Recent zbMATH articles in MSC 35Lhttps://zbmath.org/atom/cc/35L2023-05-31T16:32:50.898670ZWerkzeugComparing the dual phase lag, Cattaneo-Vernotte and Fourier heat conduction models using modal analysishttps://zbmath.org/1508.350052023-05-31T16:32:50.898670Z"van der Merwe, A. J."https://zbmath.org/authors/?q=ai:van-der-merwe.alna"van Rensburg, N. F. J."https://zbmath.org/authors/?q=ai:van-rensburg.nicolaas-f-j"Sieberhagen, R. H."https://zbmath.org/authors/?q=ai:sieberhagen.r-hSummary: This paper deals with phase lag (or time-lagged) heat conduction models: the Cattaneo-Vernotte (or thermal wave) model and the dual phase lag model. The main aim is to show that modal analysis of these second order partial differential equations provides a valid and effective approach for analysing and comparing the models. It is known that reliable values for the phase lags of the heat flux and the temperature gradient are not readily available. The modal solutions are used to determine a range of realistic values for these lag times. Furthermore, it is shown that using partial sums of the series solutions for calculating approximate solutions is an efficient procedure. These approximate solutions converge in terms of a so-called energy norm which is stronger than the maximum norm. A model problem where a single heat pulse is applied to a specimen, is used for comparing these models with the Fourier (or parabolic) heat conduction model.On a semilinear wave equation in anti-de Sitter spacetime: the critical casehttps://zbmath.org/1508.350222023-05-31T16:32:50.898670Z"Palmieri, Alessandro"https://zbmath.org/authors/?q=ai:palmieri.alessandro"Takamura, Hiroyuki"https://zbmath.org/authors/?q=ai:takamura.hiroyukiSummary: In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.
{\copyright 2022 American Institute of Physics}Solution of a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized typehttps://zbmath.org/1508.350232023-05-31T16:32:50.898670Z"Assanova, Anar T."https://zbmath.org/authors/?q=ai:assanova.anar-turmaganbetkyzy"Uteshova, Roza"https://zbmath.org/authors/?q=ai:uteshova.roza-e(no abstract)Low regularity ill-posedness for non-strictly hyperbolic systems in three dimensionshttps://zbmath.org/1508.350242023-05-31T16:32:50.898670Z"An, Xinliang"https://zbmath.org/authors/?q=ai:an.xinliang"Chen, Haoyang"https://zbmath.org/authors/?q=ai:chen.haoyang"Yin, Silu"https://zbmath.org/authors/?q=ai:yin.siluSummary: 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.
{\copyright 2022 American Institute of Physics}Global existence for the wave equation with nonlinear boundary damping and source termshttps://zbmath.org/1508.350252023-05-31T16:32:50.898670Z"Vitillaro, Enzo"https://zbmath.org/authors/?q=ai:vitillaro.enzoSummary: The paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is
\[
\begin{cases}
u_{tt} - \Delta u = 0 & \text{in } (0, \infty)\times \mathit{\Omega}, \\
u = 0 & \text{on } [0, \infty)\times \mathit{\Gamma}_0, \\
\frac{\partial u}{\partial \nu} = -|u_t|^{m-2}u_t + |u|^{p-2} u & \text{on } [0, \infty) \times \mathit{\Gamma}_1, \\
u(0, x) = u_0(x), \, u_t (0, x) = u_1(x) & \text{on } \mathit{\Omega}, \\
\end{cases}
\]
where \(\mathit{\Omega} \subset \mathbb{R}^n\) \((n \geqslant 1)\) is a regular and bounded domain, \(\partial \mathit{\Omega} = \mathit{\Gamma}_0 \cup \mathit{\Gamma}_1\), \(m>1\), \(2 \leqslant p < r\), where \(r = 2(n -1)/(n -2)\) when \(n \geqslant 3\), \(r=\infty\) when \(n=1, 2\), and the initial data are in the energy space. We prove local existence of the solutions in the energy space when \(m>r / (r + 1 - p)\) or \(n=1,2\), and global existence when \(p \leqslant m\) or the initial data are inside the potential well associated to the stationary problem.Scattering of the \(\varphi^8\) kinks with power-law asymptoticshttps://zbmath.org/1508.350262023-05-31T16:32:50.898670Z"Belendryasova, Ekaterina"https://zbmath.org/authors/?q=ai:belendryasova.ekaterina"Gani, Vakhid A."https://zbmath.org/authors/?q=ai:gani.vakhid-aSummary: We study the scattering of the \(\varphi^8\) kinks off each other, namely, we consider those \(\varphi^8\) kinks that have power-law asymptotics. The slow power-law fall-off leads to a long-range interaction between