The first part of this paper describes some important underlying themes in the mathematical theory of continuum mechanics that are distinct from formulating and analyzing governing equations. The main part of this paper is devoted to a survey of some concrete, conceptually simple, pretty problems that help illuminate the underlying themes. The paper concludes with a discussion of the crucial role of invariant constitutive equations in computation.

We study an asymptotic estimate on the number of negative eigenvalues of the Schr\"odinger operators on unbounded fractal spaces which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpi\'nski gasket and unbounded generalized Sierpi\'nski carpets. Final, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

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This paper concerns the physical behaviors of any solutions to the one dimensional compressible Navier-Stokes equations for viscous and heat conductive gases with constant viscosities and heat conductivity for fast decaying density at far fields only. First, it is shown that the specific entropy becomes not uniformly bounded immediately after the initial time, as long as the initial density decays to vacuum at the far field at the rate not slower than $O\left(\frac1{|x|^{\ell_\rho}}\right)$ with $\ell_\rho>2$. Furthermore, for faster decaying initial density, i.e., $\ell_\rho\geq4$, a sharper result is discovered that the absolute temperature becomes uniformly positive at each positive time, no matter whether it is uniformly positive or not initially, and consequently the corresponding entropy behaves as $O(-\log(\varrho_0(x)))$ at each positive time, independent of the boundedness of the initial entropy. Such phenomena are in sharp contrast to the case with slowly decaying initial density of the rate no faster than $O(\frac1{x^2})$, for which our previous works \cite{LIXINADV,LIXINCPAM,LIXIN3DK} show that the uniform boundedness of the entropy can be propagated for all positive time and thus the temperature decays to zero at the far field. These give a complete answer to the problem concerning the propagation of uniform boundedness of the entropy for the heat conductive ideal gases and, in particular, show that the algebraic decay rate $2$ of the initial density at the far field is sharp for the uniform boundedness of the entropy. The tools to prove our main results are based on some scaling transforms, including the Kelvin transform, and a Hopf type lemma for a class of degenerate equations with possible unbounded coefficients.

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and NavierStokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation

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The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions

The main result of this paper, Theorem 1.5, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on the space of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.

Let G=(V,E) be a finite weighted graph, and Ω⊆V be a domain such that Ω^∘≠∅. In this paper, we study the following initial boundary problem for the non-homogenous wave equation ∂^2_t u(t,x) − Δ_Ω u(t,x) = f(t,x), (t,x)∈[0,∞)×Ω^∘ u(0,x)=g(x), x∈Ω^∘, ∂_t u(0,x)=h(x), x∈Ω^∘, u(t,x)=0, (t,x)∈[0,∞)×∂Ω, where Δ_Ω denotes the Dirichlet Laplacian on Ω^∘. Using Rothe's method, we prove that the above wave equation has a unique solution.