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Using phase-field simulations, we investigate the bulk coarsening dynamics of ternary polymer solutions undergoing a glass transition for two models of phase separation: diffusion only and with hydrodynamics. The glass transition is incorporated in both models by imposing mobility and viscosity contrasts between the polymer-rich and polymer-poor phases of the evolving microstructure. For microstructures composed of polymer-poor clusters in a polymer-rich matrix, the mobility and viscosity contrasts significantly hinder coarsening, effectively leading to structural arrest. For microstructures composed of polymer-rich clusters in a polymer-poor matrix, the mobility and viscosity contrasts do not impede domain growth; rather, they change the transient concentration of the polymer-rich phase, altering the shape of the discrete domains. This effect introduces several complexities to the coarsening process, including percolation inversion of the polymer-rich and polymer-poor phases—a phenomenon normally attributed to viscoelastic phase separation. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

For the incompressible Navier-Stokes equations in R 3 with low viscosity ν > 0 , we consider the Cauchy problem with initial vorticity ω 0 that represents an infinitely thin vortex filament of arbitrary given strength Γ supported on a circle. The vorticity field ω (x , t) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness ν t that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that ω (x , t) is well-approximated on a large time interval by ω lin (x − a (t) , t) , where ω lin (⋅ , t) = exp (ν t Δ) ω 0 is the solution of the heat equation with initial data ω 0 , and a ˙ (t) is the instantaneous velocity given by Kelvin's formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold's geometric stability methods developed in the inviscid case ν = 0 to the slightly viscous situation. It turns out that although the geometric structures behind Arnold's approach are no longer preserved by the equation for ν > 0 , the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold's theory and interact favorably with the viscous terms. [ABSTRACT FROM AUTHOR]

The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases. • Created a variational framework based on the Timoshenko-Ehrenfest beam theory. • Proposed one-dimensional electromechanical model. • The model was solved by the spectral element method. • Verified the robustness of 1D model by comparing it to 3D models and experiments. [ABSTRACT FROM AUTHOR]

The size-dependent nonlinear vibration model of the magneto-electro-elastic (MEE) sandwich nanobeam on Winkler-Pasternak foundation is built based on nonlocal elasticity theory and Reddy's third-order shear deformation theory. Employing the Hamilton variational principle and von Karman's geometrically nonlinear theory, the nonlinear governing equation system of MEE sandwich nanobeam is deduced. The expressions of nonlinear frequency ratio and critical external electric and magnetic potential are derived with the perturbation method. Several cases are studied in order to show the influence of nonlocal parameter, Winkler and Pasternak elastic foundation, stacking sequence, external electric and magnetic potential on size-dependent nonlinear vibration behaviors of MEE simply supported sandwich nanobeam. [ABSTRACT FROM AUTHOR]

This paper studies the zero-viscosity limit for the Navier-Stokes equations with rotation, an additive white noise and nonslip boundary condition in dimension three. By establishing the well-posedness of the limit problems and constructing approximate solutions, we obtain the convergence in the mean square for the Oseen type equation with nonlinear source term, and the convergence in probability of the martingale solution of the stochastic Navier-Stokes equations with rotation. [ABSTRACT FROM AUTHOR]

This paper investigates the free vibration of functionally graded material (FGM) sandwich plates supported by different boundary conditions and influenced by a three-parameter viscoelastic foundation and hygro-thermal changes. Three types of FGM sandwich plates are studied and discussed, in which the FGM layers vary according to the power law rule and consist of ceramic and metal materials. An efficient and simple four-variable integral higher-order shear deformation theory (HSDT) is employed to model the analytical solution of the considered problem. Hamilton principle is implemented to obtain the plates' governing equations and to derive the eigenvalue equation for the free vibration study. The model is verified by comparing numerical results with previous studies on the vibration of exponentially graded plates. New results are presented in this paper showing the influences of different boundary conditions, hygro-thermal changes, viscoelastic parameters, vibration modes, material exponents, and geometric dimensions. [ABSTRACT FROM AUTHOR]