the kink and the antikink. We investigate how the scattering scenarios depend on the initial velocities of the colliding kinks. In particular, we observe the `escape windows' -- the escape of the kinks after two or more collisions, explained by the resonant energy exchange between the translational and vibrational modes. In order to elucidate this phenomenon, we also analyze the excitation spectra of a solitary kink and of a composite kink+antikink configuration.Exact solution and the multidimensional Godunov scheme for the acoustic equationshttps://zbmath.org/1508.350572023-05-31T16:32:50.898670Z"Barsukow, Wasilij"https://zbmath.org/authors/?q=ai:barsukow.wasilij"Klingenberg, Christian"https://zbmath.org/authors/?q=ai:klingenberg.christianSummary: The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.Non conservative products in fluid dynamicshttps://zbmath.org/1508.350642023-05-31T16:32:50.898670Z"Colombo, Rinaldo M."https://zbmath.org/authors/?q=ai:colombo.rinaldo-m"Guerra, Graziano"https://zbmath.org/authors/?q=ai:guerra.graziano"Holle, Yannick"https://zbmath.org/authors/?q=ai:holle.yannickThe authors justify the approximation of solutions of problems in fluid dynamics with nonconservative products in sources. Motivations stem from fluid flows in pipes with discontinuous cross section. Applications of the existence results shown are not limited, however, to fluid dynamics. They apply to abstract balance laws with nonconservative source terms in the nonresonant case, in general BV geometry.
Reviewer: Piotr Biler (Wrocław)Global mild solutions to three-dimensional magnetohydrodynamic equations in Morrey spaceshttps://zbmath.org/1508.350812023-05-31T16:32:50.898670Z"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng"Xi, Shuai"https://zbmath.org/authors/?q=ai:xi.shuai"Zeng, Zirong"https://zbmath.org/authors/?q=ai:zeng.zirong"Zhu, Shengguo"https://zbmath.org/authors/?q=ai:zhu.shengguoSummary: In this paper, we consider the Cauchy problem of three-dimensional incompressible magnetohydrodynamic equations. Some uniform estimates with respect to time for the coupling terms between the fluid and the magnetic field will be presented, under the condition that the initial \(\mathcal{M}^{1 , 1}\) norms of the vorticity and the current density are both sufficiently small. By the above estimates, we can obtain a global-in-time well-posedness of mild solutions in Morrey spaces via some effective arguments. The asymptotic behaviours of the solutions are also obtained.Exact Riemann solutions for the drift-flux equations of two-phase flow under gravityhttps://zbmath.org/1508.350912023-05-31T16:32:50.898670Z"Shen, Chun"https://zbmath.org/authors/?q=ai:shen.chun"Sun, Meina"https://zbmath.org/authors/?q=ai:sun.meinaThe authors consider the system of one-dimensional drift-flux equations describing a two-phase flow and written as:
\((\alpha_{g}\rho_{g})_{t}+(\alpha_{g}\rho_{g}u_{g})_{x}=0\),
\((\alpha_{l}\rho_{l})_{t}+(\alpha_{l}\rho_{l}u_{l})_{x}=0\),
\((\alpha_{g}\rho_{g}u_{g}+\alpha_{l}\rho_{l}u_{l})_{t}+(\alpha_{g}\rho_{g}u_{g}^{2}+\alpha_{l}\rho_{l}u_{l}^{2}+p)_{x}=-q\),
where \(\alpha_{g}\), \(\alpha_{l}\in \lbrack 0,1]\) are the volume fractions of gas and liquid which satisfy \(\alpha_{g}+\alpha_{l}=1\), \(\rho_{g}\) the gas density, \(u_{g}\) the gas velocity, \(\rho_{l}\) the liquid density, \(u_{l}\) the liquid velocity, \(p\) the pressure for both gas and liquid, and \(q\) the external forces such as gravity and friction. Considering a 1D inviscid, compressible and isentropic liquid-gas two-phase flow on an inclined pipeline under gravity, which leads to special expressions for the external forces \(q\) and pressure \(p\), the authors simplify the above system as the following one written in a conservative form as:
\(m_{t}+(m(v-\mu t))_{x}=0\),
\(n_{t}+(n(v-\mu t))_{x}=0\),
\(((m+n)v)_{t}+((m+n)v(v-\mu t)+(\gamma -1)(m+n)^{\gamma })_{x}=0\),
