A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such ‘gauge invariance’ is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by noncausal dual problems.

We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.

We establish various qualitative properties of liquid Lane-Emden stars in R d \mathbb {R}^d , including bounds for its density profile ρ \rho and radius R R . Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when γ ≥ 2 ( d − 1 ) / d \gamma \geq 2(d-1)/d ; linearly stable when γ > 2 ( d − 1 ) / d \gamma >2(d-1)/d for stars with small relative central density ρ ( 0 ) − ρ ( R ) \rho (0)-\rho (R) ; and linearly unstable when γ > 2 ( d − 1 ) / d \gamma >2(d-1)/d for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.

We study a two-dimensional quaternary inhibitory system. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which three species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geometry of local structures as a three-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. Geometrical descriptions of limit configurations are derived.

In this paper a hyperbolic system of partial differential equations for two-phase mixture flows with N N components is studied. It is derived from a more complicated model involving diffusion and exchange terms. Important features of the model are the assumption of isothermal flow, the use of a phase field function to distinguish the phases and a mixture equation of state involving the phase field function as well as an affine relation between partial densities and partial pressures in the liquid phase. This complicates the analysis. A complete solution of the Riemann initial value problem is given. Some interesting examples are suggested as benchmarks for numerical schemes.

This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called σ 2 , p \sigma _{2,p} -energy, in polar coordinates. Let X ⊂ R n {\mathbb {X}}\subset \mathbb {R}^n be a bounded Lipschitz domain and consider the energy functional ( 1.1 ) (1.1) whose integrand is defined by W ( ∇ u ( x ) ) ≔ ( σ 2 ( u ) ) p 2 + Φ ( det ∇ u ) {\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u) over an appropriate space of admissible maps, A p ( X ) \mathcal {A}_p({\mathbb {X}}) . Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method.

In this paper, we rigorously derive the compressible one-fluid Navier–Stokes equations from the scaled compressible two-fluid Navier–Stokes–Maxwell equations under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by using the elaborate energy estimates.

In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric.

Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness. These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding special velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.