More Like this

1. Uniqueness in a Navier–Stokes-nonlinear-Schrödinger model of superfluidity*

   https://doi.org/10.1088/1361-6544/ac72e5  

   Jayanti, Pranava Chaitanya; Trivisa, Konstantina (June 2022, Nonlinearity)

   Abstract In Jayanti and Trivisa (2022 J. Math. Fluid Mech. 24 46), the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived in Pitaevskii (1959 Sov. Phys. JETP 8 282–287) and couples the nonlinear Schrödinger equation and the Navier–Stokes equations. In this article, we prove a weak–strong type uniqueness theorem for these weak solutions. Only some of their regularity properties are used, allowing room for improved existence theorems in the future, with compatible uniqueness results. 

   more » « less
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. 

   more » « less
3. Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows

   https://doi.org/10.1142/S0218202520500475  

   Liang, Zhilei; Wang, Dehua (November 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent. 

   more » « less
4. A direct forcing, immersed boundary method for conjugate heat transport

   https://doi.org/10.1016/j.jcp.2025.114135  

   Koponen, Kimmo; Pal_Singh_Bhalla, Amneet; Sprinkle, Brennan; Wu, Ning; Tilton, Nils (October 2025, Journal of Computational Physics)

   Motivated by applications to fluid flows with conjugate heat transfer and electrokinetic effects, we propose a direct forcing immersed boundary method for simulating general, discontinuous, Dirichlet and Robin conditions at the interface between two materials. In comparison to existing methods, our approach uses smaller stencils and accommodates complex geometries with sharp corners. The method is built on the concept of a “forcing pair,” defined as two grid points that are adjacent to each other, but on opposite sides of an interface. For 2D problems this approach can simultaneously enforce discontinuous Dirichlet and Robin conditions using a six-point stencil at one of the forcing points, and a 12-point stencil at the other. In comparison, prior work requires up to 14-point stencils at both points. We also propose two methods of accommodating surfaces with sharp corners. The first locally reduces stencils in sharp corners. The second uses the signed distance function to globally smooth all corners on a surface. The smoothing is defined to recover the actual corners as the grid is refined. We verify second-order spatial accuracy of our proposed methods by comparing to manufactured solutions to the Poisson equation with challenging dis- continuous fields across immersed surfaces. Next, to explore the performance of our method for simulating fluid flows with conjugate heat transport, we couple our method to the incompressible Navier–Stokes and continuity equations using a finite-volume projection method. We verify the spatial-temporal accuracy of the solver using manufactured solutions and an analytical solution for circular Couette flow with conjugate heat transfer. Finally, to demonstrate that our method can model moving surfaces, we simulate fluid flow and conjugate heat transport between a stationary cylinder and a rotating ellipse or square. 

   more » « less
5. Global regularity for critical SQG in bounded domains

   https://doi.org/10.1002/cpa.22221  

   Constantin, Peter; Ignatova, Mihaela; Nguyen, Quoc‐Hung (January 2025, Communications on Pure and Applied Mathematics)

   Abstract We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system. 

   more » « less