This is the first research on the smart control and frequency analysis of a cylindrical sandwich nanoshell in the framework of the numerical-based generalized differential quadrature method (2D-GDQM). The current sandwich smart nanostructure is made of a honeycomb core and piezoelectric face sheets as sensor and actuator (PFSA). For modeling the size-dependent nanoshell, nonlocal stress-strain gradient theory (NSGT) is presented. Also, this nanostructure is under conveying viscous fluid, and the related force is calculated by the modified formulation of Navier–Stokes. Also, the current structure rotates around its axial direction. The stresses and strains are obtained using the higher-order shear deformation theory (HSDT). The external voltage is applied to the sensor layer, and a Proportional-Derivative (PD) controller is used for sensor output control. Governing equations and boundary conditions of the cylindrical sandwich nanoshell are obtained by implementing Hamilton's principle. The results show that the geometry of honeycomb core, PD controller, velocity of fluid flow, length to radius ratio (L/R), and applied voltage and have a significant influence on the frequency characteristics of a cylindrical sandwich nanoshell. Another important consequence is that applying the PD controller leads to an increase in the critical velocity of fluid flow in the smart nanostructure. [ABSTRACT FROM AUTHOR]

In the present study, the sound transmission loss (STL) through the air-filled rectangular double-walled cross-ply fiber metal laminated (FML) nanoplates under simply supported and clamped boundary conditions is studied using the nonlocal strain gradient theory (NSGT) and third-order shear deformation theory (TSDT). NSGT is complemented with hardening and softening material effects, which can significantly enhance the accuracy of small-scale results. The sound velocity potential and Hamilton's principle are employed to derive the coupled size-dependent vibroacoustic equations. The Galerkin method is exploited to solve vibroacoustic equations and obtain the STL. The developed solution is examined in terms of its accuracy and precision via a comparison with other available data in existing research. The effects of different parameters such as boundary conditions, nonlocal and strain gradient parameters, lay-ups, incident angles, and acoustic cavity depth on the STL through the double-walled FML nanoplates are investigated. [ABSTRACT FROM AUTHOR]

The authors of this research paper have introduced a new method for solving Hamilton–Jacobi (HJ) equations called third-order weighted essentially nonoscillatory (WENO) method. This method uses exponential polynomials to construct numerical fluxes and smoothness indicators, which helps distinguish between singular and smooth regions more efficiently. The smoothness indicators are created using a finite difference operator that eliminates exponential polynomials. The numerical flux is constructed using an interpolation method based on exponential polynomials, which results in better outcomes around steep gradients. The new method maintains a high level of accuracy (i.e. three) in smooth regions, even near critical points. The authors have presented some numerical results to demonstrate the effectiveness of the new method and compared it with other WENO methods. [ABSTRACT FROM AUTHOR]

The free vibration analysis of a rotating sandwich conical shell with a reentrant auxetic honeycomb core and homogenous isotropic face layers reinforced with a ring support is studied. The shell is modeled utilizing the first-order shear deformation theory (FSDT) incorporating the relative, centripetal, and Coriolis accelerations alongside the initial hoop tension created by the rotation. The governing equations, compatibility conditions, and boundary conditions are attained using Hamilton's principle. Utilizing trigonometric functions, an analytical solution is derived in the circumferential direction, and a numerical one is presented in the meridional direction via the differential quadrature method (DQM). The effects of various factors on the critical rotational speeds and forward and backward frequencies of the shell are studied. The present work is the first theoretical work regarding the dynamic analysis of a rotating sandwich conical shell with an auxetic honeycomb core strengthened with a ring support. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity ν>0$\nu >0$, when the perturbations are in the critical spaces Hx1Ly2$H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ν12−δ0$\nu ^{\frac{1}{2}-\delta _0}$ with any small δ0>0$\delta _0>0$, which implies that ν12$\nu ^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo‐spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues. [ABSTRACT FROM AUTHOR]

