Quasilinear Systems Research Articles

Global classical solutions of a kind of boundary value problem for quasilinear hyperbolic systems

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global‐in‐time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string.

Simulated QLCS Vortices in a High-Shear, Low-CAPE Environment

Abstract Tornadoes produced by quasi-linear convective systems (QLCSs) in low instability environments present distinctive challenges for forecasters. This study analyzes a population of 56 vortices (all cyclonic) in a full-physics, case study simulation to examine vortex characteristics and their relationships to the pre-line environment. Peak surface vortex intensity correlates with peak vortex depth, peak surface wind speed, and vortex pathlength. The strongest vortices are the deepest and longest lived, implying that they would be most detectable. The modeled surface vortices are primarily associated with gust front cusps and bow echoes, line breaks, and supercell-like features. Strong vortices frequently have sustained, superposed surface vorticity and near-ground updrafts for several minutes. Although weak vortices lack this superposition, they often exhibit impressive midlevel vorticity and midlevel updrafts. The environments of the weak and strong vortices are similar with small, yet statistically significant, differences in several thermodynamic and kinematic fields. The profiles near strong vortices have more low-level CAPE, steeper lapse rates, and stronger deep-layer vertical wind shear. However, the small magnitudes of the differences imply that forecasters might struggle to discriminate well between nontornadic and tornadic environments in high-shear, low-CAPE events. Despite the similarities, the profiles produce distinct reflectivity, updraft, and vertical vorticity distributions in idealized cloud model simulations. The most intense updrafts and vortices in the idealized runs occur when the environmental profiles from the strong vortex cases are combined with a QLCS orientation more normal to the lower-tropospheric vertical wind shear.

Storm-Scale Predictability and Analysis of the 13 April 2020 Central Savannah River Area Tornado Outbreak

Abstract An early morning tornado outbreak occurred on 13 April 2020 in the Central Savannah River Area. Multiple significant tornadoes were reported, resulting in fatalities and injuries. While the operational tornado warnings had positive lead times, the convective mode (quasi-linear convective system) increased the warning decision complexity. The timing of the event [0500–0600 local time (LT)] also made NWS-to-public communication difficult. The experimental NSSL Warn-on-Forecast System (WoFS) was run retrospectively for this case. The WoFS consists of 3–6-h ensemble forecasts initialized every 30 min, and the goals of the system are to bridge the gap between severe weather watches and warnings and to increase warning lead times. Multiple WoFS forecasts were initialized leading up to the first tornado report; those initialized prior to tornado warning issuance have high ensemble probabilities of low-level rotation in the appropriate areas based on subsequent tornado reports. This case highlights another example of the usefulness of WoFS before its eventual transition to operations. Using the WoFS forecasts, kinematic and thermodynamic storm–environment relationships are analyzed using ensemble sensitivity analysis (ESA). The analyses suggest variations in the mesoscale environmental vertical wind profile are not as influential on mesovortex intensity as variations in the thermodynamic environment. Surface observations recorded prior to the tornado outbreak reveal subtle temperature and moisture gradients that may be the impetus for mesovortex intensification and tornadogenesis.

Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution

In this article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve inx−yx-yplane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.

Rigorous justification of the Whitham modulation equations for equations of Whitham type

AbstractWe prove that the modulational instability criterion of the formal Whitham modulation theory agrees with the spectral stability of long‐wavelength perturbations of periodic traveling wave solutions to the generalized Whitham equation. We use the standard WKB procedure to derive a quasilinear system of three Whitham modulation equations, written in terms of the mass, momentum, and wave number of a periodic traveling wave solution. We use the same quantities as parameters in a rigorous spectral perturbation of the linearized operator, which allows us to track the bifurcation of the zero eigenvalue as the Floquet parameter varies. We show that the hyperbolicity of the Whitham system is a necessary condition for the existence of purely imaginary eigenvalues in the linearized system, and hence also a prerequisite for modulational stability of the underlying wave. Since the generalized Whitham equation has a Hamiltonian structure, we conclude that strict hyperbolicity is a sufficient condition for modulational stability.

Existence of solutions to some quasilinear degenerate elliptic systems when the datum has an intermediate degree of integrability

We prove the existence of a solution to a quasilinear system of degenerate equations, when the datum has an intermediate degree of summability. The main assumption asks the off-diagonal coefficients to have a crossed ‘party-flags’ support.

Low regularity ill-posedness for non-strictly hyperbolic systems in three dimensions

In this paper, we survey a new approach combining algebraic and geometric ideas, with which we prove low regularity ill-posedness for quasilinear hyperbolic systems with non-strict hyperbolicity in three dimensions. These systems are also associated with multiple wave-speeds.

Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth

In this paper, we consider the following coupled gradient-type quasilinear elliptic system -div(a(x,u,∇u))+At(x,u,∇u)=Gu(x,u,v)inΩ,-div(b(x,v,∇v))+Bt(x,v,∇v)=Gvx,u,vinΩ,u=v=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{*{20}l} - {\ ext{div}} ( a(x, u, \ abla u) ) + A_t (x, u, \ abla u) = G_u(x, u, v) &{}{\\hbox { in }}\\Omega ,\\\\ - {\ ext{div}} ( b(x, v, \ abla v) ) + B_t(x, v, \ abla v) = G_v\\left( x, u, v\\right) &{}{\\hbox { in }}\\Omega ,\\\\ u = v = 0 &{}{\\hbox { on }}\\partial \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega is an open bounded domain in {mathbb {R}}^N, Nge 2. We suppose that some mathcal {C}^{1}–Carathéodory functions A, B:Omega times {mathbb {R}}times {mathbb {R}}^Nrightarrow {mathbb {R}} exist such that a(x,t,xi ) = nabla _{xi } A(x,t,xi ), A_t(x,t,xi ) = frac{partial A}{partial t} (x,t,xi ), b(x,t,xi ) = nabla _{xi } B(x,t,xi ), B_t(x,t,xi ) =frac{partial B}{partial t}(x,t,xi ), and that G_u(x, u, v), G_v(x, u, v) are the partial derivatives of a mathcal {C}^{1}–Carathéodory nonlinearity G:Omega times {mathbb {R}}times {mathbb {R}}rightarrow {mathbb {R}}. Roughly speaking, we assume that A(x,t,xi ) grows at least as (1+|t|^{s_1p_1})|xi |^{p_1}, p_1 > 1, s_1 ge 0, while B(x,t,xi ) grows as (1+|t|^{s_2p_2})|xi |^{p_2}, p_2 > 1, s_2 ge 0, and that G(x, u, v) can also have a supercritical growth related to s_1 and s_2. Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.

Ground state solution for quasi-linear elliptic systems with general source terms on unbounded domain

Ground state solution for quasi-linear elliptic systems with general source terms on unbounded domain

Lyapunov functions and finite-time stabilization in optimal time for homogeneous linear and quasilinear hyperbolic systems

Hyperbolic systems in one-dimensional space are frequently used in the modeling of many physical systems. In our recent works we introduced time-independent feedbacks leading to finite stabilization in optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work we present Lyapunov’s functions for these feedbacks and use estimates for Lyapunov’s functions to rediscover the finite stabilization results.

Generalized critical Kirchhoff-type potential systems with Neumann Boundary conditions

In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais–Smale condition.

Predictive feedback boundary control of semilinear and quasilinear 2 × 2 hyperbolic PDE–ODE systems

We present a control design for semilinear and quasilinear 2 × 2 hyperbolic partial differential equations with the control input at one boundary and a nonlinear ordinary differential equation coupled to the other. The controller can be designed to asymptotically stabilize the system at an equilibrium or relative to a reference signal. Two related but different controllers for semilinear and general quasilinear systems are presented and the additional challenges in quasilinear systems are discussed. Moreover, we present an observer that estimates the distributed PDE state and the unmeasured ODE state from measurements at the actuated boundary only, which can be used to also solve the output feedback control problem.

Using Ensemble Data Assimilation to Explore the Environmental Controls on the Initiation and Predictability of Moist Convection

Abstract Atmospheric deep moist convection has emerged as one of the most challenging topics for numerical weather prediction, due to its chaotic process of development and multiscale physical interactions. This study examines the dynamics and predictability of a weakly organized linear convective system using convection permitting EnKF analysis and forecasts with assimilating all-sky satellite radiances from a water vapor sensitive band of the Advanced Baseline Imager on GOES-16. The case chosen occurred over the Gulf of Mexico on 11 June 2017 during the NASA Convective Processes Experiment (CPEX) field campaign. Analysis of the water vapor and dynamic ensemble covariance structures revealed that meso-α-scale (2000–200 km) and meso-β-scale (200–20 km) initial features helped to constrain the general location of convection with a few hours of lead time, contributing to enhancing convective activity, but meso-γ-scale (20–2 km) or even-smaller-scale features with less than 30-min lead time were identified to be essential for capturing individual convective storms. The impacts of meso-α-scale initial features on the prediction of particular individual convective cells were found to be classified into two regimes; in a relatively dry regime, the meso-α-scale environment needs to be moist enough to support the development of the convection of interest, but in a relatively wet regime, a drier meso-α-scale environment is preferable to suppress the surrounding convective activity. This study highlights the importance of high-resolution initialization of moisture fields for the prediction of a quasi-linear tropical convective system, as well as demonstrating the accuracy that may be necessary to predict convection exactly when and where it occurs.

