Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).

We study the residual diffusion phenomenon in chaotic advection computationally via adaptive orthogonal basis. The chaotic advection is generated by a class of time periodic cellular flows arising in modeling transition to turbulence in Rayleigh-Bnard experiments. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, the solutions of the advection-diffusion equation develop sharp gradients, and demand a large number of Fourier modes to resolve, rendering computation expensive. We construct adaptive orthogonal basis (training) with built-in sharp gradient structures from fully resolved spectral solutions at few sampled molecular diffusivities. This is done by taking snapshots of solutions in time, and performing singular value decomposition of the matrix consisting of these

We study detecting a boundary corrosion damage in the inaccessible part of a rectangular shaped electrostatic conductor from a one set of Cauchy data specied on an accessible boundary part of conductor. For this nonlinear ill-posed problem, we prove the uniqueness in a very general framework. Then we establish the conditional stability of Holder type based on some a priori assumptions on the unknown impedance and the electrical current input specied in the accessible part. Finally a regularizing scheme of double regularizing parameters, using the truncation of the series expansion of the solution, is proposed with the convergence analysis on the explicit regularizing solution in terms of a practical average norm for measurement data.

In this paper we study the effects of small viscosity term and the far-field boundary conditions for systems of convection-diffusion equations in the zero viscosity limit. The far-field boundary conditions are classified and the corresponding solution structures are analyzed. It is confirmed that the Neumann type of far-field boundary condition is preferred. On the other hand, we also identify a class of improperly coupled boundary conditions which lead to catastrophic reflection waves dominating the inlet in the zero viscosity limit. The analysis is performed on the linearized convection-diffusion model which well describes the behavior at the far field for many physical and engineering systems such as fluid dynamical equations and electro-magnetic equations. The results obtained here should provide some theoretical guidance for designing effective far field boundary conditions.

We prove the existence of global solutions to a coupled system of NavierStokes, and reaction-diffusion equations (for temperature and mass fraction) with prescribed front data on an infinite vertical strip or tube. This system models a one-step exothermic chemical reaction. The heat release induced volume expansion is accounted for via the Boussinesq approximation. The solutions are time dependent moving fronts in the presence of fluid convection. In the general setting, the fronts are subject to intensive Rayleigh-Taylor and thermal-diffusive instabilities. Various physical quantities, such as fluid velocity, temperature, and front speed, can grow in time. We show that the growth is at most for large time <i>t</i> by constructing a nonlinear functional on the temperature and mass fraction components. These results hold for arbitrary order reactions in two space dimensions and for quadratic and cubic reactions in