Analysis of PDEs (math.AP)

Abstract: In a recent series of important works \cite{wei-zhang-1,wei-zhang-2,wei-zhang-3}, Wei-Zhang proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.

Abstract: We consider the Cauchy problem for the fifth-order modified Korteweg-de Vries equation (mKdV) under the periodic boundary condition. The fifth-order mKdV is an asymptotic model for shallow surface waves, and (in the perspective of integrable systems) the second equation in the mKdV hierarchy as well. In strong contrast with the non-periodic case, periodic solutions for dispersive equations do not have a (local) smoothing effect, and this observation becomes a major obstacle to considering the Cauchy problem for dispersive equations under the periodic condition, consequently, the periodic fifth-order mKdV shows a quasilinear phenomenon, while the non-periodic case can be considered as a semilinear equation. In this paper, we mainly establish the global well-posedness of the fifth-order mKdV in the energy space ($H^2(\mathbb T)$), which is an improvement of the former result by the first author (2018). The main idea to overcome the lack of (local) smoothing effect is to introduce a suitable (frequency dependent) short-time space originally motivated by the work by Ionescu, Kenig, and Tataru (2008). The new idea is to combine the (frequency) localized modified energy with additional weight in the spaces, which eventually handles the logarithmic divergence appearing in the energy estimates. Moreover, by using examples localized in low and very high frequencies, we show that the flow map of the fifth-order mKdV equation is not $C^3$, which implies that the Picard iterative method is not available for the local theory. This weakly concludes the quasilinear phenomenon of the periodic fifth-order mKdV.

Abstract: We continue our study in \cite{FL} on viscosity solutions to a one-phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We first prove that viscosity solutions are locally Lipschitz continuous, which is the optimal regularity for the problem. Then we prove that Lipschitz free boundaries of viscosity solutions are $C^{1,\alpha}$. We also present some applications of our results. Moreover, we obtain new results for the operator under consideration that are of independent interest, such as a Harnack inequality.

Abstract: We consider the inverse problem of the simultaneous identification of the coefficients $\sigma$ and $q$ of the equation div$(\sigma\nabla u) + qu=0$ from the knowledge of the complete Cauchy data pairs. We assume that $\sigma=\gamma A$ where $A$ is a given matrix function and $\gamma, q$ are unknown piecewise affine scalar functions. No sign, nor spectrum condition on $q$ is assumed. We derive a result of global Lipschitz stability in dimension $n\geq 3$. The proof relies on the method of singular solutions and on the quantitative estimates of unique continuation.

Abstract: We prove quantitative growth estimates for large data solutions to the Boltzmann equation in one spatial dimension, for a collision kernel with angular cutoff. We prove in full the results presented in the note of Biryuk, Craig, and Panferov, and obtain improved bounds for large data. We show that solutions are unique, and propagate $W^{k,1}_\ell$ norms by a Beale-Kato-Majda criterion. We obtain quantitative uniform bounds on the density $\rho$ by $Ce^{C\sqrt{t}}$ for initial data with small relative entropy, and by $Ce^{Ct}$ for general initial data.

Abstract: We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $\Omega$ whose coefficients can be degenerate or singular of the type $\text{dist}(x, \partial \Omega)^\alpha$, where $\partial \Omega$ is the boundary of $\Omega$ and $\alpha \in (-1, \infty)$ is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.