Relaxation Limit Research Articles

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Pollinator competition and the contingency of nectar depletion during an early spring resource pulse.

Concerns about competition between pollinators are predicated on the assumption of floral resource limitation. Floral resource limitation, however, is a complex phenomenon involving the interplay of resource production by plants, resource demand by pollinators, and exogenous factors-like weather conditions-that constrain both plants and pollinators. In this study, we examined nectar limitation during the mass flowering of rosaceous fruit trees in early spring. Our study was set in the same region as a previous study that found severe nectar limitation in summer grasslands. We used this seasonal contrast to evaluate two alternative hypotheses concerning the seasonal dynamics of floral resource limitation: either (H1) rates of resource production and consumption are matched through seasonal time to maintain a consistent degree of resource limitation, or (H2) a mismatch of high floral resource production and low pollinator activity in early spring creates a period of relaxed resource limitation that intensifies later in the year. We found generally much lower depletion in our spring study compared to the near 100% depletion found in the summer study, but depletion rates varied markedly through diel time and across sampling days, with afternoon depletion rates sometimes exceeding 80%. In some cases, there were also pronounced differences in depletion rates across simultaneously sampled floral species, indicating different degrees of nectar exploitation. These findings generally support the seasonal mismatch hypothesis (H2) but underscore the complex contingency of nectar depletion. The challenge of future work is to discern how the fluctuation of resource limitation across diel, inter-diel, and seasonal time scales translates into population-level outcomes for pollinators.

Quantitative derivation of a two-phase porous media system from the one-velocity Baer–Nunziato and Kapila systems

We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer–Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer–Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood–Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold explicitly in terms of the time-relaxation parameter, the introduction of an effective flux in the low-frequency region to overcome the loss of parameters due to the overdamping phenomenon, and renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several auxiliary unknowns allowing us to recover crucial O(ε) bounds.

Global solutions and relaxation limit to the Cauchy problem of a hydrodynamic model for semiconductors

In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E and the a-priori L∞ estimate, 0<2δ≤ρε,δ≤M(t),|uε,δ|≤M(t), where M(t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (ρε,δ,uε,δ); Second, the compensated compactness theory is applied to prove the pointwise convergence of (ρε,δ,uε,δ) as ε,δ go to zero, and that the limit (ρ(x,t),u(x,t)) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫−∞xρε,δ(x,t)−2δdx, deduces the uniform L∞ estimate, 0<2δ≤ρε,δ≤M,|uε,δ|≤M, independent of the time t and ε,δ; Finally, as a by-product, the known compactness framework [14,29] is applied to show the relaxation limit, as the relation time τ and ε,δ go to zero, for general pressure P(ρ).

A mode of convergence arising in diffusive relaxation

Abstract In this work, a mode of convergence for measurable functions is introduced. A related notion of Cauchy sequence is given, and it is proved that this notion of convergence is complete in the sense that Cauchy sequences converge. Moreover, the preservation of convergence under composition is investigated. The origin of this mode of convergence lies in the path of proving that the density of a Euler system converges almost everywhere (up to subsequences) towards the density of a non-linear diffusion system, as a consequence of the convergence in the relaxation limit.

Experimentally corroborated model of pressure relaxation limit cycle oscillations in the vicinity of the transition to high confinement in tokamaks

An analytical formula systematically predicts the observed frequency of pressure relaxation limit cycle oscillations in the vicinity of the transition to high confinement in four tokamaks (JET, ASDEX Upgrade, COMPASS, Globus-M). The experimental dataset spans the widest available range of frequencies, machine sizes and plasma ion species. The machine size dependence is explained by the connection length scale of plasma flows parallel to the magnetic field. The model also explains the observed up-down poloidal current asymmetry and the impact of the plasma ion species mass and charge.

A Central Scheme for Two Coupled Hyperbolic Systems

A novel numerical scheme to solve two coupled systems of conservation laws is introduced. The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems, which simplifies the computation of suitable coupling data. The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit. The role of this condition is discussed, a consistency concept with respect to the original problem is introduced, the well-posedness is analyzed and explicit, nodal Riemann solvers are provided. Based on a case study considering the p-system of gas dynamics, a strategy for the design of the relaxation coupling condition within the new scheme is provided.

