Quasilinear Systems Research Articles

Quasi-Linear Convective Systems and Derechos across Europe: Climatology, Accompanying Hazards, and Societal Impacts

Abstract In this work, we use 8 years (2014–21) of Operational Programme for the Exchange of Weather Radar Information (OPERA) radar data, ESWD severe weather reports, and arrival time difference (ATD) lightning detection network (ATDnet) data to create a climatology of quasi-linear convective systems (QLCSs) across Europe. In the first step, 15-min radar scans were used to identify 1475 QLCS polygons. Severe weather reports, lightning data, and morphological properties were used to classify QLCSs according to their intensity into 1151 marginal (78.0%), 272 moderate (18.5%), and 52 derecho (3.5%) events. The manual evaluation led to the recognition of QLCS morphological and precipitation archetypes, areal extent, duration, speed, forward motion, width, length, accompanying hazards, injuries, and fatalities. Results indicate that QLCSs are the most frequent during summer in central Europe, while in southern Europe, their occurrence is extended to late autumn. A bow echo feature occurred in around 29% of QLCS cases, while a mesoscale convective vortex occurred in almost 9%. Among precipitation modes, trailing and embedded stratiform types accounted for around 50% of QLCSs. The most frequent hazard accompanying QLCSs was lightning (taking up on average 94.4% of the area impacted by QLCS), followed by severe wind gusts (7.9%), excessive precipitation (6.1%), large hail (2.9%), and tornadoes (0.5%). Derechos had the largest coverage of severe wind reports (49.8%), while back-building QLCSs were the most prone to excessive precipitation events (13.5%). QLCSs caused 104 fatalities and 886 injuries. Severe wind gusts were responsible for 87.6% of fatalities and 73.6% of injuries. Nearly half of all fatalities and injuries were associated with only the 10 most impactful QLCS events, mostly warm-season derechos.

Comparison of Tornado Damage Characteristics to Low-Altitude WSR-88D Radar Observations and Implications for Tornado Intensity Estimation

Abstract Given the obvious difficulties in directly sampling tornadic wind fields, ongoing work continues to improve estimates of near-ground wind speeds in tornadoes. This study builds upon a recently proposed empirical relationship between radar-observed velocities in the lowest 150 m above ground level (AGL) and the theoretical peak 15-m AGL wind speed. We create and analyze a dataset of 194 velocity observations within tornadoes in the lowest 150 m AGL. These observations are drawn from 105 individual tornadoes that occurred across a diverse range of EF-scale ratings (EF0–4), convective modes (discrete supercell and quasi-linear convective system), geographical regions, and housing unit densities (HUDs). Comparing the radar-estimated and damage-estimated tornado wind speeds, and corresponding EF- and F-scale ratings, is the primary focus of the ensuing analysis. Consistent with recent work, damage-estimated tornado wind speeds tend to be lower than radar-estimated near-surface wind speeds, especially for stronger tornadoes. Damage- and radar-estimated wind speed differences are not strongly related to the availability of damage indicators (as measured by HUD). While some relationship exists—particularly underestimates of peak wind speeds for strong–violent tornadoes in low HUD areas—the tendency of radar-based strong/violent tornado intensity estimates to be meaningfully higher than EF-scale-based damage estimates exists across the HUD spectrum. The legacy F-scale wind speed ranges may ultimately provide a better estimate of peak tornado wind speeds at 10–15 m AGL for strong–violent tornadoes and a better damage-based intensity rating for all tornadoes. These results are contextualized with regard to ongoing community efforts to improve tornado intensity estimation. Significance Statement Due to the numerous difficulties in collecting direct observations of tornado wind speeds, we compare methods that are used to estimate near-ground wind speeds using both radar measurements of wind speeds aloft and damage. Our results show that 1) the damage-estimated intensities of stronger tornadoes are more likely to be underestimates of true wind speeds than weaker tornadoes based on radar observations, 2) this bias is present for tornadoes that occur in more built-up areas as well as more sparsely populated ones, and 3) the legacy Fujita scale may provide better wind speed estimates in stronger tornadoes. These findings contribute to community-wide efforts to improve damage-based estimates of peak tornado wind speeds.

Two-Layer Equilibrium Model of Miscible Inhomogeneous Fluid Flow

Two-layer flow of a density-stratified fluid with mass transfer between the layers is considered. In the Boussinesq approximation, the equations of motion are reduced to a homogeneous quasilinear system of partial differential equations of mixed type. The flow parameters in the intermediate mixed layer are determined from the equilibrium conditions in a more general model of three-layer flow of a miscible fluid. In particular, the equilibrium conditions imply the constancy of the interlayer Richardson number in velocity-shift flows. A self-similar solution to the problem of breakdown of an arbitrary discontinuity (the lock-exchange problem) in the domain of hyperbolicity of the system under consideration is constructed. The transcritical flow regimes over a local obstacle are studied and the conditions under which the obstacle determines the upstream flow are determined. A comparison of steady-state and time-dependent solutions with the solutions obtained for the original three-layer models of miscible fluid flow is carried out.

