Quasilinear Systems Research Articles

An investigation of optimality and controllability for quasilinear impulsive evolution systems in abstract spaces

An investigation of optimality and controllability for quasilinear impulsive evolution systems in abstract spaces

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture

Abstract The Neumann problem for the Keller-Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = − ∫ Ω u d x , is considered in n-dimensional balls Ω with n ⩾ 2 , with suitably regular and radially symmetric, radially nonincreasing initial data u 0. The functions D and S are only assumed to belong to C 2 ( [ 0 , ∞ ) ) and to satisfy D > 0 and S ⩾ 0 on [ 0 , ∞ ) as well as S ( 0 ) = 0 ; in particular, diffusivities with arbitrarily fast decay are included. In this general context, it is shown that it is merely the asymptotic behavior as ξ → ∞ of the expression I ( ξ ) := S ( ξ ) ξ 2 n D ( ξ ) , ξ > 0 , which decides about the occurrence of blow-up: Namely, it is seen that • if lim ξ → ∞ I ( ξ ) = 0 , then any such solution is global and bounded, that • if lim sup ξ → ∞ I ( ξ ) < ∞ and ∫ Ω u 0 is suitably small, then the corresponding solution is global and bounded, and that • if lim inf ξ → ∞ I ( ξ ) > 0 , then at each appropriately large mass level m, there exist radial initial data u 0 such that ∫ Ω u 0 = m , and that the associated solution blows up either in finite or in infinite time. This especially reveals the presence of critical mass phenomena whenever lim ξ → ∞ I ( ξ ) ∈ ( 0 , ∞ ) exists.

Interesting class of quasilinear systems

With this work, I want to draw attention to a class of differential equations that encompasses a number of important equations of mathematical physics and is convenient for constructing a mathematical theory.

A lower bound estimate of solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation

This paper investigates the Cauchy problems for a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system via Riemannian invariants. Based on the first a priori estimate of the solutions, the second a priori estimate of the derivation of the solutions, and the third a priori estimate of the continuous mould of the first‐order partial derivatives of the solutions to quasilinear hyperbolic system, a lower bound estimate of classical solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation is derived.

Stability analysis of quasilinear systems on time scale based on a new estimation of the upper bound of the time scale matrix exponential function

In this paper, the stability of quasilinear systems on time scale is analyzed based on a new estimation of the upper bound of the time scale matrix exponential function. First, some new upper bounds for the norm of the matrix exponential function eA(t,t0) are derived, where A is a regressive square matrix, t,t0∈T, T being a time scale. The matrix exponential function generalizes the usual matrix exponential as well as the integer power of a matrix. It is shown that the obtained bounds are more accurate compared to the existing bounds of the norm of the matrix exponential function. One of the upper bounds is then used for stability investigation of quasilinear systems evolving on arbitrary time domains.

The Propagation, Evolution, and Rotation in Linear Storms (PERiLS) Project

Abstract Quasi-linear convective systems (QLCSs) are responsible for approximately a quarter of all tornado events in the United States, but no field campaigns have focused specifically on collecting data to understand QLCS tornadogenesis. The Propagation, Evolution, and Rotation in Linear Storms (PERiLS) project was the first observational study of tornadoes associated with QLCSs ever undertaken. Participants were drawn from more than 10 universities, laboratories, and institutes, with over 100 students participating in field activities. The PERiLS field phases spanned 2 years, late winters and early springs of 2022 and 2023, to increase the probability of intercepting significant tornadic QLCS events in a range of large-scale and local environments. The field phases of PERiLS collected data in nine tornadic and nontornadic QLCSs with unprecedented detail and diversity of measurements. The design and execution of the PERiLS field phase and preliminary data and ongoing analyses are shown.

Simple waves for anti-van der Waals modified Chaplygin gas in 2-D magnetohydrodynamics

Abstract This paper presents essential findings on the reducible equations introduced by Courant and Friedrichs in their seminal work, Supersonic Flow and Shock Waves. In this paper, we discuss the presence of simple waves in a 2-D magnetohydrodynamic system with an anti-van der Waals-modified Chaplygin gas. Following the approach of Hu and Sheng (characteristic decomposition of the 2 × 2 quasilinear strictly hyperbolic systems). Appl. Math. Lett. 25(3), 262–267 (2012), and (simple waves and characteristic decompositions of quasilinear hyperbolic systems in two independent variables). Math. Methods Appl. Sci. 38(8), 1494–1505 (2015) for the characteristic decomposition of a strictly hyperbolic system, we establish the existence of simple waves for a non-reducible system. This extends Courant and Friedrichs’s fundamental finding, which was initially proposed for reducible system (R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, New York, Interscience Publishers, Inc, 1948, p. 464). These results enhance our understanding of simple wave behaviour in magnetohydrodynamic systems with modified Chaplygin gas, expanding the applicability of Courant and Friedrichs’s theoretical framework.