where \(m\) and \(n\) are the gas and liquid masses defined as \(m=\alpha_{g}\rho_{g}\), \(n=\alpha_{l}\rho_{l}\), and \(v\) is the velocity \(v(x,t)=u(x,t)+\mu t\). Riemann initial conditions \((m,n,v)(x,t=0)=(m_{-},n_{-},u_{-})\), \(x<0\), \((m,n,v)(x,t=0)=(m_{+},n_{+},u_{+})\), \(x>0\), where \(m_{-},n_{-},m_{+},n_{+}>0\) are naturally deduced from physical considerations are imposed. The authors rewrite this last problem in a quasi-linear form for which they derive the associated characteristic equation: \((m+n)(\lambda -v+\mu t)((\lambda -v+\mu t)^{2}-(\gamma -1)\gamma (m+n)^{\gamma -1})=0\) and the three eigenvalues. They draw computations to determine and analyze the properties of the Riemann solutions to the last problem. The first main result proves that for any given Riemann initial conditions, there exists a unique solution to the last problem for fixed parameters \(\gamma\) and \(\mu\). Moving to the system: \(m_{t}+(m(v-\mu t))_{x}=0\), \(n_{t}+(n(v-\mu t))_{x}=0\), \(((m+n)v)_{t}+((m+n)v(v-\mu t))_{x}=0\), with the same initial conditions as above, the authors define the notion of generalized delta shock wave solution and they prove the existence of such solution if \(u_{+}<u_{-}\). They give its expression. In the last part of the paper, the authors describe the limits of Riemann solutions to the second system as \(\gamma \rightarrow 1+\).
Reviewer: Alain Brillard (Riedisheim)Thermodynamically consistent modeling for complex fluids and mathematical analysishttps://zbmath.org/1508.350942023-05-31T16:32:50.898670Z"Suzuki, Yukihito"https://zbmath.org/authors/?q=ai:suzuki.yukihito"Ohnawa, Masashi"https://zbmath.org/authors/?q=ai:ohnawa.masashi"Mori, Naofumi"https://zbmath.org/authors/?q=ai:mori.naofumi"Kawashima, Shuichi"https://zbmath.org/authors/?q=ai:kawashima.shuichiSummary: The goal of this paper is to derive governing equations for complex fluids in a thermodynamically consistent way so that the conservation of energy and the increase of entropy is guaranteed. The model is a system of first-order partial differential equations on density, velocity, energy (or equivalently temperature), and conformation tensor. A barotropic model is also derived. In the one-dimensional case, we express the barotropic model in the form of hyperbolic balance laws, and show that it satisfies the stability condition. Consequently, the global existence of solutions around equilibrium states is proved and the convergence rates is obtained.Global control aspects for long waves in nonlinear dispersive mediahttps://zbmath.org/1508.351162023-05-31T16:32:50.898670Z"Capistrano-Filho, Roberto de A."https://zbmath.org/authors/?q=ai:capistrano-filho.roberto-de-a"Gomes, Andressa"https://zbmath.org/authors/?q=ai:gomes.andressaSummary: A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain \(\mathbb{T}\) are studied. We used two distributed controls, supported in \(\omega \subset \mathbb{T}\) and assumed to be generated by a linear feedback law conserving the \textit{``mass'' (or ``volume'')}, to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in \(H^s(\mathbb{T})\), for \(s \geq 0\). After that, by certain properties of Bourgain spaces, we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the \textit{bilinear estimates} using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg-de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.On 1d quadratic Klein-Gordon equations with a potential and symmetrieshttps://zbmath.org/1508.351202023-05-31T16:32:50.898670Z"Germain, Pierre"https://zbmath.org/authors/?q=ai:germain.pierre"Pusateri, Fabio"https://zbmath.org/authors/?q=ai:pusateri.fabio"Zhang, Katherine Zhiyuan"https://zbmath.org/authors/?q=ai:zhang.katherine-zhiyuanSummary: This paper is a continuation of the previous work [Forum Math. Pi 10, Paper No. e17, 172 p. (2022; Zbl 1495.35126)] by the first two authors. We focus on 1-dimensional quadratic Klein-Gordon equations with a potential, under some assumptions that are less general than (Pusateri, in: Forum of mathematics, Cambridge University Press), but that allow us to present some simplifications in the proof of the global existence with decay for small solutions. In particular, we can propagate a stronger control on a basic \(L^2\)-weighted-type norm while providing some shorter and less technical proofs for some of the arguments.On quasilinear Maxwell equations in two dimensionshttps://zbmath.org/1508.351772023-05-31T16:32:50.898670Z"Schippa, Robert"https://zbmath.org/authors/?q=ai:schippa.robert"Schnaubelt, Roland"https://zbmath.org/authors/?q=ai:schnaubelt.rolandThe