Fractional Dissipation Research Articles

Convergence rates and central limit theorem for 3-D stochastic fractional Boussinesq equations with transport noise

This work investigates the 3-D stochastic modified fractional Boussinesq equations driven by transport noise on the torus. The existence of weak solutions is initially established through Galerkin approximation and compactness techniques. Subsequently, we demonstrate that the weak solutions of stochastic modified fractional Boussinesq equations converge to the unique solution of a deterministic Boussinesq model with fractional dissipation, subject to an appropriate noise scaling, along with providing the quantitative estimates with explicit convergent rates. Finally, the central limit theorem with an explicit convergence rate is established based on the aforementioned scaling limit. It is noteworthy that the scaling limit result of stochastic modified fractional Boussinesq equations to deterministic Boussinesq models with additional fractional dissipation implies that the transport noise regularizes the modified fractional Boussinesq equations, thus leading to “approximate weak uniqueness”.

Polynomial decay of the energy of solutions of coupled wave equations with locally boundary fractional dissipation

In this paper, we investigate a system of coupled wave equations featuring boundary fractional damping applied to a portion of the domain. We first establish the well-posedness of the system, proving the existence and uniqueness of solutions through semi-group theory. While the system does not exhibit exponential stability, we demonstrate its strong stability. Furthermore, leveraging Arendt and Batty’s general criteria and certain geometric conditions, we prove a polynomial rate of energy decay for the solutions.

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller–Segel system

This paper is concerned with the fractional Keller–Segel system in temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag–Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller–Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition on the initial data.

Stability for a System of the 2D Incompressible MHD Equations with Fractional Dissipation

Stability for a System of the 2D Incompressible MHD Equations with Fractional Dissipation

Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation

Abstract In this article, we focus on the global regularity of n-dimensional liquid crystal equations with fractional dissipation terms ( − Δ ) α u {\left(-\Delta )}^{\alpha }u and ( − Δ ) β d {\left(-\Delta )}^{\beta }d . We show that the equations have a unique global smooth solution if α ≥ 1 2 + n 4 \alpha \ge \frac{1}{2}+\frac{n}{4} and β ≥ 1 2 + n 4 \beta \ge \frac{1}{2}+\frac{n}{4} .

Numerical energy dissipation for time fractional volume-conserved Allen–Cahn model based on the ESAV and R-ESAV approaches

In this study, we consider volume-conserved numerical schemes for the volume-conserved time fractional Allen–Cahn equation. We start with the L1 scheme based on a modified exponential scalar auxiliary variable (ESAV) approach for handling the nonlinear potential term. Then we introduce the fast L1 scheme to significantly reduce the CPU time and to make the long-term computation possible. Further to reduce the error between the discrete energy and the original energy of the ESAV approach, we introduce the fast L1 scheme based on a relaxed modified ESAV (R-ESAV) idea. We rigorously show the discrete energy dissipation properties for the above three schemes, including the discrete energy boundedness law, the discrete fractional energy dissipation law, and the discrete weighted energy dissipation law. Theoretical findings are validated by a few numerical experiments.

Development of a novel fractional magneto-viscoelastic dynamic model for an adaptive beam featuring functional composite magnetoactive elastomers: Simulations and experimental studies

Recently, the rapid emergence of functional composite magnetoactive elastomers with integrated hard magnetic particles (known as hard magnetoactive elastomers) has attracted significant interest in fields such as soft robotics and material science. It is of paramount importance to develop an efficient model that can accurately predict the nonlinear dynamic behavior of adaptive structures for their practical application and the synthesis of effective control strategies. In this study, a novel nonlinear fractional magneto-viscoelastic model of an adaptive cantilevered beam featuring hard magnetoactive elastomers has been developed. This model effectively characterizes the beam’s nonlinear and rate-dependent response behavior under magnetic stimuli of varying frequencies and magnitudes. Considering the fractional Kelvin-Voigt energy dissipation model and large deformation nonlinearity, the governing equations for the cantilevered hard-magnetic soft beam were derived using the Hamilton principle. The finite difference method, combined with the Galerkin modal decomposition scheme, was subsequently utilized to discretize the time and space domains, respectively. A hardware-in-the-loop experimental framework was finally designed to experimentally investigate the beam’s nonlinear response behavior and validate the simulation results. The simulated nonlinear quasi-static responses of the beam, subjected to magnetic loading up to 30 mT, exhibited excellent agreement with the experimental data collected. Moreover, the simulated dynamic responses of the beam, subjected to various amplitudes of the applied magnetic field and magnetic frequencies up to 2 Hz, were thoroughly investigated. These included time-histories, phase-plane, and hysteretic responses, all demonstrating strong alignment with the experimental observations.

