Global solutions of quasi-linear Hamiltonian mKdV equation

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space-time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.Comment: 25 page

A higher dispersion KdV equation on the half-line

The initial-boundary value problem (ibvp) for the $m$-th order dispersion Korteweg-de Vries (KdV) equation on the half-line with rough data and solution in restricted Bourgain spaces is studied using the Fokas Unified Transform Method (UTM). Thus, this work advances the implementation of the Fokas method, used earlier for the KdV on the half-line with smooth data and solution in the classical Hadamard space, consisting of function that are continuous in time and Sobolev in the spatial variable, to the more general Bourgain spaces framework of dispersive equations with rough data on the half-line. The spaces needed and the estimates required arise at the linear level and in particular in the estimation of the linear pure ibvp, which has forcing and initial data zero but non-zero boundary data. Using the iteration map defined by the Fokas solution formula of the forced linear ibvp in combination with the bilinear estimates in modified Bourgain spaces introduced by this map, well-posedness of the nonlinear ibvp is established for rough initial and boundary data belonging in Sobolev spaces of the same optimal regularity as in the case of the initial value problem for this equation on the whole line

The Korteweg-de Vries equation on an interval

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 051507 (2019) and may be found at https://doi.org/10.1063/1.5080366.The initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas’ unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega

Initial-boundary value problems for a reaction-diffusion equation

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 081509 (2019); doi: 10.1063/1.5118767 and may be found at https://aip.scitation.org/doi/10.1063/1.5118767.A novel approach that utilizes Fokas’s unified transform is employed for studying a reaction-diffusion equation with power nonlinearity formulated either on the half-line or on a finite interval with data in Sobolev spaces. This approach was recently introduced for initial-boundary value problems involving dispersive nonlinear equations such as the nonlinear Schrödinger and the Korteweg-de Vries equations. Thus, the present work extends the new approach from dispersive equations to diffusive ones, demonstrating the universality of the unified transform in the analysis of nonlinear evolution equations on domains with a boundary

Determination of the Plastic Stress–Strain Relationship of a Rupture Disc Material with Quasi-Static and Dynamic Pneumatic Bulge Processes

Rupture discs, manufactured using a hydraulic or pneumatic bulge process, are widely used to protect vessels from over-pressuring. The stress–strain relationship of the material in the bulge process plays a major role in understanding the performance of rupture discs. Moreover, both the theoretical analyses and numerical simulations of rupture discs demand a reliable stress–strain relationship of the material in a real bulge process. In this paper, an approach for determining the plastic stress–strain relationship of a rupture disc material in the bulge process is proposed based on plastic membrane theory and force equilibrium equations. Pressures of compressed air and methane/air mixture explosions were used for the bulge pressure to accomplish the quasi-static and dynamic bulge processes of tested discs. Experimental apparatus for the quasi-static bulge test and the dynamic bulge test were built. The stress–strain relations of 316L material during bulge tests were obtained, compared, and discussed. The results indicated that the bulge height at the top of the domed disc increased linearly with an increase in bulge pressure in the quasi-static and dynamic bulge processes, and the effective strain increased exponentially. The rate of pressure rise during the bulge process has a significant effect on the deformation behavior of disc material and hence the stress–strain relationship. At the same bulge pressure, a disc tested with a larger pressure rise rate had smaller bulge height and effective strain. At the same effective stress at the top of the domed disc, discs subjected to a higher pressure rise rate had smaller effective strain, and hence they are more difficult to rupture. Hollomon’s equation is used to represent the relationship between the effective stress and strain during bulge process. For pressure rise rates in the following range of 0 (equivalent to quasi-static condition), 2–10 MPa/s, 10–50 MPa/s, and 50–100 MPa/s, the relation of stress and strain is σe = 1259.4·εe0.4487, σe = 1192.4·εe0.3261, σe = 1381.2·εe0.2910, and σe = 1368.4·εe0.1701, respectively