The incompressible magnetohydrodynamic equations driven by additive fractional Brownian motions are considered. We firstly establish the local existence and uniqueness of the mild solution in Lp ...space on a smooth bounded domain in Rd(d=2,3). The proof is based on the semigroup theory, fixed point theorem and the results of stochastic PDEs of linear parabolic type. In the proof, the eigenvalue problem with perfectly conducting wall condition is considered to weaken the requirements of noise terms. Finally, the global existence of mild solutions is also established by energy estimate.
We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a crucial point in order to prove the uniqueness of the kinetic solution to the ...referred problem we establish a new strong trace theorem for stochastic conservation laws, which extends to the stochastic context the pioneering theorem by Vasseur. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.
In this article, we study the Cauchy problem of three‐dimensional (3‐D) incompressible magnetohydrodynamic system with infinite energy initial data. Via some elaborate analysis of the time evolution ...of both the vorticity
ω:=∇×u and the current density
j=:∇×b, the local‐in‐time well‐posedness of mild solutions with arbitrarily large initial data in Morrey spaces is established.
We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class LtγWxs,p, where the viscosity and resistivity can be larger than the Lions exponent 5/4 and (s,γ,p) ...lies in two supercritical regimes with respect to the Ladyženskaja-Prodi-Serrin (LPS) condition. The constructed weak solutions admit the partial regularity outside a small fractal singular set in time with zero Hη⁎-Hausdorff dimension, with η⁎ being any given small positive constant. In particular, for the canonical viscous and resistive MHD, the non-uniqueness is sharp near one endpoint of the LPS condition, which extends the recent result in 22 for Navier-Stokes equations. The partial regularity for MHD equations is also new. Furthermore, the strong vanishing viscosity and resistivity result is obtained, it yields the failure of Taylor's conjecture along some sequence of weak solutions to the hyper viscous and resistive MHD equations. Our proof utilizes the spatial-temporal intermittent convex integration scheme, the temporal building blocks feature the almost optimal intermittency, which improves the recent ones constructed in 58.
We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given ...smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, for any weak solutions in Ht,xβ˜ to the ideal MHD, where β˜>0, we prove that they are the strong vanishing viscosity and resistivity limit of the weak solutions to MHD equations. This shows that, in contrast to the weak ideal limits, Taylor's conjecture does not hold along the vanishing viscosity and resistivity limits. Inspired by the works on the NSE 1, ideal MHD 2 and transport equations 3, new types of velocity and magnetic flows, featuring both the refined spatial and temporal intermittency, are constructed to respect the geometry of MHD and to control the strong viscosity and resistivity. Compatible algebraic structure is derived in the convex integration scheme. More interestingly, the new intermittent flows indeed enable us to prove the aforementioned results for the hyper-viscous and hyper-resistive MHD equations up to the sharp exponent 5/4, which coincides exactly with the Lions exponent for 3D hyper-viscous NSE.
Nous prouvons la non-unicité des solutions faibles des équations magnétohydrodynamiques 3D (en abrégé : MHD). Les solutions faibles construites ne conservent pas l'hélicité magnétique et peuvent être proches des données régulières sur les champs magnétiques et les vitesse de divergence et de moyenne nulles. De plus, nous prouvons que toute solution faible des équations MHD idéales dans Ht,xβ˜, où β˜>0, est la limite forte en zéro de la viscosité et de la résistivité d'une solution faible des équations MHD. Cela montre que, contrairement aux limites faibles idéales, la conjecture de Taylor ne tient pas pour les limite en zéro de la viscosité et de la résistivité. Inspiré par des travaux sur la NSE 1, les MHD idéales 2 et les équations de transport 3, nouveaux types de vitesse et des flux magnétiques sont construits, présentant à la fois l'intermittence raffinée spatiale et temporelle , pour resperter la géométrie des MHD et pour contrôler la viscosité et la résistivité fortes. La structure algébrique compatible est déduite dans le schéma d'intégration convexe. Plus intéressant, les nouveaux flux intermittents nous permettent de prouver les résultats susmentionnés pour les équations MHD hyper-visqueuses et hyper-résistives jusqu'à l'exposant optimal 5/4, ce qui coïncide exactement avec l'exposant de Lions pour la NSE hyper-visqueuses en 3D.
In this paper, we consider the Cauchy problem of three-dimensional incompressible magnetohydrodynamic equations. Some uniform estimates with respect to time for the coupling terms between the fluid ...and the magnetic field will be presented, under the condition that the initial M1,1 norms of the vorticity and the current density are both sufficiently small. By the above estimates, we can obtain a global-in-time well-posedness of mild solutions in Morrey spaces via some effective arguments. The asymptotic behaviours of the solutions are also obtained.
We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain O=O′×O″ where a Neumann boundary ...condition is imposed on ∂O′×O″, the hyperbolic boundary, and a Dirichlet condition is imposed on O′×∂O″, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem.
•A phase-domain modulated hybrid phase-shifting structured light measurement method is proposed.•Combining stereo phase matching and speckle matching algorithms for efficient phase ...unwrapping.•Efficient and highly accurate 3D measurement using only three patterns.
Phase-shifting profilometry (PSP) measurement technology remains a research hotspot in 3D imaging. However, existing PSP methods require projecting additional patterns, which limits measurement space, or involves compressing fringe amplitudes to eliminate phase ambiguity. These practices result in reduced measurement efficiency and robustness. To overcome these challenges, we present a novel phase-domain modulated hybrid phase-shifting (PDM-HPS) structured light 3D measurement approach to solve the phase error of existing spatial-domain modulated phase-shifting (SDM-PS) methods without imposing additional pattern or space constraints. The method integrates phase-shifting speckle and phase-shifting fringe methodologies to create a hybrid phase-shifting coding pattern. In the decoding phase, the offset speckle phase is detected and corrected by the low-precision phase extracted in the frequency domain. The separation of binary speckle and high-precision wrapped phase is realized. Ultimately, an efficient binary speckle-aided stereo phase matching process is employed to achieve unambiguous phase unwrapping. We test the PDM-HPS method in multiple complex real-world scenarios using binocular structured light systems, and compare its accuracy and efficiency with three existing advanced methods. The qualitative and quantitative results show that the proposed method can achieve the measurement accuracy similar to that of the multi-frequency (MF) method by using only 1/3 projection patterns.
The goal of this work was to find the best method for cleaning gold in order to deposit polyethyleneglycol (PEG) thiols effectively, with applications to retinal implants. Gold-coated Si wafers were ...contaminated by 0.3 mM palm oil solution and cleaned by wet chemistry methods, oxygen plasma, hydrogen plasma, or UV/ozone. A PEG-b-p(G1u50-co-Cys10) polymer was selected for comparative studies of the coating effectiveness after cleaning. Samples were characterized with attenuated total reflectance Fourier transform infrared spectroscopy (ATR-FTIR), reflection adsorption infrared spectroscopy (RAIRS), Raman, and contact angle. All methods were evaluated for cleaning efficiency, polymer coating efficiency, safety, cost, and time requirement. Plasma methods were best for cleaning, and a wet chemistry or plasma methods were best for coating. Plasma methods were generally safest, cheapest, and shortest in time. The hydrogen plasma method was recommended as the best overall method.
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