Well-Posedness and Infinite Propagation Speed for the Fifth-Order Camassa-Holm Equation with Weakly Dissipative Term

In this paper, we study the fifth-order Camassa-Holm equation with weakly dissipative term. We first give the local well-posedness result and the blow up criterion. Then, we establish sufficient conditions to guarantee that the solution exists globally in time. Finally, the infinite propagation speed of this equation is also investigated.

TWO-DIMENSIONAL VISUALIZATION OF THE PROPAGATION SPEED OF CORTICAL SPREADING DEPRESSION IN RAT CORTEX

Cortical spreading depression(CSD),which is a significant pathological phenomenon that correlates with migraines and cerebral ischemia,has been characterized by a wave of depolarization among neuronal cells and propagates across the cortex at a rate of 2–5mm/min.Although the propagation pattern of CSD was well-investigated using high-resolution optical imaging technique,the variation of propagation speed of CSD across different regions of cortex was not well-concerned,partially because of the lack of ideal approach to visualize two-dimensional distribution of propagation speed of CSD over the whole imaged cortex.Here,we have presented a method to compute automatically the propagation speed of CSD throughout every spots in the imaged cortex.In this method,temporal clustering analysis(TCA)and least square estimation(LSE)were first used to detect origin site where CSD was induced.Taking the origin site of CSD as the origin of coordinates,the data matrix of each image was transformed into the corresponding points based on the polar-coordinate representation.Then,two fixed-distance regions of interest(ROIs)are sliding along with the radial coordinate at each polar angle within the image for calculating the time lag with correlating algorithm.Finally,we could draw a twodimensional image,in which the value of each pixel represented the velocity of CSD when it spread through the corresponding area of the imaged cortex.The results demonstrated that the method can reveal the heterogeneity of propagation speed of CSD in the imaged cortex with high fidelity and intuition.

FINITE SPEED OF PROPAGATION OF SOLUTIONS FOR SOME QUASILINEAR HIGHER ORDER PARABOLIC EQUATIONS WITH DOUBLY STRONG DEGENERATION

Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.

Multiscale simulation of nanometric cutting of single crystal copper——effect of different cutting speeds

A multiscale simulation has been performed to determine the effect of the cutting speed on the deformation mechanism and cutting forces in nanometric cutting of single crystal copper. The multiscale simulation model, which links the finite element method and the molecular dynamics method, captures the atomistic mechanisms during nanometric cutting from the free surface without the computational cost of full atomistic simulations. Simulation results show the material deformation mechanism of single crystal copper greatly changes when the cutting speed exceeds the material static propagation speed of plastic wave. At such a high cutting speed, the average magnitudes of tangential and normal forces increase rapidly. In addition, the variation of strain energy of work material atoms in different cutting speeds is investigated.

Simulation and Experimental Analysis of Super High-Speed Grinding of Ductile Material

This study aims to reduce the work-affected layer of the machined surface by carrying out the grinding at the speed over static pr o pagation speed of plastic wave of ductile materials and also aims to clarify suc h super high-speed machining mechanism.This paper reports on the result obtain ed through the molecular dynamics simulations and experiments on the super-spee d grinding below and beyond static propagation speed of aluminum.From the simul ation results,it is verified that the plastic deformation is reduced when the m a chining speed exceeds the material static propagation speed of plastic wave and its mechanism is completely different from that of the ordinary grinding process .Experimental results also show the improvement of the surface integrity when t he machining speed exceeds the material static propagation speed of plastic wave .

THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION

In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.

Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes

We construct new fifth-order alternative WENO(A-WENO)schemes for the Euler equations of gas dynamics.The new scheme is based on a new adaptive diffusion centralupwind Rankine-Hugoniot(CURH)numerical flux.The CURH numerical fluxes have been recently proposed in[Garg et al.J Comput Phys 428,2021]in the context of secondorder semi-discrete finite-volume methods.The proposed adaptive diffusion CURH flux contains a smaller amount of numerical dissipation compared with the adaptive diffusion central numerical flux,which was also developed with the help of the discrete RankineHugoniot conditions and used in the fifth-order A-WENO scheme recently introduced in[Wang et al.SIAM J Sci Comput 42,2020].As in that work,we here use the fifth-order characteristic-wise WENO-Z interpolations to evaluate the fifth-order point values required by the numerical fluxes.The resulting one-and two-dimensional schemes are tested on a number of numerical examples,which clearly demonstrate that the new schemes outperform the existing fifth-order A-WENO schemes without compromising the robustness.

Asymptotic Speed of Wave Propagation for A Discrete Reaction-Diffusion Equation

We deal with asymptotic speed of wave propagation for a discrete reactlon-diffusion equation. We find the minimal wave speed c★ from the characteristic equation and show that c★ is just the asymptotic speed of wave propagation. The isotropic property and the existence of solution of the initial value problem for the given equation are also discussed.

