Analysis of PDEs (math.AP)

Abstract: We introduce the spaces $A^p_{\alpha, \beta}(\Omega)$ of $L^p$-solutions to the Vekua equation (generalized monogenic functions) $D w=\alpha\overline{w}+\beta w$ in a bounded domain in $\mathbb{R}^n$, where $D=\sum_{i=1}^n e_i \partial_i$ is the Moisil-Teodorescu operator, $\alpha$ and $\beta$ are bounded functions on $\Omega$. The main result of this work consists of a Hodge decomposition of the $L^2$ solutions of the Vekua equation, from this orthogonal decomposition arises an operator associated with the Vekua operator, which in turn factorizes certain Schr\"odinger operators. Moreover, we provide an explicit expression of the ortho-projection over $A^p_{\alpha, \beta}(\Omega)$ in terms of the well-known ortho-projection of $L^2$ monogenic functions and an isomorphism operator. Finally, we prove the existence of component-wise reproductive Vekua kernels and the interrelationship with the Vekua projection in Bergman's sense.

Abstract: We build a smooth time-dependent real potential on the two-dimensional torus, decaying as time tends to infinity in Sobolev norms along with all its time derivative, and we exhibit a smooth solution to the associated Schr\"odinger equation on the two-dimensional torus whose $H^s$ norms nevertheless grow logarithmically as time tends to infinity. We use Fourier decomposition in order to exhibit a discrete resonant system of interactions, which we are further able to reduce to a sequence of finite-dimensional linear systems along which the energy propagates to higher and higher frequencies. The constructions are very explicit and we can thus obtain lower bounds on the growth rate of the solution.

Abstract: This paper presents a novel first-order mean-field game model that includes a prescribed incoming flow of agents in part of the boundary (Neumann boundary condition) and exit costs in the remaining portion (Dirichlet boundary condition). Our model is described by a system of a Hamilton-Jacobi equation and a stationary transport (Fokker-Planck) equation equipped with mixed and contact-set boundary conditions. We provide a rigorous variational formulation for the system, allowing us to prove the existence of solutions using variational techniques. Moreover, we establish the uniqueness of the gradient of the value function, which is a key result for the analysis of the model. In addition to the theoretical results, we present several examples that illustrate the presence of regions with vanishing density.

Abstract: In this paper, we introduce a new method to establish existence of geometric flows with surgery. In contrast to all prior constructions of flows with surgery in the literature our new approach does not require any a priori estimates in the smooth setting. Instead, our approach is based on a hybrid compactness theorem, which takes smooth limits near the surgery regions but weak limits in all other regions. For concreteness, here we develop our new method in the classical setting of mean-convex surfaces in $\mathbb{R}^3$, thus giving a new proof of the existence results due to Brendle-Huisken and Haslhofer-Kleiner. Other settings, including in particular free boundary surfaces, will be addressed in subsequent work.

Abstract: The main purpose of this article is the study of an inverse problem for nonlocal porous medium equations (NPMEs) with a linear absorption term. More concretely, we show that under certain assumptions on the time-independent coefficients $\rho,q$ and the time-independent kernel $K$ of the nonlocal operator $L_K$, the (partial) Dirichlet-to-Neumann map uniquely determines the three quantities $(\rho,K,q)$ in the nonlocal porous medium equation $\rho \partial_tu+L_K(u^m)+qu=0$, where $m>1$. In the first part of this work we adapt the Galerkin method to prove existence and uniqueness of nonnegative, bounded solutions to the homogenoeus NPME with regular initial and exterior conditions. Additionally, a comparison principle for solutions of the NPME is proved, whenever they can be approximated by sufficiently regular functions like the one constructed for the homogeneous NPME. These results are then used in the second part to prove the unique determination of the coefficients $(\rho,K,q)$ in the inverse problem. Finally, we show that the assumptions on the nonlocal operator $L_K$ in our main theorem are satisfied by the fractional conductivity operator $\mathcal{L}_{\gamma}$, whose kernel is $\gamma^{1/2}(x)\gamma^{1/2}(y)/|x-y|^{n+2s}$ up to a normalization constant.

Abstract: Generalizing results of \cite{MC,S} and \cite{HSZ} for certain model reaction-diffusion and reaction-convection-diffusion equations, we derive and rigorously justify weakly nonlinear amplitude equations governing general Turing bifurcation in the presence of conservation laws. In the nonconvective, reaction-diffusion case, this is seen similarly as in \cite{MC,S} to be a real Ginsburg-Landau equation coupled with a diffusion equation in a large-scale mean-mode vector comprising variables associated with conservation laws. In the general, convective case, by contrast, the amplitude equations as noted in \cite{HSZ} consist of a complex Ginsburg-Landau equation coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed $\sim 1/\eps$ where $\eps$ measures amplitude of the wave as a disturbance from a background steady state. Different from the partially coupled case considered in \cite{HSZ} in the context of B\'enard-Marangoni convection/inclined flow, the Ginzburg Landau and mean-mode equations are here fully coupled, leading to substantial new difficulties in the analysis. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models