Unpredictable solutions of impulsive quasi-linear systems

Unpredictable solutions of impulsive quasi-linear systems

Weak shock wave interactions in isentropic Cargo‐LeRoux model of flux perturbation

In this paper, we explore the weak shock wave interactions in isentropic Cargo‐LeRoux model of flux perturbation which describes strictly hyperbolic quasilinear system of conservation laws. For the Riemann problem, we provide a global existence and uniqueness result of exact solution. Further, we establish the exact solution explicitly in terms of one parameter family of elementary wave curves. We devise a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of either a shock wave or a simple wave according to one and three family of characteristic curves. Lastly, we analyze the Von Neumann result related to collision of weak shock waves of same family.

Application of the perturbation method to the problem of optimizing the transient process in a quasi-linear dynamic system

The problem of optimizing the transient process in a quasi-linear dynamic system with a performance index, being a linear combination of the energy costs and the process duration, is considered. Asymptotic approximations of a given order to the solution of this problem are constructed.

Boundedness in a quasilinear two-species chemotaxis system with nonlinear sensitivity and nonlinear signal secretion

In this paper, we consider the quasilinear two-species chemotaxis model with nonlinear signal production and nonlinear sensitivity. Under appropriate regularity assumption on the initial data, the global boundedness of classical solution is obtained. Our results partially generalize and improve some results in the literature.

On effects of the nonlinear signal production to the boundedness and finite-time blow-up in a flux-limited chemotaxis model

We study herein the initial–boundary value problem for the flux-limited chemotaxis model with nonlinear signal production [Formula: see text] subject to no-flux boundary conditions in a ball [Formula: see text], where [Formula: see text], [Formula: see text]. For radially symmetric and positive initial data [Formula: see text], it is proved that the corresponding solution is globally bounded when [Formula: see text]. This result not only fills up a gap left in [M. Mizukami, T. Ono and T. Yokota, Extensibility criterion ruling out gradient blowup in a quasilinear degenerate chemotaxis system, J. Differ. Equ. 267(2019) 5115–5164; Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta. Appl. Math. 167 (2020) 231–259] when [Formula: see text], but also extend the boundedness result from the special case [Formula: see text] to the general case [Formula: see text]. Moreover, under the condition [Formula: see text] it is shown that if [Formula: see text] is sufficiently large, then there exists initial data [Formula: see text] such that the corresponding solution blows up at finite time [Formula: see text] in the sense that [Formula: see text] This blow-up result generalizes the works for the linear signal production case in (N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Am. Math. Soc. Ser. B 4 (2017) 31–67; Y. Chiyoda, M. Mizukami and T. Yokota, Finite-time blow-up in a quasilinear degenerate chemotaxis system with flux limitation, Acta. Appl. Math. 167 (2020) 231–259) to the nonlinear case [Formula: see text].

Kinetic Equation for Soliton Gas: Integrable Reductions

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2times 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

Tropical cyclogenesis associated with premonsoon climatological dryline over the Bay of Bengal

Tropical cyclones of the Bay of Bengal (BoB) that formed near the synoptic-scale dryline usually intensified over a distance of 600–800 km within 3 days and caused severe destruction after landfall. High-resolution simulations of very severe cyclonic storms in association with dryline indicate that the meridional shear aids in the development of a linear-shaped group of convective cells that mature as an east–west oriented quasi-linear convective system (QLCS) within the boundary between the dry-moist air masses. The leading edge deep convections are supported by low-level moist southwesterly inflow; however, the typical mid-level mesoscale convective vortex (MCV) associated with these QLCS is unremarkable due to a very narrow trailing stratiform region within the QLCS. Supercells are likely to be organized within the QLCS due to extremely unstable atmospheric conditions resulting from a strong vertical shear of 27–39 m s−1 between 0 and 6 km and large convective available potential energy of > 3000 J kg−1. The vertical shear veering with height causes several numbers of low-level mesovortices having diameters less than 10 km at the leading edge in the different convective stages of the QLCS. The dryline aloft in the BoB produces horizontal positive shear vorticity of the order 10–5 s−1 with higher values in the levels 850–600 hPa. The advection of intense cloud-scale cyclonic mesovortices (~ 10–3 s−1) assists and enhances a cyclonic vortex to the rear side of the QLCS that performs as an MCV for cyclogenesis over the BoB.