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We exploit the parquet formalism to derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and typical response functions, circumventing the reliance on higher-point vertices. This includes a concise, algebraic derivation of the multiloop flow equations, which have previously been obtained by diagrammatic considerations. Integrating the multiloop flow for a given input of the totally irreducible vertex is equivalent to solving the parquet equations with that input. Hence, one can tune systems from solvable limits to complicated situations by variation of one-particle parameters, staying at the fully self-consistent solution of the parquet equations throughout the flow. Furthermore, we use the resulting differential form of the Schwinger-Dyson equation for the self-energy to demonstrate one-particle conservation of the parquet approximation and to construct a conserving two-particle vertex via functional differentiation of the parquet self-energy. Our analysis gives a unified picture of the various many-body relations and exact renormalization group equations.

For various rock engineering, injection of fluids into rock fractures through boreholes is quite common. It is of great significance to investigate the characteristics of radial flow (RF) in rock fractures for these activities. In this study, macroscopic and mesoscopic characteristics of RF in rough rock fractures were investigated and compared with those of unidirectional flow (UF) by theoretical analysis, tests and simulations. An equation for nonlinear RF was derived for rock fractures according to conservation law of mass and Izbash’s law. Four scanned rough rock fracture models were established and used to experimentally investigate the macroscopic flow characteristics in both UF and RF. Numerical simulations were performed to clarify the mesoscopic differences in fluid pressure distributions and the flowlines of RF and UF in rock fractures. The parameters of hydraulic aperture and equivalent width for RF were obtained and correlated to those for UF. A method to calculate fracture roughness coefficient of fractures for RF related to the flow direction was proposed. The characteristic parameters, i.e., critical Reynolds numbers for the flow transition from linear to nonlinear flow, effective hydraulic apertures and non-Darcy coefficients, were obtained for the UF and RF based on the test results. It was indicated that the fracture roughness plays a critical role in the macroscopic and mesoscopic characteristics of both RF and UF. According to the test results, the macroscopic characteristic parameters for RF are related to those for UF, and the nonlinearity of RF was stronger than that of UF at a specified flow rate, which was consistent with the mesoscopic characteristics observed in the simulation that the distribution of water pressure, flow velocity and the streamlines in RF were more non-uniform than that in UF. The study results were useful to describe the RF characteristic in rock fractures with the characteristic parameters for UF, which have been investigated extensively in literature.HighlightsA nonlinear flow equation for radial flow in rock fractures was derived to describe the relationship between the hydraulic head and flow rate.The differences and relations between radial and unidirectional flow were studied from macroscopic and mesoscopic aspects.The parameters of hydraulic aperture and equivalent width for radial flow were obtained and correlated to those for unidirectional flow.The effect of fracture roughness on radial and unidirectional flow was related to the flow direction and was incorporated in the Forchheimer equation.

This paper focuses on a one-dimensional fourth-order nonlinear dispersive partial differential equation for curve flows on a Kähler manifold. The equation arises as a fourth-order extension of the one-dimensional Schrödinger flow equation, with physical and geometrical backgrounds. First, this paper presents a framework that can transform the equation into a system of fourth-order nonlinear dispersive partial differential-integral equations for complex-valued functions. This is achieved by developing the so-called generalized Hasimoto transformation, which enables us to handle general higher-dimensional compact Kähler manifolds. Second, this paper demonstrates the computations to obtain the explicit expression of the derived system for three examples of the compact Kähler manifolds, dealing with the complex Grassmannian as an example in detail. In particular, the result of the computations when the manifold is a Riemann surface or a complex Grassmannian verifies that the expression of the system derived by our framework actually unifies the ones derived previously. Additionally, the computation when the compact Kähler manifold has a constant holomorphic sectional curvature, the setting of which has not been investigated, is also demonstrated.

In recent studies of atmospheric turbulent surface exchange in complex terrain, questions arise concerning velocity-sensor tilt corrections and the governing flow equations for coordinate systems aligned with steep slopes. The standard planar-fit method, a popular tilt-correction technique, must be modified when applied to complex mountainous terrain. The ramifications of these adaptations have not previously been fully explored. Here, we carefully evaluate the impacts of the selection of sector size (the range of flow angles admitted for analysis) and planar-fit averaging time. We offer a methodology for determining an optimized sector-wise planar fit (SPF), and evaluate the sensitivity of momentum fluxes to varying these SPF input parameters. Additionally, we clarify discrepancies in the governing flow equations for slope-aligned coordinate systems that arise in the buoyancy terms due to the gravitational vector no longer acting along a coordinate axis. New adaptions to the momentum equations and turbulence kinetic energy budget equation allow for the proper treatment of the buoyancy terms for purely upslope or downslope flows, and for slope flows having a cross-slope component. Field data show that new terms in the slope-aligned forms of the governing flow equations can be significant and should not be omitted. Since the optimized SPF and the proper alignment of buoyancy terms in the governing flow equations both affect turbulent fluxes, these results hold implications for similarity theory or budget analyses for which accurate flux estimates are important.

This study proposes Chebyshev wavelet collocation method for partial differential equation and applies to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Approximate solutions of velocity and induced magnetic field are obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. Numerical results of the MHD flow problem show that the accuracy of proposed method is quite good even in the case of a small number of grid points. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number Ha ≤ 1000.

In this paper we have studied the class of Finsler metrics, called C3like metrics which satisfy the un-normal and normal Ricci flow equation and proved that such metrics are Einstein.

We address the inverse problem of identifying from some pumping tests unknown spatially distributed transmissivity and storativity occurring in 2 D depth-averaged groundwater flow equations. We employ two phases pumping rate to ensure significant effects of the unknown storativity on the transient state at early times before driving this latest to its steady state. Afterwards, the term involving the unknown storativity switches off which in turn uncouples the underlined inverse problem by allowing to focus firstly on identifying the unknown transmissivity distribution in steady flow. Based on impulse response techniques, we relax the common necessary condition into requiring the knowledge of the state at well selected observation locations. Under some reasonable requirements, we prove that the relaxed necessary condition becomes sufficient to ensure identifiability for a wide class of unknown transmissivity distributions. We develop a constructive identification approach using specific adjoint functions defined from the impulse responses at the state observation locations. Some numerical experiments are presented in Sect. 6.

The bandwidth of multimode W-type plastic optical fibers (POFs) with graded-index (GI) core distribution is investigated by solving the time-dependent power flow equation. The multimode W-type GI POF is designed from a multimode single-clad (SC) GI POF fiber upon modification of the cladding layer of the latter. Results show how the bandwidth in W-type GI POFs can be enhanced by increasing the wavelength for different widths of the intermediate layer and refractive indices of the outer cladding. These fibers are characterized according to their apparent efficiency to reduce modal dispersion and increase bandwidth.

The authors consider the energy super critical semilinear heat equation \\partial _{t}u=\\Delta u+u^{p}, x\\in \\mathbb{R}^3, p>5. The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over