authors establish new Strichartz estimates for the Maxwell equations in two dimensions with rough permittivity. After localizing in space and frequency and using the Fourier-Bros-Iagolnitzer transform to transfer the problem to phase space, they reduce the estimates to prove to dyadic estimates for the half-wave equation. The latter are proved by following the approach proposed by \textit{D. Tataru} [Am. J. Math. 122, No. 2, 349--376 (2000; Zbl 0959.35125); ibid. 123, No. 3, 385--423 (2001; Zbl 0988.35037); J. Am. Math. Soc. 15, No. 2, 419--442 (2002; Zbl 0990.35027)] for the derivation of Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. The authors next use the so obtained Strichartz estimates for proving an improved version of the local well-posedness for quasilinear Maxwell equations in two dimensions. (Previously, well-posedness for hyperbolic systems was obtained by using energy methods.)
Reviewer: Catalin Popa (Iaşi)Continuity of dynamical behaviors for strongly damped wave equations with perturbationhttps://zbmath.org/1508.371022023-05-31T16:32:50.898670Z"Chang, Qingquan"https://zbmath.org/authors/?q=ai:chang.qingquan"Li, Dandan"https://zbmath.org/authors/?q=ai:li.dandanSummary: We explore the convergence of the global attractors for a class of perturbed severely damped wave equations with the Dirichlet boundary condition in the 3D bounded domain. With respect to the perturbation parameter, it is shown that the global attractors are both upper and lower semicontinuous.
{\copyright 2022 American Institute of Physics}Well-posedness of systems of 1-D hyperbolic partial differential equationshttps://zbmath.org/1508.470932023-05-31T16:32:50.898670Z"Jacob, Birgit"https://zbmath.org/authors/?q=ai:jacob.birgit"Kaiser, Julia T."https://zbmath.org/authors/?q=ai:kaiser.julia-tSummary: We consider the well-posedness of a class of hyperbolic partial differential equations on a one-dimensional spatial domain. This class includes in particular infinite networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. We consider these equations on a finite interval as well as on a semi-axis.Finite-difference method for solving a multidimensional pseudoparabolic equation with boundary conditions of the third kindhttps://zbmath.org/1508.650992023-05-31T16:32:50.898670Z"Beshtokov, Murat Khamidbievich"https://zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievichSummary: We study an initial-boundary value problem for a multidimensional pseudoparabolic equation with variable coefficients and boundary conditions of the third kind. The multidimensional pseudoparabolic equation is reduced to an integro-differential equation with a small parameter. It is shown that as the small parameter tends to zero, the solution of the resulting modified problem converges to the solution of the original problem. For an approximate solution of the obtained problem, a locally one-dimensional difference scheme by A. A. Samarsky is constructed. An a priori estimate is obtained by the method of energy inequalities, from which the uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow. For a two-dimensional problem, an algorithm for the numerical solution of the initial-boundary value problem for a pseudoparabolic equation with conditions of the third kind is developed.A simple WENO-AO method for solving hyperbolic conservation lawshttps://zbmath.org/1508.651032023-05-31T16:32:50.898670Z"Huang, Cong"https://zbmath.org/authors/?q=ai:huang.cong"Chen, Li Li"https://zbmath.org/authors/?q=ai:chen.liliSummary: In this paper, we propose a simple weighted essentially non-oscillatory method with adaptive order(SWENO-AO). The SWENO-AO consists of a high order and a second-order subreconstructions with a new weight, in which the second-order subreconstruction is always smooth. Comparing to WENO-AO method developed by \textit{J. Zhu} and \textit{J. Qiu} [J. Comput. Phys. 318, 110--121 (2016; Zbl 1349.65365)], the SWENO-AO has the advantage of simplicity. First, the SWENO-AO does not need to regroup the candidate cells, but uses the weighted-least-squares to obtain a smooth second-order subreconstruction directly, which is more flexible