Effect of the neutral beam injector operational regime on the Alfven eigenmode saturation phase in DIII-D plasma

The aim of this study is to analyze the effect of the neutral beam injector (NBI) operation regime on the saturation phase of the Alfven Eigenmodes (AEs) in DIII-D plasma. The analysis is done using the linear and nonlinear versions of the gyro-fluid code FAR3d. A set of parametric analyses are performed modifying the nonlinear simulation EP β (NBI injection power), EP energy (NBI voltage) and the radial location of the EP density profile gradient (NBI radial deposition). The analysis indicates a transition from the soft (local plasma relaxation) to the hard MHD (global plasma relaxation) limit if the simulation EP , leading to bursting MHD activity caused by radial AEs overlapping. MHD bursts cause an enhancement of the EP transport showing ballistic-like features as avalanche-like events. Simulations in the soft MHD limit show an increment of the EP density gradient as the EP β increases. On the other hand, there is a gradient upper limit in the hard MHD limit, consistent with the critical-gradient behavior. AEs induce shear flows and zonal current leading to the deformation of the flux surfaces and the safety factor profile, respectively, particularly strong for the simulation in the hard MHD limit. Simulations in the hard MHD regime show a decrease of the AE frequency in the saturation phase; this is caused by the destabilization of a transitional mode between a TAE and a RSAE that may explain the AE frequency down-sweeping observed in some DIII-D discharges. Reducing the EP energy in the nonlinear simulations leads to a weakening of the plasma perturbation. On the other hand, increasing the EP energy causes the opposite effect. Nonlinear simulations of off-axis NBI profiles indicate a lower plasma perturbation as the EP density gradient is located further away from the magnetic axis.

A remark on the multi-dimensional compressible Euler system with damping in the 𝐿^{𝑝} critical Besov spaces

We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in L p L^p -type critical Besov spaces. In comparison with the results obtained by Crin-Barat and Danchin [Pure Appl. Anal. 4 (2022), pp. 85–125; Math. Ann. 386 (2023), pp. 2159–2206], the more general pressure law satisfying P ′ ( ρ ¯ ) &gt; 0 P’(\bar {\rho })&gt;0 is allowed. To achieve it, a new composition estimate is established in the L 2 L^2 - L p L^p hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies.

Decoupling limit of diffusion and structural relaxation predicted by a fragility-tunable glassy model

Decoupling limit of diffusion and structural relaxation predicted by a fragility-tunable glassy model

Accelerated stress relaxation with simultaneously enhanced strength of titanium alloy by phase transformation and stress-induced twinning

Stress relaxation with synchronous aging for titanium alloys was proposed to reduce the stress relaxation limit and enhance the mechanical strength simultaneously. Stress relaxation tests at 600 °C of solution treated sample and aged sample were performed to study the interplay between microstructure evolution and stress relaxation. Kohlrausch-Williams-Watts equation was employed to analyze the relaxation kinetics. Results show that the applied stress promotes the rapid precipitation and growth of the αs (secondary α) phase. Meanwhile, the applied stress also induces the presence of twinning in the αs phase, leading to the relaxation of the elastic energy in the αs. In return, both phase transformation and stress-induced twinning reduce the stress relaxation limit by 56 %. Moreover, the tensile strength of Ti-6.5Al-2Zr-1Mo-1 V alloy at room and service temperature was improved by 3.4 % and 9.2 %, respectively.

On strictly convex entropy functions for the reactive Euler equations

This work is concerned with entropy functions of the reactive Euler equations describing inviscid compressible flow with chemical reactions. In our recent work (W. Zhao, Math. Comput. 91 (2022) 735–760.) we point out that for these equations as a hyperbolic system, the classical entropy function associated with the thermodynamic entropy is no longer strictly convex under the equation of state (EoS) for the ideal gas. In this work, we propose two strategies to address this issue. The first one is to correct the entropy function. Namely, we present a class of strictly convex entropy functions by adding an extra term to the classical one. Such strictly entropy functions contain that constructed in (W. Zhao, Math. Comput. 91 (2022) 735–760.) as a special case. The second strategy is to modify the EoS. We show that there exists a family of EoS (for the nonideal gas) such that the classical entropy function is strictly convex. Under these new EoS, the reactive Euler equations are proved to satisfy the Conservation-Dissipation Conditions for general hyperbolic relaxation systems, which guarantee the existence of zero relaxation limit. Additionally, an elegant eigen-system of the Jacobian matrix is derived for the reactive Euler equations under the proposed EoS. Numerical experiments demonstrate that the proposed EoS can also generate ZND detonations. Extension of the present results to high dimensions is direct.

Hyperbolic systems of balance laws with stiff source

The method of redistribution of damping is employed for solving the Cauchy problem, in the BV setting, for hyperbolic systems of balance laws with partially dissipative source that becomes stiff as the relaxation time shrinks to zero. One may then pass to the zero relaxation limit.

