Genetic variations in the COVID19 virus are one of the main causes of the COVID19 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
This paper studies a family of generalized surface quasigeostrophic (SQG) equations for an active scalar
 Award ID(s):
 1818754
 NSFPAR ID:
 10337310
 Date Published:
 Journal Name:
 Communications on Pure & Applied Analysis
 Volume:
 0
 Issue:
 0
 ISSN:
 15340392
 Page Range / eLocation ID:
 0
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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and removed\begin{document}$ S $\end{document} populations by ODEs and the infected\begin{document}$ R $\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for\begin{document}$ I $\end{document} and\begin{document}$ S $\end{document} contains terms that are related to the measure\begin{document}$ R $\end{document} . We establish analytically the wellposedness of the coupled ODEMDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODEMDE model coincides with the classical SIR model in case of constant or timedependent parameters as special cases.\begin{document}$ I $\end{document} 
We consider the wellknown LiebLiniger (LL) model for
bosons interacting pairwise on the line via the\begin{document}$ N $\end{document} potential in the meanfield scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the timedependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the onedimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [\begin{document}$ \delta $\end{document} 3 ] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65 ,66 ,67 ] and Knowles and Pickl [44 ]. To overcome difficulties stemming from the singularity of the potential, we introduce a new shortrange approximation argument that exploits the Hölder continuity of the\begin{document}$ \delta $\end{document} body wave function in a single particle variable. By further exploiting the\begin{document}$ N $\end{document} subcritical wellposedness theory for the 1D cubic NLS, we can prove meanfield convergence when the limiting solution to the NLS has finite mass, but only for a very special class of\begin{document}$ L^2 $\end{document} body initial states.\begin{document}$ N $\end{document} 
Abstract A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, HanKwan 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 of
N particles interacting in ,$${\mathbb {T}}^d$$ ${T}^{d}$ , via Newton’s second law through a$$d\ge 2$$ $d\ge 2$supercritical meanfield limit . Namely, the coupling constant in front of the pair potential, which is Coulombic, scales like$$\lambda $$ $\lambda $ for some$$N^{\theta }$$ ${N}^{\theta}$ , in contrast to the usual meanfield scaling$$\theta \in (0,1)$$ $\theta \in (0,1)$ . Assuming$$\lambda \sim N^{1}$$ $\lambda \sim {N}^{1}$ , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$\theta \in (1\frac{2}{d(d+1)},1)$$ $\theta \in (1\frac{2}{d(d+1)},1)$ . HanKwan and Iacobelli asked if their range for$$N\rightarrow \infty $$ $N\to \infty $ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$\theta $$ $\theta $ for$$N\rightarrow \infty $$ $N\to \infty $ . Our proof is based on Serfaty’s modulatedenergy method, but compared to that of HanKwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta \in (1\frac{2}{d},1)$$ $\theta \in (1\frac{2}{d},1)$ . Additionally, we show that for$$\theta $$ $\theta $ , one cannot, in general, expect convergence in the modulated energy notion of distance.$$\theta \le 1\frac{2}{d}$$ $\theta \le 1\frac{2}{d}$ 
We consider the linear third order (in time) PDE known as the SMGTJequation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control
. Optimal interior and boundary regularity results were given in [\begin{document}$ g $\end{document} 1 ], after [41 ], when , which, moreover, in the canonical case\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document} , were expressed by the wellknown explicit representation formulae of the wave equation in terms of cosine/sine operators [\begin{document}$ \gamma = 0 $\end{document} 19 ], [17 ], [24 ,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether or\begin{document}$ \gamma = 0 $\end{document} , since\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator basedexplicit representation formulae to provide optimal interior and boundary regularity results with\begin{document}$ \gamma \neq 0 $\end{document} "smoother" than\begin{document}$ g $\end{document} , qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [\begin{document}$ L^2(\Sigma) $\end{document} 17 ]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22 ], [23 ], [37 ] for control smoother than , and [\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document} 44 ] for control less regular in space than . In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [\begin{document}$ L^2(\Gamma) $\end{document} 42 ], [24 ,Section 9.8.2]. 
By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular MongeAmpère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singularlooking equations, such as
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