For a class of reaction–convection–diffusion systems studied by Sattinger, notably including Majda’s model of reacting flow, we rigorously characterize the transition from stability to time-periodic “galloping” instability of traveling-wave solutions as a relative Poincaré–Hopf bifurcation arising in ODE with a group invariance– in this case, translational symmetry. More generally, we show how to construct a finite-dimensional center manifold for a second-order parabolic evolution equation inheriting an underlying group invariance of the PDE, by working with a canonical integro-differential equation induced on the quotient space. This reduces the questions of existence and stability of bounded solutions of the PDE to existence and stability of solutions of the reduced, finite-dimensional ODE on the center manifold, which may then be studied by more standard, finite-dimensional bifurcation techniques.

The unique local existence is established for the Cauchy problem of the incompressible Navier-Stokes equations with the Coriolis force for a class of initial data nondecreasing at space infinity. The Coriolis operator restricted to divergence free vector fields is a zero order pseudodifferential operator with the skew-symmetric matrix symbol related to the Riesz operator. It leads to the additional term in the Navier-Stokes equations which has real parameter being proportional to the speed of rotation. For initial datum as Fourier preimage of finite Radon measures having no-point mass at the origin we show that the length of existence time-interval of mild solution is independent of the rotation speed.

The approximate solution of optimization and control problems for systems governed by linear, elliptic partial differential equations is considered. Such problems are most often solved using methods based on the application of the Lagrange multiplier rule followed by discretization through, e.g., a Galerkin finite element method. As an alternative, we show how least-squares finite element methods can be used for this purpose. Penalty-based formulations, another approach widely used in other settings, have not enjoyed the same level of popularity in the partial differential equation case perhaps because naively defined penalty-based methods can have practical deficiencies. We use methodologies associated with modern least-squares finite element methods to develop and analyze practical penalty methods for the approximate solution of optimization problems for systems governed by linear, elliptic partial differential equations. We develop an abstract theory for such problems; along the way, we introduce several methods based on least-squares notions, and compare and constrast their properties.

We explore a level set representation of vorticity in the study of the singularity problems for incompressible fluid models. This representation exists for all initial vorticity fields. We further apply it to study the 3D Lagrangian averaged Euler equations and the 3D Euler equations, and obtain new global existence conditions.

We consider how the finite mass shock wave cosmology introduced by the authors in Shock-wave cosmology inside a black hole could connect up with Guth’s original theory of inflation.