Optical Observations on Propagation Characteristics of Leaders in Cloud-to-Ground Lightning Flashes

Using 2 high-speed cameras, we have recorded 14 negative cloud-to-ground （CG） lightning flashes, half of which are natural and the others are artificially triggered. The two-dimensional （2D） propagation speed of different type leaders and the luminosity of lightning channel are analyzed in detail. Bidirectional leader processes are observed during the initial processes of two altitude triggered negative lightning （ATNL） flashes. The analysis shows： the propagation speed of the upward positive leader （UPL） before the initiation of the downward negative leader （DNL） is at the order of 10^4-10^5 m s-1; the UPL can be intensified by the initiation and development of the DNL in the way that the luminosity is enhanced and the speed is sped up; after initiation, the DNL in one ATNL flash propagates downward three times intermittently with interval of about 1 ms, while that in the other ATNL flash propagates downward continuously with a speed at the order of 10^5 m s^-1. In the five classical triggered negative lightning （CTNL） flashes, the propagation speeds of the UPLs vary between 0.35×10^5 and 7.71×10^5 m s-1, and the variations of their luminosities and speeds are quite complex during the development processes. Among the four observed natural negative lightning flashes occurred on the land, three have only one return stoke （RS） each and all of their DNLs have many branches with an average speed at the order of 10^5 m s-l; while the another one has 13 RSs. In the CG flash with 13 RSs, the DNL before the first RS has no obvious branch below 1.4 km above the ground, and its speed ranges from 2.2×10^5 to 2.3×10^6 m s-1 between the heights of 0.7 and 1.4 km and exceeds 3.9×10^6 m s-1 below 0.7 km; preceding the 4th RS, an attempted leader is observed with a speed ranging from 1.1× 10^5 to 1.1×10^6 m s-1 between 0.8 and 1.5 km. As for the three observed natural negative lightning flashes occurred on the sea, each has only one RS, and each DNL preceding the RS has a few branches, two of which have an average propagation speed at the order of 10^5 m s-1, and the other of 10^6 m s-1, respectively. All the DNLs contained in the observed natural negative lightning flashes, except the attempted leader, propagate with gradually increasing luminosity and increasing speed in whole.

Experimental hydrodynamic study of the Qiantang River tidal bore

To study the hydrodynamics of tidal bore, a physical modeling study is carried out in a rectangular flume with considera- tions of the tidal bore heights, the propagation speeds, the tidal current velocities, the front steepness, and the bore shapes. After the validation with the field observations, the experimental results are analyzed, and it is shown that： （1） the greater initial ebb velocity or the larger initial water depth impedes the tidal bore propagation, （2） the maximum bore height appears at an initial ebb velocity in the range of 0.5 m/s-l.5 m/s. （3） when the Froude number exceeds 1.2, an undular bore appears, atter it exceeds 1.3, a breaking bore occurs, and after it exceeds 1.7, the bore is broken.

Heat Equation with Memory in Anisotropic and Non-Homogeneous Media

A modified Fourier＇s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.

Transient process of methane-oxygen diffusion flame-street establishment in a microchannel

“Flame-street”is an interesting diffusion flame behavior in which a series of flame-segments is separately distributed along the mixing layer in a narrow channel.This experimental phenomenon was experimentally and numerically investigated with the focus on the steady-state,thermo-chemical flame structures in previous literature.In the present paper,the dynamic formation process of a methane-oxygen diffusion flame-street structure was simulated with a reacting flow solver developed based on the open-source framework OpenFOAM.By imposing a certain amount of ignition-energy near the channel outlet,a reaction-kernel was formed and bifurcated.Subsequently,three separate flames were consecutively generated from this kernel and propagated within the channel.The whole process was completed within 15 ms and all the discrete flames were eventually in a steady-state.Interestingly,different propagation features were observed for the three flame segments:The leading flame experienced a flame shape/type change from a tribrachial structure in its fastpropagating phase to a long,trailing diffusion tail after being anchored to the inlet.The successive flame had a much lower propagation speed,keeping its two wing-like(fuel-lean premixed and fuel-rich premixed)structure while moving toward its stabilization location,which was approximately in the middle of the channel.The last flame,after the ignition source was turned-off,was immediately convected a bit downstream,and eventually featured a similar two-branch-like structure as the second one.Moreover,chemical insights for the premixed and diffusion branches of the leading flame were also provided with the change of significance of some key elementary reactions focused on,in order to attain a detailed profiling of the flame-type transition.This paper is a first-ever one discussing the transient formation of flame-streets in literature and is believed to be useful for obtaining a comprehensive understanding of this unique flame characteristics from a dynamic point of view.