Singularity formation for the 1D compressible Euler equations with variable damping coefficient

Recent standardisation work on open plan offices acoustics is characterized by a shift in the way room acoustics is handled. Spatial decay (expressed through rate of spatial decay per doubling of distance, DL2) is preferred to temporal decay (expressed through reverberation time, RT). The complexity of open plan office acoustic design results from the fundamental contradiction of communication and concentration. DL2 reflects better the challenges of sound control in such rooms, which is basically sound propagation control. Beyond, this contradiction, specific space analysis models and tools can help to visualize the acoustic interactions between workstations, teams and departments accommodated in the same room. These models are now ready to be included in standardization work. International technical standard for field measurements of DL2 integrates new descriptive models for open plan office acoustics, taking into account geometric proportions, presence of screens and furniture as well as group behaviour and speech characteristics, such as increased energy contents at low frequencies, voice levels, raised hearing sensitivity at high frequencies. Also, normative guidelines from Netherlands and France integrating this approach will be presented.

We obtain spatial and temporal decay rates of weak solutions of the Navier–Stokes equations, and for strong solutions. For the spatial decay rate of the weak solutions, the power of the weight given by He and Xin in 2001 does not exceed 3/2;. However, we show the power can be extended up to 5/2;.

We consider solutions u to the 3D nonlinear Schrödinger equation i∂tu + Δu + |u|2u = 0. In particular, we are interested in finding criteria on the initial data u0 that predict the asymptotic behaviour of u(t), e.g., whether u(t) blows up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for H1 data) (Duyckaerts–Holmer–Roudenko) if M[u]E[u] ⩽ M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u and Q denotes the ground state solution to −Q + ΔQ + |Q|2Q = 0. Here we consider the complementary case M[u]E[u] > M[Q]E[Q]. In the first (analytical) part of the paper, we present a result due to Lushnikov, based on the virial identity and the generalized uncertainty principle, giving a sufficient condition for blow-up. By replacing the uncertainty principle in his argument with an interpolation-type inequality, we obtain a new blow-up condition that in some cases improves upon Lushnikov's condition. Our approach also allows for an adaptation to radial infinite-variance initial data that has a conceptual interpretation: for real-valued initial data, if a certain fraction of the mass is contained within the ball of radius M[u], then blow up occurs. We also show analytically (if one takes the numerically computed value of ) that there exist Gaussian initial data u0 with negative quadratic phase such that but the solution u(t) blows up. In the second (numerical) part of the paper, we examine several different classes of initial data—Gaussian, super Gaussian, off-centred Gaussian, and oscillatory Gaussian—and for each class give the theoretical predictions for scattering or blow-up provided by the above theorems as well as the results of numerical simulation. We find that depending upon the form of the initial conditions, any of the three analytical criteria for blow-up can be optimal. We formulate several conjectures, among them that for real initial data, the quantity provides the threshold for scattering.

Blow-up and global existence for solutions to the porous medium equation with reaction and slowly decaying density

The Euler–Boltzmann equations are an important class of mathematical models that describe the coupling between particle transport and macroscopic fluid dynamics. They find broad applications in plasma physics, rarefied gas dynamics, and astrophysics. In these fields, incorporating a time-dependent damping term is crucial for modeling real-world scenarios, as opposed to idealized inviscid conditions. In recent years, there has been growing interest in the long-time behavior of their solutions. This paper focuses on the initial value problem for the three-dimensional Euler–Boltzmann equations with time-dependent damping, aiming to investigate the finite-time blowup behavior of classical solutions. We use an integration method with general test function f and show that if the initial data are sufficiently large, classical solutions of the Euler–Boltzmann equations with time-dependent damping in R3 will blowup on or before the finite time T*>0.