for solving multi-dimensional problem. However the WENO-AO needs to regroup the candidate cells properly for obtaining different second-order subreconstructions, so that near discontinuity, at least one of them is smooth and can be used for avoiding the spurious oscillations. Second, the SWENO-AO only consists of two subreconstructions, so the implementation is simple. However the WENO-AO uses more subreconstructions for higher dimensional problem, which increases the computational complexity. Finally, the weight of SWENO-AO is simpler than the one of WENO-AO. Numerical tests also show that, the SWENO-AO gives comparable solution as WENO-AO, but uses less computational cost, thus has higher efficiency.A note on WENO-Z schemehttps://zbmath.org/1508.651042023-05-31T16:32:50.898670Z"Hu, Fuxing"https://zbmath.org/authors/?q=ai:hu.fuxingSummary: In this paper we recover a latent advantage of WENO-Z schemes. Taking the fifth-order WENO-Z scheme for instance, we realize that the scheme can be regarded as a nonlinear combination of a five-cell stencil and three three-cell stencils. The five-cell stencil is allotted a global higher-order indicator of smoothness than three-cell stencils. Then the five-cell stencil dominates the nonlinear combination and ensures the optimal accuracy in the smooth regions even at extremal points. In non-smooth regions, the three-cell stencils dominate the combination and compress the nonphysical oscillations. As the adaptive order WENO schemes which release the requirement of linear optimal weights, we will show that there is no requirement of linear optimal weights for the WENO-Z schemes as well, and even it is unnecessary to require the sum of linear optimal weights to be one.Two conservative and linearly-implicit compact difference schemes for the nonlinear fourth-order wave equationhttps://zbmath.org/1508.651132023-05-31T16:32:50.898670Z"Zhang, Gengen"https://zbmath.org/authors/?q=ai:zhang.gengenSummary: Two conservative and linearly-implicit difference schemes are presented for solving the nonlinear fourth-order wave equation with the periodic boundary condition. These schemes are based on two different compact finite difference discretization, and they are shown to be fourth-order accurate in space and second-order accurate in time. The discrete energy conservation is obtained for the developed schemes, which preserve the original energy conservation. Meanwhile, two implicit conservative compact difference schemes are also presented. Numerical experiments are given to confirm the theoretical results.A class of non-oscillatory direct-space-time schemes for hyperbolic conservation lawshttps://zbmath.org/1508.651152023-05-31T16:32:50.898670Z"Yeganeh, Solmaz Mousavi"https://zbmath.org/authors/?q=ai:yeganeh.solmaz-mousavi"Farzi, Javad"https://zbmath.org/authors/?q=ai:farzi.javadSummary: The main concern of this paper is to develop a class of non-oscillatory direct-space-time (DST) schemes for hyperbolic conservation laws. This class of DST schemes have optimal order of accuracy, however, similar to the standard schemes the naive implementation of these schemes produce oscillatory and unstable solutions. To study the nonlinear stability of DST schemes, a TVD flux limiter is introduced, and it is proven that the overall method is a TVD scheme. The numerical illustrations justify the non-oscillatory behaviour of the developed class of schemes in presence of shocks and discontinuities. It is worth to note that the underlying DST schemes include both of the upwind and symmetric schemes and the resulting TVD DST scheme produce comparable results with the standard non-oscillatory schemes like WENO schemes.Numerical upscaling for wave equations with time-dependent multiscale coefficientshttps://zbmath.org/1508.651322023-05-31T16:32:50.898670Z"Maier, Bernhard"https://zbmath.org/authors/?q=ai:maier.bernhard"Verfürth, Barbara"https://zbmath.org/authors/?q=ai:verfurth.barbaraThe authors consider the classical wave equation with time-dependent, spatially multiscale coefficients. They propose a fully discrete computational multiscale method in the spirit of the localized orthogonal decomposition in space with a backward Euler scheme in time.