Dissipative Dark Matter on FIRE. II. Observational Signatures and Constraints from Local Dwarf Galaxies

We analyze the first cosmological baryonic zoom-in simulations of galaxies in dissipative self-interacting dark matter (dSIDM). The simulations utilize the FIRE-2 galaxy formation physics with the inclusion of dissipative dark matter self-interactions modeled as a constant fractional energy dissipation (f diss = 0.75). In this paper, we examine the properties of dwarf galaxies with M * ∼ 105–109 M ⊙ in both isolation and within Milky Way–mass hosts. For isolated dwarfs, we find more compact galaxy sizes and promotion of disk formation in dSIDM with (σ/m) ≤ 1 cm2 g−1. On the contrary, models with (σ/m) = 10 cm2 g−1 produce puffier stellar distributions that are in tension with the observed size–mass relation. In addition, owing to the steeper central density profiles, the subkiloparsec circular velocities of isolated dwarfs when (σ/m) ≥ 0.1 cm2 g−1 are enhanced by about a factor of 2, which are still consistent with the kinematic measurements of Local Group dwarfs but in tension with the H i rotation curves of more massive field dwarfs. Meanwhile, for satellites of Milky Way–mass hosts, the median circular velocity profiles are marginally affected by dSIDM physics, but dSIDM may help promote the structural diversity of dwarf satellites. The number of satellites is slightly enhanced in dSIDM, but the differences are small compared with the large host-to-host variations. In conclusion, the dSIDM models with (σ/m) ≳ 0.1 cm2 g−1, f diss = 0.75 are in tension in massive dwarfs (M halo ∼ 1011 M ⊙) due to circular velocity constraints. However, models with lower effective cross sections (at this halo mass/velocity scale) are still viable and can produce nontrivial observable signatures.

Dynamics of a one-dimensional non-autonomous laminated beam

In this paper is analyzed the pullback dynamics of a laminated beam model subject to fractional Laplacian dissipation, nonlinear source terms and non autonomous external forces. For each gamma exponent of the Laplacian in the open interval with endpoints 0 and 1/2, the model is well-posedness and the evolution process associated to solutions of problem possesses a pullback attractor for a general basin of attraction. To conclude, we prove that the attractors are upper semicontinuous as γ tends to 0. The result is new and it is the first time when the non-autonomous laminated beam is studied.

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Well-Posedness and $$L^2$$-Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

Well-Posedness and $$L^2$$-Decay Estimates for the Navier–Stokes Equations with Fractional Dissipation and Damping

Stability of Wave Equation with Variable Coefficients by Boundary Fractional Dissipation Law

Stability of Wave Equation with Variable Coefficients by Boundary Fractional Dissipation Law

Regularity criterion of three dimensional magneto-micropolar fluid equations with fractional dissipation

<p>In this paper, we investigate the regularity criterion of weak solutions to three-dimensional magneto-micropolar fluid equations with fractional dissipation. A regularity criterion is established via the third component of the velocity fields, the micro-rotational velocity fields, and the magnetic fields.</p>

Suppression of blow up by mixing in generalized Keller-Segel system with fractional dissipation and strong singular kernel

Suppression of blow up by mixing in generalized Keller-Segel system with fractional dissipation and strong singular kernel

Random perturbations for the chemotaxis-fluid model with fractional dissipation: Global pathwise weak solutions

Random perturbations for the chemotaxis-fluid model with fractional dissipation: Global pathwise weak solutions

Asymptotic Behavior of a transmission Heat/Piezoelectric smart material with internal fractional dissipation law

Asymptotic Behavior of a transmission Heat/Piezoelectric smart material with internal fractional dissipation law

Data-driven characterization of viscoelastic materials using time-harmonic hydroacoustic measurements

Any numerical procedure in mechanics requires choosing an appropriate model for the constitutive law of the material under consideration. The most common assumptions regarding linear wave propagation in a viscoelastic material are the standard linear solid model, (generalized) Maxwell, Kelvin-Voigt models or the most recent fractional derivative models. Usually, once the frequency-dependent constitutive law is fixed, the intrinsic parameters of the mathematical model are estimated to fit the available experimental data with the mechanical response of that model. This modelling methodology potentially suffers from the epistemic uncertainty of an inadequate a priori model selection. However, in this work, the mathematical modelling of linear viscoelastic materials and the choice of their frequency-dependent constitutive laws is performed based only on the available experimental measurements without imposing any functional frequency dependence. This data-driven approach requires the numerical solution of an inverse problem for each frequency. The acoustic response of a viscoelastic material due to the time-harmonic excitations has been calculated numerically. In these numerical simulations, the non-planar directivity pattern of the transducer has been taken into account. Experimental measurements of insertion loss and fractional power dissipation in underwater acoustics have been used to illustrate the data-driven methodology that avoids selecting a parametric viscoelastic model.

Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces

In this article, we prove the existence of a unique global solution for the critical case of the generalized Navier-Stokes equations in Lei-Lin and Lei-Lin-Gevrey spaces, by assuming that the initial data is small enough. Moreover, we obtain a unique local solution for the subcritical case of this system, for any initial data, in these same spaces. It is important to point out that our main result is obtained by discussing some properties of the solutions for the heat equation with fractional dissipation.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/78/abstr.html

Blow-Up of Solution of Lamé Wave Equation with Fractional Damping and Logarithmic Nonlinearity Source Terms

In this work, by the use of a semigroup theory approach, we provide a global solution for an initial boundary value problem of the wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. In addition to this, we establish a blow-up result for the solution under the condition of non-positive initial energy.

Existence and nonexistence of solution of fractional Lamé wave equation with polynomial nonlinearity source terms

In this work, we examine the existence and nonexistence of solution for lamé wave equation with a nonlinear source terms and fractional boundary dissipation. By using the semi group theory approach, we prove the existence of a global solution. Furthermore, we established the blow-up results under the non-positive initial energy condition.