In this paper we consider a new rod equation derived recently by Dai [Acta Mech. 127 No. 1–4, 193–207 (1998)] for a compressible hyperelastic material. We establish local well‐posedness for regular initial data and explore various sufficient conditions of the initial data which guarantee the blow‐up in finite time both for periodic and non‐periodic case. Moreover, the blow‐up time and blow‐up rate are given explicitly. Some interesting examples are given also. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In \cite{ChCa}, Califano and Chiuderi conjectured that the energy of incompressible Magnetic hydrodynamical system is dissipated at a rate that is independent of the ohmic resistivity. The goal of this paper is to mathematically justify this conjecture in three space dimension provided that the initial magnetic field and velocity is a small perturbation of the equilibrium state $(e_3,0).$ In particular, we prove that for such data, 3-D incompressible MHD system without magnetic diffusion has a unique global solution. Furthermore, the velocity field and the difference between the magnetic field and $e_3$ decay to zero in both $L^\infty$ and $L^2$ norms with explicit rates. We point out that the decay rate in the $L^2$ norm is optimal in sense that this rate coincides with that of the linear system. The main idea of the proof is to exploit H$\ddot{o}$rmander's version of Nash-Moser iteration scheme, which is very much motivated by the seminar papers \cite{Kl80, Kl82, Kl84} by Klainerman on the long time behavior to the evolution equations.

In this paper, we study the asymptotic decay properties in both spatial and temporal variables for a class of weak and strong solutions, by constructing the weak and strong solutions in corresponding weighted spaces. It is shown that, for the strong solution, the rate of temporal decay depends on the rate of spatial decay of the initial data. Such rates of decay are optimal.

This paper is concerned with the Cauchy problem for one-dimensional unipolar Euler-Poisson equations with time-dependent damping, where the time-asymptotically degenerate damping in the form of - (1+t) u for > 0 with > 0 plays a crucial role for the structure of solutions. The main issue of the paper is to investigate the critical case with = 1. We first prove that, for all cases with > 0 and > 0 (including the critical case of = 1), once the initial data is steep at a point, then the solutions are locally bounded but their derivatives will blow up in finite time, by means of the method of Riemann invariants and the technical convex analysis. Secondly, for the critical case of = 1 with > 7/3, we prove that there exists a unique global solution, once the initial perturbation around the constant steady-state is sufficiently small. In particular, we derive the algebraic convergence rates of the solution to the constant steady-state, which are piecewise, related to the parameter for 7/3 < 3, 3 < 4 and > 4. The adopted method of proof in this critical case is the technical time-weighted energy method and the time-weight depends on the parameter . Finally, we carry out some numerical simulations in two cases for blow-up and global existence, respectively, which numerically confirm our theoretical results.

A transfer matrix approach to the analysis of the dynamics of mistuned bladed disks is presented. The study focuses on mono-coupled systems, in which each blade is coupled to its two neighboring blades, and bi-coupled systems, where each blade is coupled to its four nearest neighbors. Transfer matrices yield the free dynamics, both the characteristic free wave and the normal modes - in closed form for the tuned assemblies. Mistuned assemblies are represented by random transfer matrices and an examination of the effect of mistuning on harmonic wave propagation yields the localization factor - the average rate of spatial wave amplitude decay per blade - in the mono-coupled assembly. Based on a comparison of the wave propagation characteristics of the mono- and bi-coupled assemblies, important conclusions are drawn about the effect of the additional coupling coordinate on the sensitivity to mistuning and the strength of mode localization predicted by a mono-coupled analysis.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

ABSTRACTThis paper is dedicated to study the Cauchy problem of the nonlinear Schrödinger equation without gauge invariance where and . When , we first prove local well-posedness of the equation in . If in addition, , we prove the global well-posedness with small initial data in . Under a suitable condition on the initial data and , we prove that the -norm of the solution would blow up in finite time although the initial data are arbitrarily small. Meanwhile, we also give a large initial data blow-up result when in . Finally, we show the non-existence of local weak solution for some -data with when .

Global classical solution to two-dimensional compressible Navier–Stokes equations with large data in [formula omitted

In distributed optimization (DO), the designed algorithms are expected to have a fast convergence rate but less computation cost. Moreover, the boundedness of the control inputs is generally required for practical networking agent systems with actuator limitations. Motivated by these observations, we first revisit the well-known finite-time distributed average tracking (FTDAT) problem where a novel sufficient condition on the control gain and the finite settling time estimation are derived based on the minimum cut of the underlying topology. Then, based on FTDAT, three types of discontinuous dynamics with bounded inputs are designed for solving unconstrained and constrained DO problems, respectively. The first algorithm can successfully find the optimal solution for an unconstrained DO in finite time. For DO problems with a common constraint set or separated equality constraints, two projection-based algorithms are designed and utilized to first make the agents' states achieve consensus in finite time, and then drive the common state to the optimum exponentially. Finally, several case studies and extensive numerical simulations are conducted to testify the designed algorithms.

Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data