The aim of this paper is to present a function field analogue of the classical Kronecker limit formula. We first introduce a “non-holomorphic” Eisenstein series on the Drinfeld half plane, and connect its “second term” with Gekeler’s discriminant function. One application is to express the Taguchi height of rank 2 Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-type L-function associated to two “Drinfeld-type” newforms, we then derive a formula for a non-central special derivative of the L-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell’s equation over function fields.

For the physical vacuum free boundary problem with the sound speed being C^1/2-Holder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates, and elliptic estimates.

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We prove uniqueness of ground state solutions$Q$=$Q$(|$x$|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$ where 0 <$s$< 1 and 0 <$α$< 4$s$/(1−2$s$) for $${s<\frac{1}{2}}$$ and 0 <$α$<$∞$for $${s\geq \frac{1}{2}}$$ . Here (−Δ)^{$s$}denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $${s=\frac{1}{2}}$$ and$α$= 1 in [5] for the Benjamin–Ono equation.

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