In this paper, we prove two existence results of solutions to mean field equations Δu+e^u=ρδ_0 and Δu=λe^u(e^u−1)+4π\sum_{j=1}^M δ_{p_j} on an arbitrary connected finite graph, where ρ>0 and λ>0 are constants, M is a positive integer, and p_1,…,p_M are arbitrarily chosen distinct vertices on the graph.

We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal Q-composition) for Q0<α<2−1(Rn≥2), and yet demonstrate that if f:R2→R2 is a homeomorphism then the boundedness of u↦u∘f on Q2−1<α<1(R2)⊂BMO(R2) yields the quasiconformality of f.

Abstract We prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak solution to the b-equation when the initial data is a nonnegative Radon measure with compact support.

In this paper, we study a class of analytical solutions to the 3-D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity h ( ) = 0 , and the bulk viscosity g ( ) = ( &gt; 0 ). By constructing a class of radial symmetric and self-similar analytical solutions in R N ( N 2 ) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity

In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend to many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.