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1. Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

   McEneaney, William M; Kaise, Hidehiro; Dower, Peter M; Zhao, Ruobing. (July 2020, 21st IFAC World Congress)

   Existence and uniqueness results for stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of a strong solution is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solution of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to the degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential and the resulting solutions to the dequantized Schrödinger equation. The first type consists of discontinuities that may occur on a possibly high-dimensional manifold (up to codimension one). The second consists of singularities that may occur on a lower-dimensional manifold (up to codimension two). 

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2. Weak solutions for a poro-elastic plate system

   https://doi.org/10.1080/00036811.2021.1953483  

   Gurvich, Elena; Webster, Justin T. (July 2021, Applicable Analysis)

   null (Ed.)

   We consider a recent plate model obtained as a scaled limit of the three-dimensional Biot system of poro-elasticity. The result is a ‘2.5’-dimensional linear system that couples traditional Euler–Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We allow the permeability function to be time dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so-called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions for implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability function. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate and exposits a variety of interesting related models and associated analytical investigations. 

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3. Twisted Self-Similarity and the Einstein Vacuum Equations

   Rothman-Slapentokh, Yakov (January 2023, Communications in mathematical physics)

   In the previous works [RSR19, SR22] we have introduced a new type of self-similarity for the Einstein vacuum equations characterized by the fact that the homothetic vector field may be spacelike on the past light cone of the singularity. In this work we give a systematic treatment of this new self-similarity. In particular, we provide geometric characterizations of spacetimes admitting the new symmetry and show the existence and uniqueness of formal expansions around the past null cone of the singularity which may be considered analogues of the well-known Fefferman–Graham expansions. In combination with results from [RSR19] our analysis will show that the twisted self-similar solutions are sufficiently general to describe all possible asymptotic behaviors for spacetimes in the small data regime which are selfsimilar and whose homothetic vector field is everywhere spacelike on an initial spacelike hypersurface. We present an application of this later fact to the understanding of the global structure of Fefferman–Graham spacetimes and the naked singularities of [RSR19, SR22]. Lastly, we observe that by an amalgamation of the techniques from [RSR18, RSR19], one may associate true solutions to the Einstein vacuum equations to each of our formal expansions in a suitable region of spacetime. 

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4. Uniqueness and stability of Ricci flow through singularities

   https://doi.org/DOI: https://dx.doi.org/10.4310/ACTA.2022.v228.n1.a1  

   Bamler, Richard; Kleiner, Bruce (July 2022, Acta mathematica)

   We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper. 

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5. Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential

   https://doi.org/10.1090/qam/1600  

   Hsieh, Dai-Ni; Arguillère, Sylvain; Charon, Nicolas; Younes, Laurent (March 2022, Quarterly of Applied Mathematics)

   This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D. 

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