We consider the Cauchy problem for a strictly hyperbolic, n × n system in one-space dimension: ut + A(u)ux = 0, assuming that the initial data have small total variation. We show that the solutions of the viscous approximations ut + A(u)ux = euxx are defined globally in time and satisfy uniform BV estimates, independent of e. Moreover, they depend continuously on the initial data in the L 1 distance, with a Lipschitz constant independent of t, e. Letting e → 0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where A = Df is the Jacobian

/pdf/vanishing-viscosity-solutions-of-nonlinear-hyperbolic-xoy9mm0dyu.pdf

Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems

Conductive diamond possesses unique features as compared to other solid electrodes, such as a wide electrochemical potential window, a low and stable background current, relatively rapid rates of electron-transfer for soluble redox systems without conventional pretreatment, long-term responses, stability, biocompatibility, and a rich surface chemistry. Conductive diamond microcrystalline and nanocrystalline films, structures and particles have been prepared using a variety of approaches. Given these highly desirable attributes, conductive diamond has found extensive use as an enabling electrode across a variety of fields encompassing chemical and biochemical sensing, environmental degradation, electrosynthesis, electrocatalysis, and energy storage and conversion. This review provides an overview of the fundamental properties and highlights recent progress and achievements in the growth of boron-doped (metal-like) and nitrogen and phosphorus-doped (semi-conducting) diamond and hydrogen-terminated undoped diamond electrodes. Applications in electroanalysis, environmental degradation, electrosynthesis electrocatalysis, and electrochemical energy storage are also discussed. Diamond electrochemical devices utilizing micro-scale, ultramicro-scale, and nano-scale electrodes as well as their counterpart arrays are viewed. The challenges and future research directions of conductive diamond are discussed and outlined. This review will be important and informative for chemists, biochemists, physicists, materials scientists, and engineers engaged in the use of these novel forms of carbon.

Conductive diamond: synthesis, properties, and electrochemical applications

We rigorously justify in $3D$ the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves. The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.

Large time existence for 3D water-waves and asymptotics

We consider the Cauchy problem for a strictly hyperbolic, $n\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation. 
We show that the solutions of the viscous approximations $u_t+A(u)u_x=\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\ve$. Moreover, they depend continuously on the initial data in the $\L^1$ distance, with a Lipschitz constant independent of $t,\ve$. Letting $\ve\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\R^n\mapsto\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$.

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/e] for initial data of the form e
 Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e
 T/e
 ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation.

Almost global wellposedness of the 2-D full water wave problem

The Kawahara equation is a higher-order Korteweg-de Vries equation with an additional fifth order derivative term. It was derived by Hasimoto as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime when the Bond number is nearly one third. In this paper, we give a mathematically rigorous justification of this modeling and show that the solution of the Kawahara equation approximates that of the full problem of capillary-gravity waves in an appropriate sense for a long time interval. We also consider the case where the bottom is not flat and derive coupled Kawahara type equations whose solution approximates that of the full problem in that case.

/pdf/a-long-wave-approximation-for-capillary-gravity-waves-and-1otn2jvmje.pdf

A long wave approximation for capillary-gravity waves and the kawahara equation

We consider the initial value problem to a model system for water waves. The model system is the Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The model are nonlinear dispersive equations and the hypersurface t = 0 is characteristic for the model equations. Therefore, the initial data have to be restricted in an infinite dimensional manifold in order to the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh-Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces.

/pdf/solvability-of-the-initial-value-problem-to-a-model-system-42dm8zvvdm.pdf

Solvability of the initial value problem to a model system for water waves

Isobe–Kakinuma model for water waves as a higher order shallow water approximation

The forced Korteweg-de Vries (KdV) equation is the KdV equation with a forcing term and arises as a model for several physical situations. In this paper, we study the validity of this modeling for capillary-gravity waves in an infinitely long canal over an uneven bottom. An underlying background flow of the water together with an uneven bottom causes a deriving force in the KdV equation in some scaling limit. We will show that the solutions of the full problem for capillary-gravity waves split up into two waves moving with different propagation speeds and that the shape of each wave is governed by a forced KdV equation in a slow time scale.

A mathematical justification of the forced korteweg-de vries equation for capillary-gravity waves

We consider the initial value problem to the Isobe–Kakinuma model for water waves and the structure of the model The Isobe–Kakinuma model is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian The Isobe–Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations Since the hypersurface $$t=0$$
 is characteristic for the Isobe–Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces We also discuss the linear dispersion relation to the model

pdf/solvability-of-the-initial-value-problem-to-the-isobe-46kdaxr97v.pdf

Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves

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