This paper plans to propose a general numerical model for investigating the free and forced vibration behaviors of the multi-stepped functionally graded porous graphene platelet-reinforced composite (FGP-GPLRC) curved beam with various boundary conditions based on the first-order shear deformation Timoshenko beam theory by employing Rayleigh–Ritz method in conjunction with Jacobi polynomials. Different porosity and GPL distribution types are considered. The general boundary conditions of the multi-stepped FGP-GPLRC curved beam and connecting conditions of the FGP-GPLRC curved beam elements are simulated by employing boundary springs and connecting springs, respectively. The convergence and validation of the proposed numerical model are demonstrated by comparing the proposed results with the corresponding results which come from open literature and finite element software ABAQUS. The free and forced vibration behavior analysis of multi-stepped FGP-GPLRC curved beam is carried out by investigating the influences of material property and geometric parameters on the natural frequency and displacement response in the frequency and time domains of multi-stepped FGP-GPLRC curved beam structure. The investigation results can offer the technique guidance for the design and manufacture of the multi-stepped FGP-GPLRC curved beam. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic Fokas–Lenells equation with multiplicative white noise in the Itô sense. For this purpose, we will use the Jacobi elliptic expansion function method. The nonlinear partial differential equation is transformed into ordinary differential equations through the symmetry reduction technique. New optical solutions are constructed as dark, combined hyperbolic, periodic, bright, and rational solutions. Depending on the parameters, these optical solutions exhibit varying periods of qualitative characteristics. Furthermore, the effects of time on the solutions of the nonlinear stochastic Fokas–Lenells equation are examined. In addition, graphical representations of a few solutions are included to supplement our analysis. [ABSTRACT FROM AUTHOR]

We study the weak KAM theory for finite co-dimensional Hamiltonian with relative equilibrium. On the one hand, for the Tonelli (strictly convex, superlinear) finite co-dimensional Hamiltonian, we utilize Evans' approximate variational principle and show that the Hamilton-Jacobi equation with relative equilibrium has a weak KAM solution. It as a generating function can transform the original Hamiltonian system into an integrable one on the relative equilibrium. On the other hand, when the finite co-dimensional Hamiltonian is non-convex, we give a decomposition formula of the corresponding effective Hamiltonian. This formula consists of effective Hamiltonians of quasi-convex Hamiltonians. [ABSTRACT FROM AUTHOR]

We consider the compressible Navier-Stokes system describing the motion of a barotropic fluid with density dependent viscosity confined in a three-dimensional bounded domain Ω. We show the convergence of the weak solution to the compressible Navier-Stokes system to the strong solution to the compressible Euler system when the viscosity and the damping coefficients tend to zero. [ABSTRACT FROM AUTHOR]

The common geometrical (symplectic) structures of classical mechanics, quantum mechanics, and classical thermodynamics are unveiled with three pictures. These cardinal theories, mainly at the non-relativistic approximation, are the cornerstones for studying chemical dynamics and chemical kinetics. Working in extended phase spaces, we show that the physical states of integrable dynamical systems are depicted by Lagrangian submanifolds embedded in phase space. Observable quantities are calculated by properly transforming the extended phase space onto a reduced space, and trajectories are integrated by solving Hamilton's equations of motion. After defining a Riemannian metric, we can also estimate the length between two states. Local constants of motion are investigated by integrating Jacobi fields and solving the variational linear equations. Diagonalizing the symplectic fundamental matrix, eigenvalues equal to one reveal the number of constants of motion. For conservative systems, geometrical quantum mechanics has proved that solving the Schrödinger equation in extended Hilbert space, which incorporates the quantum phase, is equivalent to solving Hamilton's equations in the projective Hilbert space. In classical thermodynamics, we take entropy and energy as canonical variables to construct the extended phase space and to represent the Lagrangian submanifold. Hamilton's and variational equations are written and solved in the same fashion as in classical mechanics. Solvers based on high-order finite differences for numerically solving Hamilton's, variational, and Schrödinger equations are described. Employing the Hénon–Heiles two-dimensional nonlinear model, representative results for time-dependent, quantum, and dissipative macroscopic systems are shown to illustrate concepts and methods. High-order finite-difference algorithms, despite their accuracy in low-dimensional systems, require substantial computer resources when they are applied to systems with many degrees of freedom, such as polyatomic molecules. We discuss recent research progress in employing Hamiltonian neural networks for solving Hamilton's equations. It turns out that Hamiltonian geometry, shared with all physical theories, yields the necessary and sufficient conditions for the mutual assistance of humans and machines in deep-learning processes. [ABSTRACT FROM AUTHOR]