The main result of this paper consists of a rigorous fully discrete a priori error analysis of the considered scheme for the wave equation with time-dependent coefficients. They combine techniques for time-invariant multiscale and smooth time-dependent coefficients in order to prove the expected order of convergence in space and time.
Moreover, they propose an adaptive update strategy for the time-dependent multiscale basis. They propose a Petrov-Galerkin variant of the scheme to avoid products of multiscale functions, thereby reducing the communication in the assembly of the mass and stiffness matrices of the discrete system. The adaptive update strategy improves the computational efficiency.
Numerical experiments illustrate the theoretical results and showcase the practicability of the adaptive update strategy. The relative error between a reference solution and the numerical one is computed.
Reviewer: Giovanni Nastasi (Catania)Correction to: ``An energy conservative \(hp\)-method for Liouville's equation of geometrical optics''https://zbmath.org/1508.651482023-05-31T16:32:50.898670Z"van Gestel, R. A. M."https://zbmath.org/authors/?q=ai:van-gestel.r-a-m"Anthonissen, M. J. H."https://zbmath.org/authors/?q=ai:anthonissen.martijn-johannes-hermanus"ten Thije Boonkkamp, J. H. M."https://zbmath.org/authors/?q=ai:ten-thije-boonkkamp.jan-h-m"IJzerman, W. L."https://zbmath.org/authors/?q=ai:ijzerman.wilbert-lFrom the text: The original version of the article unfortunately contained mistakes. It has been corrected in this correction. The original article [ibid. 89, No. 1, Paper No. 27, 35 p. (2021; Zbl 1496.65187)] has also been corrected.Wave front tracing and asymptotic stability of planar travelling waves for a two-dimensional shallow river modelhttps://zbmath.org/1508.760202023-05-31T16:32:50.898670Z"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Yu, Shih-Hsien"https://zbmath.org/authors/?q=ai:yu.shih-hsienSummary: The propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two-dimensional shallow river model. In this paper, we consider the time-asymptotic stability of weak planar travelling waves for a two-dimensional shallow river model with Darcy's law. We derive an effective parabolic equation to analyze the wave front motion. By employing weighted energy estimates, we show that weak planar travelling waves are time-asymptotically stable under sufficiently small perturbations.Parallel kinetic scheme for transport equations in complex toroidal geometryhttps://zbmath.org/1508.820482023-05-31T16:32:50.898670Z"Boileau, Matthieu"https://zbmath.org/authors/?q=ai:boileau.matthieu"Bramas, Bérenger"https://zbmath.org/authors/?q=ai:bramas.berenger"Franck, Emmanuel"https://zbmath.org/authors/?q=ai:franck.emmanuel"Hélie, Romane"https://zbmath.org/authors/?q=ai:helie.romane"Helluy, Philippe"https://zbmath.org/authors/?q=ai:helluy.philippe"Navoret, Laurent"https://zbmath.org/authors/?q=ai:navoret.laurentA new numerical method (based on a kinetic formulation resembling the Lattice-Boltzmann approach) for solving nonhomogeneous conservative transport equations in toroidal geometries (as is in tokamaks) is proposed. This permits to handle unstructured meshes of the poloidal plane, and allow a parallelization. The algorithm has been tested in a few model problems including the diocotron case (the slipping stream plasma instability).
