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Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints. The classification of universal displacements in homogeneous linear elasticity has been completed for all the eight anisotropy classes. Here, we extend our previous work by studying universal displacements in inhomogeneous anisotropic linear elasticity assuming that the directions of anisotropy are known. We show that universality constraints of inhomogeneous linear elasticity include those of homogeneous linear elasticity. For each class and for its known universal displacements, we find the most general inhomogeneous elastic moduli that are consistent with the universality constrains. It is known that the larger the symmetry group, the larger the space of universal displacements. We show that the larger the symmetry group, the more severe the universality constraints are on the inhomogeneities of the elastic moduli. In particular, we show that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not admit universal displacements and we completely characterize the universal inhomogeneities for the other six anisotropy classes.
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Stability of Peaked Solitary Waves for a Class of Cubic Quasilinear Shallow-Water Equations
Abstract This paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space $H^1$, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in $$H^1\cap W^{1,4}$$. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.
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- Award ID(s):
- 2205910
- PAR ID:
- 10443599
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 7
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 6186 to 6218
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnac032