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution. [ABSTRACT FROM AUTHOR]

In this study, the stability and nonlinear vibrations of a thin plate in narrow aero-thermal channel are studied. According to the classic nonlinear plate theory and linear potential flow theory, the governing equation of the thin plate in narrow aero-thermal channel is obtained by employing the Hamilton's principle and discretized into nonlinear ordinary differential equations by using the assumed mode method. The effects of the temperature variation, airflow velocity and slit height on the stability and nonlinear vibrations of the plate are considered. The stability of the thin plate in flow field is obtained by generalized eigenvalue method. For the nonlinear system, the nonlinear vibrations of the plate subjected to harmonic excitation are studied by using incremental harmonic balance method. From the results, with the temperature change, the airflow velocity, and the slit height decreasing, the stability of the plate becomes worse. The effects of the temperature change, airflow velocity and slit height on the nonlinear amplitude-frequency response of the first four generalized coordinates of the plate are discussed. It is found that with the temperature change, the airflow velocity, and the slit height decreasing, the resonance frequency of the plate decreases and resonance amplitude of the plate increases. In the simulation, some novel nonlinear dynamic behaviors of the plate in narrow aero-thermal channel are also observed such as the resonance trough and the resonance peak of the unstable solution. [ABSTRACT FROM AUTHOR]

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed. [ABSTRACT FROM AUTHOR]

In this paper, we will review two analytical approaches to the construction of non-homogeneous Baryonic condensates in the low-energy limit of QCD in (3+1) dimensions. In both cases, the minimal coupling with the Maxwell U (1) gauge field can be taken explicitly into account. The first approach (which is related to the generalization of the usual spherical hedgehog ansatz to situations without spherical symmetry at a finite Baryon density) allows for the construction of ordered arrays of Baryonic tubes and layers. When the minimal coupling of the Pions to the U (1) Maxwell gauge field is taken into account, one can show that the electromagnetic field generated by these inhomogeneous Baryonic condensates is of a force-free type (in which the electric and magnetic components have the same size). Thus, it is natural to wonder whether it is also possible to analytically describe magnetized hadronic condensates (namely, Hadronic distributions generating only a magnetic field). The idea of the second approach is to construct a novel BPS bound in the low-energy limit of QCD using the theory of the Hamilton–Jacobi equation. Such an approach allows us to derive a new topological bound which (unlike the usual one in the Skyrme model in terms of the Baryonic charge) can actually be saturated. The nicest example of this phenomenon is a BPS magnetized Baryonic layer. However, the topological charge appearing naturally in the BPS bound is a non-linear function of the Baryonic charge. Such an approach allows us to derive important physical quantities (which would be very difficult to compute with other methods), such as how much one should increase the magnetic flux in order to increase the Baryonic charge by one unit. The novel results of this work include an analysis of the extension of the Hamilton–Jacobi approach to the case in which Skyrme coupling is not negligible. We also discuss some relevant properties of the Dirac operator for quarks coupled to magnetized BPS layers. [ABSTRACT FROM AUTHOR]

How does the interplay between selection, mutation and horizontal gene transfer modify the phenotypic distribution of a bacterial or cell population? While horizontal gene transfer, which corresponds to the exchange of genetic material between individuals, has a major role in the adaptation of many organisms, its impact on the phenotypic density of populations is not yet fully understood. We study an elliptic integro-differential equation describing the evolutionary equilibrium of the phenotypic density of an asexual population. In a regime of small mutational variance, we characterize the solution which results from the balance between competition for a resource, mutation and horizontal gene transfer. We show that in a certain range of parameters polymorphic equilibria exist, which means that the phenotypic density may concentrate around several dominant traits. Such polymorphic equilibria result from an antagonist interplay between horizontal gene transfer and selection, while similar models which neglect the transfer lead only to monomorphic equilibria. [ABSTRACT FROM AUTHOR]

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. [ABSTRACT FROM AUTHOR]

We study the optimal convergence rate for the homogenization problem of convex Hamilton–Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of (Tran and Yu (2021)), which means the optimal convergence rate is also O (ε). [ABSTRACT FROM AUTHOR]

