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1. 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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2. A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data

   https://doi.org/10.1016/j.amc.2025.129286  

   Abney, Ray; Le, Thuy T; Nguyen, Loc H; Peters, Cam (June 2025, Applied Mathematics and Computation)

   We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the ``reduced dimensional method.'' Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples. 

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3. On Self-Similar Converging Shock Waves

   https://doi.org/10.1007/s00205-025-02096-x  

   Jang, Juhi; Liu, Jiaqi; Schrecker, Matthew (April 2025, Archive for Rational Mechanics and Analysis)

   Abstract In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for$$\gamma \in (1,3]$$ $\gamma \in (1,3]$. These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions. 

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4. Fluid-poroviscoelastic structure interaction problem with nonlinear coupling

   Jeffrey Kuan, Suncica Canic (December 2024, Jorunal de Mathematiques Pures and Appliques)

   We prove the existence of a weak solution to a fluid-structure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot equations. The two are nonlinearly coupled over an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. The existence proof is constructive, consisting of two steps. First, the existence of a weak solution to a regularized problem is shown. Next, a weak-classical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media. 

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5. Well-Posedness of Solutions to Stochastic Fluid–Structure Interaction

   https://doi.org/10.1007/s00021-023-00839-y  

   Kuan, Jeffrey; Čanić, Sunčica (February 2024, Journal of Mathematical Fluid Mechanics)

   In this paper we introduce a constructive approach to study well-posedness of solutions to stochastic uid-structure interaction with stochastic noise. We focus on a benchmark problem in stochastic uidstructure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D time-dependent Stokes equations describing the ow of an incompressible, viscous uid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the time-dependent white noise. The uid and the structure are linearly coupled. The constructive existence proof is based on a time-discretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the Gyongy-Krylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the  rst well-posedness result for stochastic uid-structure interaction. 

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