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1. 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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2. Compressible Navier–Stokes equations with heterogeneous pressure laws

   https://doi.org/10.1088/1361-6544/ac03a1  

   Bresch, Didier; Jabin, Pierre-Emmanuel; Wang, Fei (June 2021, Nonlinearity)

   Abstract This paper concerns the existence of global weak solutions à la Leray for compressible Navier–Stokes equations with a pressure law which depends on the density and on time and space variables t and x . The assumptions on the pressure contain only locally Lipschitz assumption with respect to the density variable and some hypothesis with respect to the extra time and space variables. It may be seen as a first step to consider heat-conducting Navier–Stokes equations with physical laws such as the truncated virial assumption. The paper focuses on the construction of approximate solutions through a new regularized and fixed point procedure and on the weak stability process taking advantage of the new method introduced by the two first authors with a careful study of an appropriate regularized quantity linked to the pressure. 

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3. Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

   https://doi.org/10.1111/sapm.12426  

   Congy, T.; El, G. A.; Hoefer, M. A.; Shearer, M. (August 2021, Studies in Applied Mathematics)

   Abstract Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equationare studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation. The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit asis a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem. 

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4. Numerical Methods for Integral Equations of the Second Kind with NonSmooth Solutions of Bounded Variation

   https://doi.org/10.1137/22M1480422  

   Li, Shukai; Mehrotra, Sanjay (October 2022, SIAM Journal on Numerical Analysis)

   This paper develops a finite approximation approach to find a non-smooth solution of an integral equation of the second kind. The equation solutions with non-smooth kernel having a non-smooth solution have never been studied before. Such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available. 

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5. Regularity of weak solution of variational problems modeling the Cosserat micropolar elasticity.

   https://doi.org/10.1093/imrn/rnaa202  

   Li, Yimei; Wang, Changyou (January 2022, International mathematics research notices)

   In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $$p$$-harmonic maps (⁠#2\le p\le 3$$⁠). We show that if a weak solution is stationary, then its singular set is discrete for $2<3$ and has zero one-dimensional Hausdorff measure for $p=2$⁠. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $$p\in [2, 32/15]$$. 

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