More Like this

1. 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
2. Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations ^*

   https://doi.org/10.1088/1361-6544/ad7f18  

   Wu, Jiahong; Yamazaki, Kazuo (October 2024, Nonlinearity)

   Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations. 

   more » « less
3. Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4 (2018), no. 1, Art. 1, 119 pp.

   Esposito, Raffaele; Guo, Yan; Kim, Chanwoo; Marra, Rossana (July 2018, Annals of PDE)

   Abstract Despite its conceptual and practical importance, a rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition.We employ a recent quantitative L2–L∞ approach with new L6 estimates for the hydrodynamic part P f of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to an asymptotic stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system. 

   more » « less
4. Extensions of the constitutive relations to describe the response of compressible nonlinear Kelvin–Voigt solids

   https://doi.org/10.1007/s11012-025-01947-x  

   de_Castro_Motta, Julia; Rajagopal, Kumbakonan; Saccomandi, Giuseppe (February 2025, Meccanica)

   Abstract This paper investigates the propagation of longitudinal waves in some possible models of compressible Kelvin-Voigt viscoelastic solids. With respect to the elastic part two extensions of the classical neo-Hookean material model are proposed: the A-model, which incorporates a logarithmic volumetric function, and the B-model, based on the deviatoric invariants and a power-law volumetric function. For both models, we assume the same dissipative part given by the classical Navier–Stokes constitutive equation. These models are analyzed for their ability to describe a recovery phenomenon, ensuring conditions for monotonicity, boundedness, and uniqueness of solutions. The propagation of longitudinal traveling waves is proved. We show that the equation governing such motions is indeed a special case of the viscous p-system and a weakly nonlinear analysis demonstrates the emergence of Burgers’ equations. 

   more » « less
5. Inclined flow of a second-gradient incompressible fluid with pressure-dependent viscosity

   Balitactac, C; Rodriguez, C (June 2025, arxiv_25_d)

   Many viscous liquids behave effectively as incompressible under high pressures but display a pronounced dependence of viscosity on pressure. The classical incompressible Navier-Stokes model cannot account for both features, and a simple pressure-dependent modification introduces questions about the well-posedness of the resulting equations. This paper presents the first study of a second-gradient extension of the incompressible Navier-Stokes model, recently introduced by the authors, which includes higher-order spatial derivatives, pressure-sensitive viscosities, and complementary boundary conditions. Focusing on steady flow down an inclined plane, we adopt Barus' exponential law and impose weak adherence at the lower boundary and a prescribed ambient pressure at the free surface. Through numerical simulations, we examine how the flow profile varies with the angle of inclination, ambient pressure, viscosity sensitivity to pressure, and internal length scale. 

   more » « less