On a quasilinear two-species chemotaxis system with general kinetic functions and interspecific competition

On a quasilinear two-species chemotaxis system with general kinetic functions and interspecific competition

On environmental protection, an investigation of a quasilinear evolution system’s controllability and optimality

On environmental protection, an investigation of a quasilinear evolution system’s controllability and optimality

Multidimensional Nonhomogeneous Quasi-Linear Systems and Their Hamiltonian Structure

In this paper, we investigate multidimensional first-order quasi-linear systems and find necessary conditions for them to admit Hamiltonian formulation. The insufficiency of the conditions is related to the Poisson cohomology of the admissible Hamiltonian operators. We present in detail the examples of two-dimensional, two-components systems of hydrodynamic type and of a real reduction of the 3-waves system.

A quasilinear hyperbolic one-dimensional model of the lymph flow through a lymphangion with valve dynamics and a contractile wall

This paper presents a one-dimensional model that describes fluid flow in lymphangions, the segments of lymphatic vessels between valves, using quasilinear hyperbolic systems. The model incorporates a phenomenological pressure-cross-sectional area relationship based on existing literature. Numerical solutions of the differential equations align with known results, offering insights into lymphatic flow dynamics. This model enhances the understanding of lymph movement through the lymphatic system, driven by lymphangion contractions.

Strongly quasilinear elliptic systems with perturbations through yong measures

We use the Galerkin method to obtain weak solutions for strongly quasilinear elliptic systems in the Sobolev space W1,p0(Ω; ℝm) of the form Gϖ(x) = f(x,ϖ,Dϖ) in Ω, with ϖ(x) = 0 on ∂Ω, where the term Gϖ(x) includes perturbations. Our approach involves the use of Young measures to obtain the desired results.

Influence of Low-Level Shear Orientation and Magnitude on the Evolution and Rotation of Idealized Squall Lines. Part I: Storm Morphology and Bulk Updraft/Mesovortex Attributes

Abstract Observational and modeling efforts have explored the formation and maintenance of mesovortices, which contribute to severe hazards in quasi-linear convective systems (QLCSs). There exists an important interplay between environmental shear and cold-pool-induced circulations which, when balanced, allow for upright QLCS updrafts with maximized lift along storm outflow boundaries. Numerical simulations have primarily tested the sensitivity of squall lines to zonally varying low-level (LL) shear profiles (i.e., purely line-normal, assuming a north–south-oriented system), but observed near-storm environments of mesovortex-producing QLCSs exhibit substantial LL hodograph curvature (i.e., line-parallel shear). Therefore, previous QLCS simulations may fail to capture the full impacts of LL shear variability on mesovortex characteristics. To this end, this study employs an ensemble of idealized QLCS simulations with systematic variations in the orientation and magnitude of the ambient LL shear vector, all while holding 0–3-km line-normal shear constant. This allows for a nuanced examination of how line-parallel shear modulates system structure, as well as mesovortex strength, size, and longevity. Results indicate that hodographs with LL curvature support squall lines with prominent bowing segments and wider, more intense rotating updrafts. Shear orientation also impacts mesovortex characteristics, with curved hodographs favoring cyclonic vortices that are stronger, wider, deeper, and longer-lived than those produced with straight-line wind profiles. These results provide a more complete physical understanding of how LL shear variability influences the generation of rotation in squall lines. Significance Statement Research related to linear storms has largely focused on vertical changes in winds (i.e., shear) oriented perpendicular to squall lines given its ability to balance storm cold pools and keep updrafts upright, thus promoting long-lived storms that presumably can go on to produce rotation. However, squall lines that produce a great deal of rotation often have a component of low-level shear oriented parallel to storms. This study gauges the sensitivity of simulated squall lines to changes in the direction and strength of shear close to the surface. We find that shear oriented parallel to linear storms creates stronger and larger updrafts that in turn support the development of intense and persistent rotation with characteristics supportive of tornadoes. These insights have impacts on both our physical understanding and prediction of the rotation and associated hazards of linear storms.

Stabilization of Fractional-Order Quasilinear Systems via Event-Triggered Mechanisms

Stabilization of Fractional-Order Quasilinear Systems via Event-Triggered Mechanisms

Partial regularity for manifold constrained quasilinear elliptic systems

We consider manifold constrained weak solutions of quasilinear uniformly elliptic systems of divergence type with a source term that grows at most quadratically with respect to the gradient of the solution. As we impose that the solution lies on a Riemannian manifold, the classical smallness condition for regularity can be relaxed to an inequality relating strict convexity of the squared distance and growth of the leading order term in the tangent component of the source. As a key tool for the proof of a partial regularity result, we derive a fully intrinsic Caccioppoli inequality which may be of independent interest. Finally we show how the systems under consideration have a variational nature and arise in the context of F- or V-harmonic maps.

On quasi-linear reaction diffusion systems arising from compartmental SEIR models

The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131–1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529–12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.