Strongly hyperbolic quasilinear systems revisited, with applications to relativistic fluid dynamics

We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest.

Investigating Temporal Characteristics of Polarimetric and Electrical Signatures in Three Severe Storms: Insights from the VORTEX-Southeast Field Campaign

Abstract The dual-polarization radar characteristics of severe storms are commonly used as indicators to estimate the size and intensity of deep convective updrafts. In this study, we track rapid fluctuations in updraft intensity and size by objectively identifying polarimetric fingerprints such as ZDR and KDP columns, which serve as proxies for mixed-phase updraft strength. We quantify the volume of ZDR and KDP columns to evaluate their utility in diagnosing temporal variability in lightning flash characteristics. Specifically, we analyze three severe storms that developed in environments with low-to-moderate instability and strong 0–6-km wind shear in northern Alabama during the 2016–17 VORTEX-Southeast field campaign. In these three cases (a tornadic supercell embedded in stratiform precipitation, a nontornadic supercell, and a supercell embedded within a quasi-linear convective system), we find that the volume of the KDP columns exhibits a stronger correlation with the total flash rate. The higher covariability of the KDP column volume with the total flash rate suggests that the overall electrification and precipitation microphysics were dominated by cold cloud processes. The lower covariability with the ZDR column volume indicates the presence of nonsteady updrafts or a less prominent role of warm rain processes in graupel growth and subsequent electrification. Furthermore, we observe that the majority of cloud-to-ground (CG) lightning strikes a carried negative charge to the ground. In contrast to findings from a tornadic supercell over the Great Plains, lightning flash initiations in the Alabama storms primarily occurred outside the footprint of the ZDR and KDP column objects. Significance Statement This study quantifies the correlation between mixed-phase updraft intensity and total lightning flash rate in three severe storms in northern Alabama. In the absence of direct updraft velocity measurements, we use polarimetric signatures, such as ZDR and KDP columns, as proxies for updraft strength. Our analysis of polarimetric radar and lightning mapping array data reveals that the lightning flash rate is more highly correlated with the KDP column volume than with the ZDR column volume in all three storms examined. This contrasts with previous findings in storms over the central Great Plains, where the ZDR column volume showed higher covariability with flash rate. Interestingly, lightning initiation in the Alabama storms mainly occurred outside the ZDR and KDP column areas, contrary to previous findings.

Existence of positive solutions for an elliptic system with sign-changing weights

Using the method of sub-super solutions and comparison principle, we study the existence of positive solutions for a class of quasilinear elliptic systems with sign-changing weights.

Multiplicity of solutions for a class of nonhomogeneous quasilinear elliptic system with locally symmetric condition in ℝN

This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole‐space . By a variant of Clark's theorem without the global symmetric condition and Moser's iteration technique, we obtain the existence of multiple solutions when the nonlinear term satisfies some growth conditions only in a circle with center 0, and the perturbation term is any continuous function with a small parameter and no any growth hypothesis.

Combined effects in coupled quasilinear elliptic systems via a “like” Gagliardo–Nirenberg inequality

In this paper, we study the existence, nonexistence and multiplicity results to a class of quasilinear elliptic system involving ‐Laplacian by exploiting the competition of nonlinearities. We mainly propose a “like” Gagliardo–Nirenberg inequality for a pair of functions, and use it to combine the variational method to prove the existence, nonexistence and multiplicity of nontrivial nonnegative solutions for a quasilinear elliptic system involving ‐Laplacian under different conditions between parameters.

Nontrivial solutions for resonance quasilinear elliptic systems

Abstract We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically ( p , q ) \left(p,q) -linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C 2 {C}^{2} regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

On a class of Hardy-Sobolev critical quasilinear elliptic systems on compact Riemannian manifolds

In this work, we investigate the existence and regularity of solutions to the critical system{−Δp,gu+a(x)|u|p−2u+b(x)[(p−1)|u|p−2+|v|p−2]pv=αp⋆(s)f(x)u|u|α−2|v|βdg(x,x0)sinM,−Δp,gv+b(x)[(p−1)|v|p−2+|u|p−2]pu+c(x)|v|p−2v=βp⋆(s)f(x)v|v|β−2|u|αdg(x,x0)sinM, where (M,g) is a smooth closed Riemannian manifold of dimension n≥2, dg is the Riemannian distance, Δp,g=divg(|∇gu|p−2∇gu) is the p-Laplace-Beltrami operator on (M,g), p∈(1,n), a,b,c∈C0,ϱ(M) for some ϱ∈(0,1), with b≡0 when 1<p<2, x0∈M, s∈[0,p), f is a smooth function in M with f(x0)=maxM⁡f>0, and α>1, β>1 are two real numbers such that α+β=p⋆(s), where p⋆(s)=p(n−s)/(n−p) denotes the critical Hardy-Sobolev exponent.

Stabilization of quasilinear systems with variable exponents: A modified approach for energy decay rate determination

This article investigates the stabilization of quasilinear systems governed by elliptic partial differential equations (PDEs) with variable exponents using the modified Tikhonov approach. Our main focus is to determine the energy decay rate of the solution. The findings have significant potential for practical applications in various fields of science and engineering, where the energy decay rate plays a vital role in system stability and control.

