Analysis of PDEs (math.AP)

Abstract: We consider the two dimensional pure gravity water waves with nonzero constant vorticity in infinite depth, working in the holomorphic coordinates introduced by Hunter, Ifrim, and Tataru. We show that close to the critical velocity corresponding to zero frequency, a solitary wave exists. We use a fixed point argument to construct the solitary wave whose profile resembles a rescaled Benjamin-Ono soliton. The solitary wave is smooth and has an asymptotic expansion in terms of powers of the Benjamin-Ono soliton.

Abstract: We use a convex integration construction from \cite{ModenaSattig2020} in a Baire category argument to show that weak solutions to the transport equation with incompressible vector fields with Sobolev regularity are generic in the Baire category sense. Using the construction of \cite{BurczakModenaSzekelyhidi20} we prove an analog statement for the 3D Navier-Stokes equations.

Abstract: We address a physically-meaningful extension of the Prandtl system, also known as hyperbolic Prandtl equations. We show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order k^(1/3) in the frequencies of the tangential direction, akin the pioneering result of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion was of order k^(1/2)). We emphasise however that this growth rate does not imply ill-posedness in Gevrey-class m, with m > 3 and we relate also these aspects to the original Prandtl equations in Gevrey-class m, with m > 2.

Abstract: In this article we study convex non-autonomous variational problems with differential forms and corresponding function spaces. We introduce a general framework for constructing counterexamples to the Lavrentiev gap, which we apply to several models, including the double phase, borderline case of double phase potential, and variable exponent. The results for the borderline case of double phase potential provide new insights even for the scalar case, i.e., variational problems with $0$-forms.

Abstract: We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the $\Gamma$ convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the incompressible PKS model (which includes a density constraint $\rho\leq 1$ and a pressure term): It shows that the incompressibility constraint is not necessary to observe phase separation and surface tension phenomena.

Abstract: These Notes are intended for graduate or undergraduate students who have familiarity with Lebesgue measure theory, partial differential equations, and functional analysis. The main topics covered in this work are the study of the Cauchy problem and unique continuation properties associated with partial differential equations. The primary objective is to familiarize students with stability estimates in inverse problems and quantitative estimates of unique continuation. The treatment is presented in a self-contained manner.

Abstract: We consider initial boundary value problems with the homogeneous Neumann boundary condition. Given an initial value, we establish the uniqueness in determining a spatially varying coefficient of zeroth-order term by a single measurement of Dirichlet data on an arbitrarily chosen subboundary. The uniqueness holds in a subdomain where the initial value is positive, provided that it is sufficiently smooth which is specified by decay rates of the Fourier coefficients. The key idea is the reduction to an inverse elliptic problem and relies on elliptic Carleman estimates.