We consider the Hamilton-Jacobi equation H (x , D u) + λ (x) u = c , x ∈ M , where M is a connected, closed and smooth Riemannian manifold. The functions H (x , p) and λ (x) are continuous. H (x , p) is convex, coercive with respect to p , and λ (x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function. [ABSTRACT FROM AUTHOR]

We study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type Ω λ = (1 + r (λ)) Ω with a continuous function r and a positive constant λ. We characterize all solutions to the ergodic problem on Ω in terms of r. In addition, we demonstrate that the additive eigenvalue λ ↦ c Ω λ on a rescaled domain Ω λ = (1 + λ) Ω possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of λ ↦ c Ω λ . We provide examples where higher regularity is achieved. [ABSTRACT FROM AUTHOR]

We show the existence and the regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with density-dependent viscosity coefficients, by analyzing a fourth-order nonlinear elliptic equation for the stream function. For some stationary symmetric flows, we reformulate the Navier–Stokes equations as ordinary differential equations and give explicit examples of weak solutions. We present some further (ir-)regularity results in the case of piecewise-constant viscosity coefficients. [ABSTRACT FROM AUTHOR]

This paper studies a maximal L q -regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a L p duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold q = d (γ − 1) / γ , d being the dimension of the ambient space and γ > 1 the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range q > 1. [ABSTRACT FROM AUTHOR]

In this paper, we investigate direct and inverse problems for time-fractional pseudoparabolic equations associated with the Jacobi operator. The existence and uniqueness of the solutions are proven. Also, the stability result of the inverse source problem (ISP) is established. [ABSTRACT FROM AUTHOR]

In the setting of Carnot groups, we prove the q -logarithmic Sobolev inequality for probability measures as a function of the Carnot–Carathéodory distance. As an application, we use the Hamilton–Jacobi equation in the setting of Carnot groups to prove the p -Talagrand inequality and hypercontractivity. [ABSTRACT FROM AUTHOR]

This article considers a bidirectional functionally graded (BDFG) beam with various distributions of volume fraction. The impact of these distributions on the buckling and bending response of BDFG beam with general boundary conditions is investigated via a quasi-3D solution. It is assumed that the beam is exposed to a set of in-plane varying compressive loads and there mechanical characteristics change in both directions. Also, the BDFG beam is considered to be exposed different external mechanical loads. Hamilton's principle is employed to derive the governing equations, which are then solved using analytical solution to obtain buckling and bending characteristics. The precision of the current formulation is investigated via a comparison with available data in literature. Some numerical results are presented and discussed to assess the effect of different parameters such type of volume fraction, boundary condition, varying load, and beam geometry on the buckling and bending of BDFG beam. [ABSTRACT FROM AUTHOR]

All non-equivalent solutions of vacuum Maxwell equations are found for the case when space-time manifolds admit simply transitive four-parameter groups of motions G 4 (N). The potentials of the admissible electromagnetic fields admit the existence of the algebra of motion integrals of the Hamilton–Jacobi and Klein–Gordon–Fock equations which is isomorphic to the algebra of the group operators for the same group G 4 (N). [ABSTRACT FROM AUTHOR]

Two phase flows are common in pipeline engineering, and most times, the assumption of single phase flow is usually assumed. In this paper, the dynamics of a slightly curved Maxwell viscoelastic pipe conveying two phase flow under four boundary conditions; cantilever, simple supported, clamped simply supported and clamped clamped supported is investigated. Eigenfunction expansion was used to discretize the PDE into four modes. The linear analysis results show that the Maxwell viscoelastic parameter has no significant effect on the first and the second modes of the vibration, but on the third and fourth modes for the three boundaries conditions considered. The void fraction has effects on the frequency and critical velocity only if the densities of the components of the two phase fluid are significantly large. The nonlinear dynamics shows that Maxwell viscoelastic parameter reduces the bifurcation position and changes the bifurcation point from being sharp to a gentle curve at the bifurcation point. Further results were presented for the Maxwell parameter and velocity of fluid and temperature loading on the pipe, which are very important to understanding the behavior of Maxwell viscoelastic pipe conveying two phase flow. • Vibrations of a slightly curved Maxwell viscoelastic pipe conveying two phase flow are investigated. • The governing equation for the pipe is derived based on Hamilton's principle. • The two-phase components densities determine whether the flow can be classified as single phase or two phase. • Maxwell parameter tends to affect the response of the nonlinear system significantly when the value ranges from zero to 50. [ABSTRACT FROM AUTHOR]

