Piecewise Constant Background Research Articles

Weakly nonlinear surface waves on the plasma–vacuum interface

We consider the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement, where the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Our aim is to construct highly oscillating surface wave solutions in weakly nonlinear regime to this plasma–vacuum interface problem.Under a necessary and sufficient stability condition for a piecewise constant background state, we construct approximate solutions at any arbitrarily large order of accuracy to the free boundary problem in three space dimensions when the initial discontinuity displays high frequency oscillations. Moreover, such approximate surface waves have nontrivial residual non-oscillatory components.

Stability of Multidimensional Thermoelastic Contact Discontinuities

We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface. Mathematically, this renders a nonlinear multidimensional hyperbolic problem with a characteristic free boundary. We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. The missing normal derivatives are recovered from the estimates of several quantities relating to physical involutions. In the estimate of tangential derivatives, there is a significant new difficulty, namely the presence of characteristic variables in the boundary conditions. To overcome this difficulty, we explore an intrinsic cancellation effect, which reduces the boundary terms to an instant integral. Then we can absorb the instant integral into the instant tangential energy by means of the interpolation argument and an explicit estimate for the traces on the hyperplane.

On Bragg resonances and wave triad interactions in two-layered shear flows

The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways: (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the higher-order spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both on the theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of the study of these kinds of problems, to incorporate piecewise constant background shear.

Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by nonlinear arguments. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic–sonic limit of the solutions is also shown. Finally, through the incompressible limit, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.

Robustly reconstructing magnetic resonance images via structure decomposition

In magnetic resonance (MR) imaging, for highly under-sampled k-space data, it is typically difficult to reconstruct images and preserve their original texture simultaneously. The high-degree total variation (HDTV) regularization handles staircase effects but still blurs textures. On the other hand, the non-local TV (NLTV) regularization can preserve textures, but will introduce additional artifacts for highly-noised images. In this paper, we propose a reconstruction model derived from HDTV and NLTV for robust MRI reconstruction. First, an MR image is decomposed into a smooth component and a texture component. Second, for the smooth component with sharp edges, isotropic second-order TV is used to reduce staircase effects. For the texture component with piecewise constant background, NLTV and contourlet-based sparsity regularizations are employed to recover textures. The piecewise constant background in the texture component contributes to accurately detect non-local similar image patches and avoid artifacts introduced by NLTV. Finally, the proposed reconstruction model is solved through an alternating minimization scheme. The experimental results demonstrate that the proposed reconstruction model can effectively achieve satisfied quality of reconstruction for highly under-sampled k-space data.

Bi-component decomposition based hybrid regularization method for partly-textured CS-MR image reconstruction

For compressive sensing magnetic resonance (CS-MR) image reconstruction, it is vital to preserve edges and textures while eliminating aliasing artifacts. The total variation (TV) regularization preserves edges well but causes staircase effect and fails to preserve textures. The high-order regularization can eliminate the staircase effect but causes edge blurring. The nonlocal TV (NLTV) regularization can preserve textures well, but causes extra artifacts and is likely to preserve the texture-like artificial structures caused by aliasing artifacts mistakenly. In this paper, we assume that the image consists of cartoon component and anisotropic component. For the cartoon component, we utilize the fractional-order TV regularization to eliminate the staircase effect and avoid edge blurring, and more importantly, we can avoid its disadvantage in texture preservation because there are no textures in the cartoon component. We utilize the NLTV regularization and the shearlet based sparsity regularization for the anisotropic component with piecewise-constant background. Without the effect of intensity inhomogeneity, the NLTV regularization can avoid preserving the texture-like artificial structures while preserving the real edges. The shearlet based sparsity regularization can provide further improvement of the image quality. Numerical experiments demonstrate that our method can eliminate the aliasing artifacts and preserve the edges and textures efficiently.

A new non-iterative singular sources method for the reconstruction of piecewise constant media.

We describe a novel non-iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering, which we call the ‘singular sources method’. The basic idea of the method is to use the behaviour of the scattered field for singular incident fields (multipoles) to calculate the size of the refractive index n at some point z0 on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium. The paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium. Also, we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions, using techniques that differ from those used to prove previous well-known results.