On a class of a coupled nonlinear viscoelastic Kirchhoff equations variable-exponents: global existence, blow up, growth and decay of solutions

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, and variable exponents. Under suitable assumptions on the initial data and the relaxation functions, we obtained that the solution of the system is global and bounded. Next, the blow-up is proved with negative initial energy. After that, the exponential growth of solutions is showed with positive initial energy, and by using an integral inequality due to Komornik, the general decay result is obtained in the case of absence of the source term.

Existance of solutions for some class of system quasilinear differentials with nonlocal boundary conditions and with nonlinearity depandant on the first derivate in time scales

This work is conserned with the construction of minimal and maximal solution for some equation quasilinear differentials with nonlocal boundary condition with nonlinearity is continuous function dependent on the first derivate in the time scale. In a previous study, see[11] it was found that there exists the extrimals solutions of a quasilinear elliptic system with intergral boundary conditions in the continuous case, and from here we asked the following question using the upper and lower solutions method coupled with monotone iterative technique and we asked the following question: do we have the existence of extrimals solutions for certain classes of systems of differential equations with nonlocal boundary conditions in time scales by same method. We can classify the problem following the monotonicity of the two functions and in the true types.Type 1: Increasing qasimonotonic systems.Type 2: Decreasing qasimonotonic systems. Type 3: Mexed qasimonotonic systems. In this paper we show the existence minimuls-maximuls solutions if the problem is of the type 1. If the problem is of the type 2 , we show the existence of maximuls-minimuls solutions. If the problem is of the type 3 , we show existence of least quasilear-solution. In the end, we gave examples of the results presented in this work to prove their validity through two cases: the continuous case, represented by the set of real numbers, and the discontinuous case, represented by the set of integers.

Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad &amp; x\in \Omega,\quad t&gt;0,\\ v_t=\Delta v-uv,\quad &amp; x\in \Omega,\quad t&gt;0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m&gt;\max \{1,\,\frac {3N-2}{2N+2}\}$ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$ in an appropriate sense as $t\rightarrow \infty$ , where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$ . This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m&gt;\frac {3N-2}{2N}$ in the case $N\geq 3$ , but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B22 (2017), 669–686) to $m&gt;\frac {3N}{2N+2}$ when $N\geq 2$ .

Asymptotically unpredictable trajectories in semiflows

A special kind of Poisson stable trajectory, which is called unpredictable and leads to sensitivity in the quasi-minimal set, was proposed by Akhmet and Fen (2016) for semiflows. In the present paper we carry this finding one step further by defining a new kind of trajectory, called asymptotically unpredictable. We prove that such motions also lead to sensitivity in the dynamics. This feature is now achieved under a weaker hypothesis. Benefiting from the Bebutov dynamical system, continuous asymptotically unpredictable functions on the real axis are defined, and it is shown that the set of these functions properly includes the set of unpredictable ones. Moreover, results on producing new asymptotically unpredictable functions from a given one are obtained. The existence and uniqueness of bounded asymptotically unpredictable solutions of quasi-linear systems are also investigated, and an application to Hopfield neural networks is provided.

Blow up, growth, and decay of solutions for a class of coupled nonlinear viscoelastic Kirchhoff equations with distributed delay and variable exponents

In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay, and variable exponents. Under a suitable hypothesis the blow-up and growth of solutions are proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case of absence of the source term f1=f2=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f_{1}=f_{2}=0$\\end{document}.

Retention profiles of multiparticle filtration in porous media

A one‐dimensional deep bed filtration model of a polydisperse suspension or colloid in a porous medium is considered. The model includes a quasilinear system of 2n equations for concentrations of suspended and retained particles of n types. The problem is reduced to a closed 3 × 3 system for total concentrations of suspended and retained particles and of occupied rock surface area, which allows an exact solution. The exact solution to the n‐particle problem is derived, the existence and uniqueness of the solution are proven, and the solution in the form of a traveling wave is obtained. The retention profiles (dependence of the deposit concentration on the coordinate at a fixed time) of different size particles and the total profile are studied. It is shown that the profile of large particles decreases monotonically, while the profile of small particles is nonmonotonic. Conditions for the monotonicity/nonmonotonicity of the intermediate particle profiles and the total profile are obtained. The maximum point of the small particles profile tends to infinity with unlimited growth of time, and the maximum points of nonmonotonic profiles of intermediate particles are limited. The asymptotical expansion of the maximum points of nonmonotonic profiles is constructed.

A numerical approach for investigating multiple eigenpairs of a quasilinear elliptic system

A numerical approach for investigating multiple eigenpairs of a quasilinear elliptic system

Quasilinear Systems with Unbounded Variable Exponents and Convection Terms

Quasilinear Systems with Unbounded Variable Exponents and Convection Terms

Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs

We investigate a generalized poly-Laplacian system with a parameter on weighted finite graphs, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 52(1-2):152–160, 2010], which is an abstract critical points theorem without compactness condition, to obtain that these systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range.