Structure and Maintenance of a Quasi-Linear Mesoscale Convective System That Caused Torrential Rain in Southern Kyushu, Japan, on 11 July 2021

Abstract A quasi-linear mesoscale convective system that remained nearly stationary (hereafter referred to as a stationary QLCS) for almost 8 h in southern Kyushu, Japan, caused torrential rainfall exceeding 350 mm on 11 July 2021. The stationary QLCS consisted of several rainbands organized by back-building convection. As the cold pool intensified, the system attained a more widespread structure, leaning on the upshear side. To elucidate the mechanism responsible for the upshear tilt, numerical simulations with 250-m horizontal grid spacing were conducted, including sensitivity experiments in which the evaporative cooling rates from rain were reduced to modify the cold pool intensity. Results show that the cold pool is critical to the organization of the QLCS, and the structure normal to it is mainly governed by the balance between the low-level shear magnitude and cold pool intensity, supporting the application of so-called Rotunno–Klemp–Weisman (RKW) theory to this event. The cause of the weak updraft that accompanies the leaning system over the strong cold pool was also investigated, analyzing the trajectories, vertical momentum equation, and pressure perturbation field using the anelastic equation. It is revealed that the updraft travels a longer distance through the tilted system experiencing more mixing with the ambient air, which results in less thermal buoyancy. In addition, the updraft is decelerated by the downward perturbation gradient force owing to the vertical buoyancy gradient around the sloping surface of the cold pool and the dynamical effect caused by the baroclinicity-associated strong horizontal vorticity around the cold pool leading edge.

Dynamics of Two Episodes of High Winds Produced by an Unusually Long-Lived Quasi-Linear Convective System in South China

Abstract Using radar observation and convection-permitting simulation, this work studies the storm-scale dynamics governing the generation of two episodes of high winds by an unusually long-lived quasi-linear convective system (QLCS) in South China on 21 April 2017. The first episode of high winds occurred at the apex of a bowing segment in the southern QLCS due to the downward transport of high momentum by a descending rear-inflow jet (RIJ). The RIJ was initially elevated which was generated as low-frequency gravity wave response to the thermal forcing in the QLCS leading convective line. It descended to the surface owing to the enhancement of low-level diabatic cooling which strengthened the downdrafts at the RIJ leading edge. The vertical momentum budget revealed that the downdrafts were initiated by the negative buoyancy of the cold pool and strengthened by the weakened buoyancy-induced upward pressure gradient force in the boundary layer and enhanced hydrometeor loading above. The second episode of high winds occurred in the decaying stage of the QLCS which, however, redeveloped as its northern part interacted with an intensifying large-scale shear line to the east. A zonal convective line developed along the shear line and finally merged with the QLCS. The merger greatly enhanced the low-level convergence, leading to the downward development of the line-end vortex via vertical stretching of vertical vorticity. The area of high winds was notably increased by the superposition of the ambient translational wind with the vortex rotational flow. The findings provide new insights into the generation of high winds by the QLCS-MCS merger, highlighting the importance of low-level vortices in addition to the RIJ. Significance Statement Quasi-linear convective systems (QLCSs) are prolific producers of high winds at the surface. While conceptual models have been established for high winds produced by single QLCSs, this is not the case for the high winds generated by QLCSs merging with other convective systems. This work revealed different dynamics for two episodes of high winds produced by a long-lived QLCS merging with an MCS in South China. The first episode resulted from the descending rear-inflow jet (RIJ) as in the case of single QLCSs. The second and stronger one was due to the superposition of ambient flow with a line-end vortex that developed downward given the enhanced low-level convergence by the merger. This finding sheds light on the importance of mergers in high-wind generation which has even greater damaging potential than the RIJ.

Radar Signatures Associated with Quasi-Linear Convective System Mesovortices

Abstract Recent operationally driven research has generated a framework, known as the three ingredients method and mesovortex warning system, that can help forecasters anticipate mesovortex development and issue warnings within quasi-linear convective systems (QLCSs). However, dual-polarization radar data has not yet been incorporated into this framework. Therefore, several dual- and single-polarization radar signatures associated with QLCS mesovortices were analyzed to determine if they could provide additional information about mesovortex development and intensity. An analysis of 167 mesovortices showed that 1) KDP drops precede ∼95% of mesovortices and provide an initial indication of where a mesovortex may develop; 2) midlevel KDP cores are a potentially useful precursor signature because they precede a majority of mesovortices and have higher magnitudes for mesovortices that produce wind damage or tornadoes; 3) low-level KDP cores and areas of enhanced spectrum width have higher magnitudes for mesovortices that produce wind damage or tornadoes but tend to develop at about the same time as the mesovortex, which makes them more useful as diagnostic than as predictive signatures; and 4) as range from the radar increases, the radar signatures become less useful in anticipating mesovortex intensity but can still be used to anticipate mesovortex development or build confidence in mesovortex existence. Significance Statement The purpose of this study is to look at weather radar features that might help forecasters predict the development and intensity of tornadoes and strong winds within linear thunderstorm systems. Our results show that the intensity and trends of some radar features are helpful in showing when these hazards might develop and how strong they might be, while other radar features are less helpful. This information can help forecasters focus on the most useful radar features and ultimately provide the best possible warnings.