In recent decades, the importance of surface acoustic waves, as a biocompatible tool to integrate with microfluidics, has been proven in various medical and biological applications. The numerical modeling of acoustic streaming caused by surface acoustic waves in microchannels requires the effect of viscosity to be considered in the equations which complicates the solution. In this paper, it is shown that the major contribution of viscosity and the horizontal component of actuation is concentrated in a narrow region alongside the actuation boundary. Since the inviscid equations are considerably easier to solve, a division into the viscous and inviscid domains would alleviate the computational load significantly. The particles' traces calculated by this approximation are excellently alongside their counterparts from the completely viscous model. It is also shown that the optimum thickness for the viscous strip is about 9-fold the acoustic boundary layer thickness for various flow patterns and amplitudes of actuation. [ABSTRACT FROM AUTHOR]

In this work we study the splitting distance of a rapidly perturbed pendulum H(x, y, t) = 1/2y² + (cos(x) 1) + μ (cos(x) - 1)g(t\ω) with g(τ) = ∑|k| >1 g[k]eikτ a 2π-periodic function and μ, ω≪ 1. Systems of this kind undergo exponentially small splitting, and, when μ ≪ 1, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided g[±1] ≠ 0. Our study focuses on the case g[±1] = 0, and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as g(τ) = cos(5τ) + cos(4τ) + cos(3τ). On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency p/q in Arnold's original model for diffusion leads to the consideration of pendulum-like Hamiltonians with g(τ) = sin(p·t\ω)+cos(q·t\ω), where, for most p, q ∈ Z, the perturbation satisfies g[±1] = 0. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton--Jacobi formalism. The leading exponentially small term appears at order μn, where n is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it. [ABSTRACT FROM AUTHOR]

The study explores the free vibration characteristics of hybrid laminated nanocomposite truncated conical shells and panels incorporating functionally graded graphene platelet-reinforced (FG-GPLs) and functionally graded carbon nanotube-reinforced (FG-CNTs) materials, advancing understanding in hybrid material applications. The structures are assumed to be supported by a two-parameter elastic foundation, and the study also considers the influence of CNT agglomeration. The estimation of CNT agglomeration effect is performed using a two-parameter agglomeration model based on the Mori–Tanaka approach, considering the random orientation of carbon nanotubes. By employing a combination of various control parameters, it becomes possible to determine the optimal mode for setting the natural frequency of the system. To ensure accurate calculations for both thin and thick shells, a third-order shear deformation theory is employed. The governing equations and boundary conditions are formulated based on Hamilton's principle. Numerical solutions are obtained through the systematic differential quadrature method, wherein the Kronecker delta function is employed. By introducing subtle adjustments to the governing equations, this method effectively minimizes computational volume and complexity, proving particularly advantageous for addressing problems characterized by higher degrees of freedom. To estimate the effective mechanical properties of the CNT-reinforced nanocomposite layers, the rule of mixtures is employed. Meanwhile, the Halpin–Tsai micromechanical model is utilized for calculating the properties of the GPL-reinforced nanocomposite layers. The presented study establishes convergence and accuracy through evaluation, considering various parameters such as CNTs volume fraction, GPLs mass fraction, different distribution patterns, different geometries under various boundary conditions, vertex angle of the cone, agglomeration characteristics of CNTs, and different stiffness of the elastic foundation. [ABSTRACT FROM AUTHOR]

In this article we derive the equations that constitute the nonlinear model of extensible thermoelastic Timoshenko microbeam. The constructed mathematical model is based on the modified couple stress theory which implies prediction of size dependent effects in microbeam resonators, by applying the Hamilton principle to full von Kármán equations. This takes account of the effects of extensibility where the dissipations are entirely contributed by temperature. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By an approach based on the Gearhart‐Herbst‐Prüss‐Huang theorem, we prove that the associated linear semigroup (without extensibility) is not analytic in general. In the absence of additional mechanical dissipations, the system is often not highly stable. Then by adding a damping frictional function to the first equation of the derived model and using the multiplier method, we show that the solutions decay exponentially under a condition on the physical coefficients. [ABSTRACT FROM AUTHOR]

