skip to main content

Attention:

The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 5:00 PM ET until 11:00 PM ET on Friday, June 21 due to maintenance. We apologize for the inconvenience.


Title: On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$ \theta $\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$ u $\end{document} is of lower singularity, i.e., \begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}, where \begin{document}$ p $\end{document} is a logarithmic smoothing operator and \begin{document}$ \beta \in [0, 1] $\end{document}. We complete this study by considering the more singular regime \begin{document}$ \beta\in(1, 2) $\end{document}. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.

 
more » « less
Award ID(s):
1818754
NSF-PAR ID:
10337310
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Communications on Pure & Applied Analysis
Volume:
0
Issue:
0
ISSN:
1534-0392
Page Range / eLocation ID:
0
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$ S $\end{document} and removed \begin{document}$ R $\end{document} populations by ODEs and the infected \begin{document}$ I $\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$ S $\end{document} and \begin{document}$ R $\end{document} contains terms that are related to the measure \begin{document}$ I $\end{document}. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.

     
    more » « less
  2. We consider the well-known Lieb-Liniger (LL) model for \begin{document}$ N $\end{document} bosons interacting pairwise on the line via the \begin{document}$ \delta $\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$ \delta $\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$ N $\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$ L^2 $\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of \begin{document}$ N $\end{document}-body initial states.

     
    more » « less
  3. Abstract

    A long-standing problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system ofNparticles interacting in$${\mathbb {T}}^d$$Td,$$d\ge 2$$d2, via Newton’s second law through asupercritical mean-field limit. Namely, the coupling constant$$\lambda $$λin front of the pair potential, which is Coulombic, scales like$$N^{-\theta }$$N-θfor some$$\theta \in (0,1)$$θ(0,1), in contrast to the usual mean-field scaling$$\lambda \sim N^{-1}$$λN-1. Assuming$$\theta \in (1-\frac{2}{d(d+1)},1)$$θ(1-2d(d+1),1), they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$N\rightarrow \infty $$N. Han-Kwan and Iacobelli asked if their range for$$\theta $$θwas optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$N\rightarrow \infty $$Nfor$$\theta \in (1-\frac{2}{d},1)$$θ(1-2d,1). Our proof is based on Serfaty’s modulated-energy method, but compared to that of Han-Kwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta $$θ. Additionally, we show that for$$\theta \le 1-\frac{2}{d}$$θ1-2d, one cannot, in general, expect convergence in the modulated energy notion of distance.

     
    more » « less
  4. We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].

     
    more » « less
  5. By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as \begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document} with zero boundary data, have unexpected degenerate nature.

     
    more » « less