We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.

The zero dissipation limit of the compressible heat-conducting NavierStokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient and the viscosity coefficient satisfy = O (), = O (), c> 0, as 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the NavierStokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of . The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].

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).

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their$L$_{2}Whitney averages belongs to$L$_{2}on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to$L$_{2}on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to$L$_{$p$}generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.

The well-posedness of the three space dimensional Prandtl equations is studied under some constraint on its flow structure. Together with the instability result given in [28], it gives an almost necessary and sufficient structural condition for the stability of the three-dimensional Prandtl equations. It reveals that the classical Burgers equation plays an important role in determining this type of flow with special structure, that avoids the appearance of the secondary flow, an unstabilizing factor in the three-dimensional Prandtl boundary layers. And the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the Prandtl boundary layer equations is illustrated in the three-dimensional setting. Moreover, it is shown that this structured flow is linearly stable for any smooth three-dimensional perturbation.