We propose a new globally convergent numerical method to compute solution Hamilton-Jacobi equations defined in R d , d ≥ 1 , on a truncated bounded domain. This method is named the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented. [ABSTRACT FROM AUTHOR]

We study parameterisation-independent closed planar curve matching as a Bayesian inverse problem. The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space, leading to a large deformation diffeomorphic metric mapping (LDDMM) functional penalising the kinetic energy of the deformation. We solve Hamilton's equations for the curve matching problem using the Wu-Xu element (Wu and Xu (2019) [12]) which provides mesh-independent Lipschitz constants for the forward motion of the curve, and solve the inverse problem for the momentum using Bayesian inversion. Since this element is not affine-equivalent we provide a pullback theory which expedites the implementation and efficiency of the forward map. We adopt ensemble Kalman inversion (EKI) using a negative Sobolev norm mismatch penalty to measure the discrepancy between the target and the ensemble mean shape. We provide several numerical examples to validate the approach. [ABSTRACT FROM AUTHOR]

In this paper, we have explored the optical characteristics, namely the shadow and the deflection angle, inherent to the solution of a 4D-AdS-Einstein–Gauss–Bonnet black hole. This solution, which finds its inspiration in noncommutative geometry, had previously been established in our previous work. The radius of the shadow was determined using the Hamilton-Jacobi method and the Carter separation. Our results revealed that the presence of noncommutativity in the background of spacetime impacts the variation of the shadow radius. More specifically, we have demonstrated that an increase in the parameter θ induces a decrease in the radius of the shadow. In a similar way, analogous observations have been made by studying the variation of the electric charge Q. The noncommutative parameter θ and the electric charge Q have been constrained regarding the EHT observation data of the M87* and Sgr A* black holes. Furthermore, the angle of deflection, which is the outcome of lensing by the black hole, has been derived following the Ishihara et al. approach for a receiver and source positioned at finite distances from the black hole in an asymptotically non-flat spacetime. The impact of the noncommutative parameter θ and the charge Q of the black hole are hence analyzed, and our results depict that these parameters have a significant influence on the angle at which light is deflected by the gravitational field of the black hole. [ABSTRACT FROM AUTHOR]

Axially loaded beam-like structures, such as beam segments and piers of cable-stayed bridges, are widely used in civil engineering. This paper presents a study on the linear and nonlinear vibration of an axially loaded Timoshenko beam with elastic supports. The governing equations of the system are derived using Timoshenko beam theory and von Kármán geometric nonlinearity within the framework of Hamilton's principle. A transition parameter is introduced to describe the direction of the axial internal force during the beam's deformation. Galerkin's method is employed to reduce the nonlinear partial differential equations, from which linear solutions can be obtained. Nonlinear discretized equations are solved using the incremental harmonic balance (IHB) method. Frequency-response curves of the beam system are tracked by pseudo arc-length continuation, and Floquet's theory is used to determine the system's stability. Numerical examples are presented to validate the proposed solutions, comparing the results with the literature and the Runge-Kutta method. The effects of the transition parameter, elastic support's stiffness, and supported position are investigated for both linear and nonlinear vibration of the axially loaded Timoshenko beam. Of particular interest is the significance of the transition parameter, especially for a Timoshenko beam subjected to extreme axial compression with a small slenderness ratio. The present analysis provides critical insights into the design and analysis of such axially loaded structures in various civil engineering applications. [ABSTRACT FROM AUTHOR]

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method. [ABSTRACT FROM AUTHOR]

In this study, phenomenological observations and the Kreuer interpretation of the origin of viscosity were used to develop a computational method for solving the turbulence problem of incompressible viscous Newtonian fluids based on extended Navier–Stokes (N–S) equations. The shear process in fluid flow was hypothesized to be accompanied by eddy formation, and the effects of eddies on the convection and diffusion were considered. The classical N–S equations were improved to obtain extended N–S equations. The extended equations are closed, and the sources of the velocity fluctuations are explicitly considered to be additional convection and diffusion. The extended equations are compatible with the classical N–S equations; thus, they can describe laminar and turbulent flows in a unified manner. In fluid flow simulations, the equations describing the mean flow quantities could be directly obtained from the extended N–S equations without any additional turbulence models. A numerical investigation was carried out to verify the extended equations by exploring the flow over a cube placed in a channel. The simulation results were compared with both the large eddy simulation and experimental results. [ABSTRACT FROM AUTHOR]