Reviewer: Piotr Biler (Wrocław)A scattering theory for the wave equation on Kerr black hole exteriorshttps://zbmath.org/1508.830152023-05-31T16:32:50.898670Z"Dafermos, Mihalis"https://zbmath.org/authors/?q=ai:dafermos.mihalis"Rodnianski, Igor"https://zbmath.org/authors/?q=ai:rodnianski.igor"Shlapentokh-Rothman, Yakov"https://zbmath.org/authors/?q=ai:shlapentokh-rothman.yakovSummary: We develop a definitive physical-space scattering theory for the scalar wave equation \(\square_g\psi=0\) on Kerr exterior backgrounds in the general subextremal case \(|a|<M\). In particular, we prove results corresponding to ``existence and uniqueness of scattering states'' and ``asymptotic completeness'' and we show moreover that the resulting ``scattering matrix'' mapping radiation fields on the past horizon \(\mathcal{H}^-\) and past null infinity \(\mathcal{I}^-\) to radiation fields on \(\mathcal{H}^+\) and \(\mathcal{I}^+\) is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity \(\mathcal{I}^-\) is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalized reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity \(\mathcal{I}^+\) of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [Ann. Math. (2) 183, No. 3, 787--913 (2016; Zbl 1347.83002)] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards.The wave equation with locally distributed control in non-cylindrical domainhttps://zbmath.org/1508.930312023-05-31T16:32:50.898670Z"Cui, Lizhi"https://zbmath.org/authors/?q=ai:cui.lizhiSummary: This paper is concerned with exact internal controllability for a one-dimensional wave equation in a non-cylindrical domain. This equation characterizes the motion of a string with a fixed endpoint and the other moving one. When the speed of the moving endpoint is less than wave speed, exact internal controllability of this equation is established.Exact boundary synchronization by groups for a kind of system of wave equations coupled with velocitieshttps://zbmath.org/1508.930402023-05-31T16:32:50.898670Z"Lu, Xing"https://zbmath.org/authors/?q=ai:lu.xing"Li, Tatsien"https://zbmath.org/authors/?q=ai:li.tatsienSummary: This paper deals with the exact boundary controllability and the exact boundary synchronization for a 1-D system of wave equations coupled with velocities. These problems can not be solved directly by the usual HUM method for wave equations, however, by transforming the system into a first order hyperbolic system, the HUM method for 1-D first order hyperbolic systems, established by \textit{T. Li} and \textit{X. Lu} [ESAIM, Control Optim. Calc. Var. 28, Paper No. 34, 27 p. (2022; Zbl 1492.93021)] and \textit{X. Lu} and \textit{T. Li} [Chin. Ann. Math., Ser. B 43, No. 1, 1--16 (2022; Zbl 1487.93014)], can be applied to get the corresponding results.Practical output regulation and tracking for linear ODE-hyperbolic PDE-ODE systemshttps://zbmath.org/1508.931532023-05-31T16:32:50.898670Z"Redaud, Jeanne"https://zbmath.org/authors/?q=ai:redaud.jeanne"Bribiesca-Argomedo, Federico"https://zbmath.org/authors/?q=ai:bribiesca-argomedo.federico"Auriol, Jean"https://zbmath.org/authors/?q=ai:auriol.jeanIn this paper, the authors consider the problem of practical output regulation and output tracking for a linear \(2\times 2\) hyperbolic partial differential equation (PDE) system with actuation (governed by ordinary differential equation (ODE)) at one boundary and load dynamics (governed by ODE). and the output at the other boundary. The present paper proposes a constructive approach for the design of a dynamic, strictly proper, output-feedback control law for practical output regulation and output tracking, guaranteeing a non-zero delay margin for a large class of interconnected ODE-hyperbolic PDEs-ODE systems. The proposed approach can be summarized as follows. Based on some structural assumptions, design a state feedback controller to stabilize an output depending on the states of the unactuated ODE, solving thereby the practical output tracking and regulation problem; Following the backstepping methodology, use a general invertible integral transform to map initial system (dynamically augmented with finite-dimensional exosystems) to a target system; Using a frequency analysis to design a feedback controller; Using structural assumptions to propose a state observer design for the system and the disturbances, in which the filtering techniques to guarantee that all the dynamic error injection gains are strictly proper. Finally, the two designs are coupled to obtain a dynamic output feedback controller.
For the entire collection see [Zbl 1485.93009].
Reviewer: Gen Qi Xu (Tianjin)