Extending the coherence of spin defects in hBN enables advanced qubit control and quantum sensing

Negatively-charged boron vacancy centers ({{V}_{B}}^{-}) in hexagonal Boron Nitride (hBN) are attracting increasing interest since they represent optically-addressable qubits in a van der Waals material. In particular, these spin defects have shown promise as sensors for temperature, pressure, and static magnetic fields. However, their short spin coherence time limits their scope for quantum technology. Here, we apply dynamical decoupling techniques to suppress magnetic noise and extend the spin coherence time by two orders of magnitude, approaching the fundamental T1 relaxation limit. Based on this improvement, we demonstrate advanced spin control and a set of quantum sensing protocols to detect radiofrequency signals with sub-Hz resolution. The corresponding sensitivity is benchmarked against that of state-of-the-art NV-diamond quantum sensors. This work lays the foundation for nanoscale sensing using spin defects in an exfoliable material and opens a promising path to quantum sensors and quantum networks integrated into ultra-thin structures.

Zero relaxation limit in BV

Zero relaxation limit in BV

Diffusive relaxation limit of the multi-dimensional Jin-Xin system

Diffusive relaxation limit of the multi-dimensional Jin-Xin system

Numerical relaxation limit and outgoing edges in a central scheme for networked conservation laws

AbstractA recently introduced scheme for networked conservation laws is analyzed in various experiments. The scheme makes use of a novel relaxation approach that governs the coupling conditions of the network and does not require a solution of the Riemann problem at the nodes. We numerically compare the dynamics of the solution obtained by the scheme to solutions obtained using a classical coupling condition. In particular, we investigate the case of two outgoing edges in the Lighthill–Whitham–Richards model of traffic flow and in the Buckley–Leverett model of two phase flow. Moreover, we numerically study the asymptotic preserving property of the scheme by comparing it to its preliminary form before the relaxation limit in a 1‐to‐1 network.

Stability of Equilibria in Time-Inconsistent Stopping Problems

We investigate the stability of equilibrium-induced optimal values with respect to reward functions and transition kernels for time-inconsistent stopping problems under nonexponential discounting in discrete time. First, with locally uniform convergence of and equipped with total variation distance, we show that the optimal value is semicontinuous with respect to . We provide examples showing that continuity may fail in general, and the convergence for in total variation cannot be replaced by weak convergence. Next we show that, with the uniform convergence of and , the optimal value is continuous with respect to when we consider a relaxed limit over -equilibria. We also provide an example showing that for such continuity the uniform convergence of cannot be replaced by locally uniform convergence.

Objective Review

Leather industry uses a great number of processing chemicals to make leather from animal hides. The toxicity, health and environmental impacts of many of these chemicals have been established leading to significant regulatory certifications and programs helping in trade and policy making. The focus of this review are the hazardous chemicals and auxiliaries relevant to leather industry, their categories, possible origins, analysis and toxicity with respect to humans and the environment and the regulatory mechanisms suggested for them through manufacturing restricted substance list (MRSL) 2.0.** Informative guidelines are provided about the most probable leather processing stage for the origin of these chemicals. Some alternative technologies, chemicals and ideas gaining popularity are also suggested as probable remedies. Recommended test methods are stated for the adequate monitoring of the hazardous chemicals. The nature and severity of chemical toxicity and corresponding limits set for their allowed use in formulations are graphically expressed for clarity and ease of understanding. The maximum number of compounds / isomers belong to classes comprising chlorinated paraffins, perfluorooctanoic acids (PFOA), perfluorooctanesulfonates (PFOS) and banned aromatic amines. Similarly, most of the restricted chemicals are used during the finishing stage and may be considered for eco-friendly alternatives. Likewise, the analytical equipment covering most testing requirements is GC-MS among other hyphenated techniques. Additionally, most critical chemicals from toxicity point of view are arsenic, cadmium and chromium (VI), whereas navy blue colorants and chlorinated aromatic compounds may be considered as less toxic among the restricted chemicals under the current scope. Surprisingly, the latest UV absorbents and polymeric fatliquors are comparatively non-toxic. Similarly, the most relaxed formulation limits are given for UV absorbers and 2-methoxypropylacetate whereas strict limits have been set for PFOA, PFOS, mercury, triclosan permethrin, sensitizing dyes etc. in MRSL 2.0.

Convergence of the relaxed compressible Navier–Stokes equations to the incompressible Navier–Stokes equations

In this paper, we study the combined low Mach number and the relaxation limits of the isentropic compressible Navier–Stokes equations with revised Maxwell’s law satisfying Galilean invariance. It is shown that, for the ill-prepared initial data, the solutions of the relaxed compressible Navier–Stokes equations converge to that of the incompressible Navier–Stokes equations as the Mach number and relaxation parameters tend to zero.