We propose a hierarchy of the nonlocal mKdV (NmKdV) equation. Based on a constraint, we obtain nonlocal finite-dimensional integrable systems in a Lie–Poisson structure. By a coordinate transformation, the nonlocal Lie–Poisson Hamiltonian systems are reduced to nonlocal canonical Hamiltonian systems in the standard symplectic structure. Moreover, using the nonlocal finite-dimensional integrable systems, we give parametric solutions of the NmKdV equation and the generalized nonlocal nonlinear Schrödinger (NNLS) equation. According to the Hamilton–Jacobi theory, we obtain the action–angle-type coordinates and the inversion problems related to Lie–Poisson Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In this paper, a research on theoretical analysis and finite element simulation of the damping sandwich composite plate with fixed support condition is carried out, and the dynamic characteristics of composite plates are explored by modal experiments. The vibration equilibrium equation of damping sandwich composite plate with fixed support condition is derived by combining first-order shear deformation theory, variational principle, and Hamilton principle. A set of modal basis functions satisfying the fixed support condition are established. The analytical solution is obtained via Galerkin weighted residual theory. The influence of structural parameters on the free vibration characteristics of the sandwich composite plates is investigated based on the mutual verification of theoretical calculation, numerical simulation, and modal test results. When the damping layer (0.2 mm) is located on the surface of the skin (2 mm), the first-order mode frequency and first-order loss factor are 196.66 Hz and 0.01, respectively. When the damping layer is in the neutral layer, the first-order mode frequency and first-order loss factor are 189.99 Hz and 0.14, respectively. Compared with the damping layer in the skin, the first-order mode frequency is reduced by 3.39%, while the first-order loss factor is 14 times higher. The results show that the damping performance of the structure is optimal when the damping layer is located in the neutral layer, and the structural stiffness is higher when the damping layer is close to the skin. The work provides theoretical guidance for the development and application of lightweight composite structures with large damping and high stiffness. [ABSTRACT FROM AUTHOR]

We introduce a scalar elliptic equation defined on a boundary layer given by Π2 × [0, ztop], where Π2 is a two dimensional torus, with an eddy vertical viscosity of order zα, α∈ [0, 1], an homogeneous boundary condition at z = 0, and a Robin condition at z = ztop. We show the existence of weak solutions to this boundary problem, distinguishing the cases 0 ≤ α < 1 and α = 1. Then we carry out several numerical simulations, showing the ability of our model to accurately reproduce profiles close to those predicted by the Monin–Obukhov theory, by calculating stabilizing functions. [ABSTRACT FROM AUTHOR]

We consider the infinite horizon optimal control problems of the controlled Markov process. We verify the relationship between the controlled Markov process and its fluid limit by the viscosity solution approach. More precisely, we show that the value function of the controlled Markov process converges to one of its limit processes which is the viscosity solution of the associated Hamilton--Jacobi--Bellman equation. [ABSTRACT FROM AUTHOR]

Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schrödinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used. [ABSTRACT FROM AUTHOR]

The investigation revolved around the study of the time fractional Gardner equation, which was examined in terms of the conformable derivative. The reduction of the Gardner equation to an integer order nonlinear ordinary differential equation was carried out, and subsequently, the resulting equations were solved using the Jacobi elliptic function method. The construction of exact solutions, including solitary wave, periodic, and shock wave solutions, for the fractional order of the Gardner equation was performed. A comparison between the exact solutions and the fractional solutions was presented. This work is important because the suggested technique offers a simple and efficient way to examine a wide range of nonlinear fractional differential equations. By employing this approach, it becomes possible to solve several nonlinear time-fractional differential equations that involve conformable derivatives. The graphical representation of the resulting data simplifies the process of determining the physical significance of the equation. [ABSTRACT FROM AUTHOR]