAu skeleton in a site-selective manner, allowing us to investigate their surface plasmonic coupling phenomenon and near-field enhancement as a function of sizes of nanospheres, which are directly related to the intrananogap distance and interior volume size. The resulting 3D Au hexamer structures with octahedral arrangement were realized through precise control of the Au growth pattern. The complex 3D Au hexamers were composed of six Au nanospheres connected by thin metal conductive bridges. The standard deviation of the metal conductive bridges and Au nanospheres was within ca. 10%, exhibiting a high degree of homogeneity and precise structural tunability. Interestingly, charge transfer among the six Au nanospheres occurred along the metal conductive bridges leading to surface plasmonic coupling between Au nanospheres. Accordingly, electric near fields were strongly and effectively focused at the vertices, intrananogap regions between Au nanospheres, and interior space, exhibiting well-resolved single-particle surface-enhanced Raman spectroscopy signals of absorbed analytes.

We consider the Cauchy problem of a certain type of non-Newtonian fluids combined with Maxwell equations in three dimensions. We establish local existence of unique regular solutions for sufficiently smooth initial data. In addition, the regular solutions are globally extended in time, provided that the H 3 -norm of the initial data is small enough. Lastly, using the Fourier splitting method, we show that H l -norms of the global regular solution decay with the rate of ( 1 + t ) − ( 3 4 + l 2 ) for l ≥ 0 , as time tends to infinity.

This study is concerned with the following elliptic equation: − M ( ∫ R N 1 p ( x ) | ∇ u | p ( x ) d x ) div ( | ∇ u | p ( x ) − 2 ∇ u ) + V ( x ) | u | p ( x ) − 2 u = λ f ( x , u ) in R N , where M ∈ C ( R + ) is a Kirchhoff-type function, the potential function V : R N → ( 0 , ∞ ) is continuous, and f : R N × R → R satisfies a Caratheodory condition. The aim is to determine the precise positive interval of λ for which the problem admits at least two nontrivial solutions by using abstract critical point results for an energy functional satisfying the Cerami condition. It should be noted that the existence of at least one nontrivial weak solution is established by employing the mountain pass theorem. Moreover, the existence of an unbounded sequence of nontrivial weak solutions follows from the fountain theorem owing to the variational nature of the problem.

In pursuit of the elusive mechanism of high-TC superconductors (HTSC), spectroscopic imaging scanning tunneling microscopy (SI-STM) is an indispensable tool for surveying local properties of HTSC. ...

We investigate the local existence of smooth solutions of a 3D ideal magnetohydrodynamics (MHD) equations in a bounded domain and give a blow-up criteria to this equations with respect to vorticists.

We study the large-time dynamics of Cucker-Smale (C-S) flocking particles interacting with non-Newtonian incompressible fluids. Dynamics of particles and fluids were modeled using the kinetic Cucker-Smale equation for particles and non-Newtonian Navier-Stokes system for fluids, respectively and these two systems are coupled via the drag force, which is the main flocking (alignment) mechanism between particles and fluids. We present a global existence theory for weak solutions to the coupled Cucker-Smale-Navier-Stokes system with shear thickening. We also use a Lyapunov functional approach to show that sufficiently regular solutions approach flocking states exponentially fast in time.

We justify a modified Navier–Stokes system that includes the damping effect for a channel flow with an arbitrary irregular boundary. The effects of rough walls are modeled by a localized damping te...

We study the existence of infinitely many weak solutions for the elliptic type equation via a general nonsmooth critical point theorem.

We present an interior regularity condition for suitable weak solutions with respect to the magnetic pressure under the class of scaling invariance.

We give a weak − L p Serrin type regularity criteria of weak solutions to the 3D viscoelastic Navier–Stokes equations with damping, which is described as an incompressible non-Newtonian fluid in the whole space R 3 .

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are Holder continuous near boundary provided that the scaled mixed Lp,qx,t-norm of the velocity vector field with 3/p+2q ≤ 2, 2 < q < ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ɛ Lp,qx,t with 1 ≤ 3 p + 2 p ≤ 3 2 , 3 p ∞ , the Hausdorff dimension of its singular set is no greater than max {p,q} ( 3 p + 2 p - 1 ) .

This paper presents the first free-space optical transceiver IC for social, psychological brain–computer interface experiments with multiple mice. The proposed IC embedded in a head-mounted module (HMM) includes 8-channel neural recorder and a neural stimulator in order to record extracellular action potentials and local field potentials while injecting diverse waveforms of monophasic intracranial stimulation current into the medial forebrain bundle for the activation of the nervous system. The HMM implanted on a mouse transmits the neural recording data to an optical base station by using a mandatory 591/624-nm (amber/red) visible behavior tracking LED as an optical transmitter to save power. The HMM is controlled simultaneously via a 940 nm infrared optical downlink to reduce interference to the visible optical uplink. The proposed IC fabricated in a 0.18- $\mu \text{m}$ HV BCDMOS process consumes 90 $\mu \text{W}$ while achieving a 2.048-Mb/s transmission rate.

The aim of this paper is to establish the regularity criterion of weak solutions to the 3D micropolar fluid equations by one directional derivative of the pressure in anisotropic Lebesgue spaces. We improve the regularity criterion for weak solutions previously given by Jia, Zhang and Dong in [ 21 ].

Background The prognosis for cardiac arrest (CA) is associated with the degree of cerebral ischemia. We investigated the relationship between the wavelet coherence of cerebral oxyhemoglobin (HbO2) among different channels and outcomes after CA. Moreover, we aimed to develop a prognostication method after CA. Methods Eighty-three post-resuscitation patients were included. The HbO2 data were collected during the post-resuscitation period (median day, 1) using functional near-infrared spectroscopy. The coherence between sections of prefrontal HbO2 oscillations in five frequency intervals (I, 0.6–2 Hz; II, 0.15–0.6 Hz; III, 0.05–0.15 Hz; IV, 0.02–0.05 Hz; and V, 0.0095–0.02 Hz) were analyzed. We evaluated the outcomes using cerebral performance category (CPC) scores (good outcome, CPC ≤ 2 and poor outcome, CPC ≥ 3) at 3 months after CA. Additionally, the predictive method was developed using the biomarker and coherence value after CA. Results Among the included patients, 19 patients (22.9%) had a good outcome. Poor outcome group had significantly lower phase coherence in the myogenic frequency interval III compared to good outcome group (0.36 ± 0.14 vs. 0.54 ± 0.18, P Conclusions The predictive method using NSE and phase coherence of HbO2 in the interval III from the vascular smooth muscle cells could be a useful tool for prognosticating after CA. This suggests that evaluating cerebral ischemia using phase coherence of HbO2 might be a helpful outcome predictor following CA.