The pressureless Euler-Navier-Stokes system can be obtained formally from the Vlasov-Navier-Stokes system, under the assumption that the distribution function describing the density of particles is monokinetic. Its study has been the subject of several recent papers, which have established the global existence of solutions with high enough regularity, for small initial data. In this work, we demonstrate the global existence of strong solutions in the whole space case, without assuming the initial density to be small and regular: it suffices for it to be bounded and for the total mass to be finite. In passing, we obtain optimal decay estimates for the energy and dissipation functionals. As a corollary, we get a long-time description of the density. All these results are based on an elementary energy method, with no need of sophisticated Fourier analysis tools.

We study the large-time behavior of finite-energy weak solutions for the Vlasov-Navier-Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable-density case. In both cases, large-time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han-Kwan et al. [17] dedicated to the homogeneous, three-dimensional case, where an additional smallness condition on the velocity was required. In the non-homogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.

This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space Rd with d >= 1. We address the question of well-posedness for large data having critical Besov regularity in the spirit of previous works by the second author on the compressible Navier-Stokes equations. Compared to the pioneering work of Kawashima in [14] and to the more recent paper by Serre in [18], we take advantage of the partial parabolicity of the system to consider data in functional spaces that need not be embedded in the set of Lipschitz functions. This is in sharp contrast with the classical well-posedness theory of (multi-dimensional) hyperbolic systems where it is mandatory. A leitmotiv of our analysis is to require less regularity for the components experiencing a direct diffusion, than for the hyperbolic components. We then use an energy method that is performed on the system after spectral localization and a suitable Ga & ring;rding inequality. As examples, we consider the Navier-Stokes-Fourier and Euler-Fourier systems.

We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier–Stokes system in the whole three-dimensional space. Our primary goal is to establish that small initial velocities with critical Sobolev regularity H^1/2 and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier–Stokes equations (NS) that has been proved by Fujita and Kato in [11]. Assuming also that the initial velocity is in L^1, we establish that the total energy E_0 of the system decays to 0 with the same rate t^-3/2 as for the weak solutions of (NS), see [22, 24]. Our results partly rely on the use of a higher order energy functional E_1 that controls the regularity H^1 of the velocity. This idea seems to originate from the recent paper [18] by Li, Shou and Zhang, devoted to the inhomogeneous Vlasov-Navier–Stokes system. Here we show that E_1 decays with the rate t^-5/2 which, in particular, allows us to prove that the density of the particles has a strong limit when the time goes to infinity.

We investigate the high viscosity limit (also called inertial limit) of the barotropic compressible Navier-Stokes equations supplemented with initial data which are perturbations of a stable constant solution. In the case of constant viscosity coefficients, we establish that, after diffusive rescaling, the density tends to satisfy a transport equation with nonlinear damping which is globally well-posed, even for large data. Similar results are proved for variable viscosity coefficients. In this latter case, the damping term in the limit equation of the density is nonlocal.

We are concerned with the barotropic compressible Navier-Stokes equations on the real line. Our primary goal is to establish the global well-posedness in a critical regularity framework in the case where the initial data are small perturbations of a stable constant state. Surprisingly, even though the result in the multi-dimensional case is by now classical, the one-dimensional case has not been elucidated yet as far as we know. This is due to the fact that in the critical framework, the regularity of the velocity is so negative that some nonlinear terms are out of control. Here, we overcome the difficulty by considering the equations in the mass Lagrangian coordinates system. Granted with a global well-posedness statement, we then establish optimal time decay estimates and investigate the high viscosity limit, pointing out the convergence of the specific volume to the solution of some ordinary differential equation, after time and space rescaling.

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space W^d,1(ℝ^d) for d = 2, 3. In two dimensions, we prove global propagation of the W^2,1(ℝ^2) regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate W^3,1(ℝ^3) regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces W^d/p,p(ℝ^d) for all 1 < p < ∞, which were based on axisymmetric flows without swirl when d = 3.

In this work, we explore the global existence of strong solutions for a class of partially diffusive hyperbolic systems within the framework of critical homogeneous Besov spaces. Our objective is twofold: first, to extend our recent findings on the local existence presented in J.-P. Adogbo and R. Danchin. Local well-posedness in the critical regularity setting for hyperbolic systems with partial diffusion. arXiv:2307.05981, 2024, and second, to refine and enhance the analysis of Kawashima (S. Kawashima. Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics. PhD thesis, Kyoto University, 1983). To address the distinct behaviors of low and high frequency regimes, we employ a hybrid Besov norm approach that incorporates different regularity exponents for each regime. This allows us to meticulously analyze the interactions between these regimes, which exhibit fundamentally different dynamics. A significant part of our methodology is based on the study of a Lyapunov functional, inspired by the work of Beauchard and Zuazua (K. Beauchard and E. Zuazua. Large time asymptotics for partially dissipative hyperbolic system. Arch. Rational Mech. Anal, 199:177-227, 2011.) and recent contributions (T. Crin-Barat and R. Danchin. Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case. J. Math. Pures Appl. (9), 165:1-41, 2022). To effectively handle the high-frequency components, we introduce a parabolic mode with better smoothing properties, which plays a central role in our analysis. Our results are particularly relevant for important physical systems, such as the magnetohydrodynamics (MHD) system and the Navier-Stokes-Fourier equations.

The present paper is devoted to the proof of time decay estimates for derivatives at any order of finite energy global solutions of the Navier-Stokes equations in general two-dimensional domains. These estimates only depend on the order of derivation and on the L2 norm of the initial data. The same elementary method just based on energy estimates and Ladyzhenskaya inequality also leads to Gevrey regularity results.

We consider the evolution of two-dimensional incompressible flows with variable density, only bounded and bounded away from zero. Assuming that the initial velocity belongs to a suitable critical subspace of L2, we prove a global-in-time existence and stability result for the initial (boundary) value problem. Our proof relies on new time decay estimates for finite energy weak solutions and on a "dynamic interpolation" argument. We show that the constructed solutions have a uniformly C1 flow, which ensures the propagation of geometrical structures in the fluid and guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent. By adopting this latter formulation, we establish the uniqueness of the solutions for prescribed data and the continuity of the flow map in an energy-like functional framework. In contrast with prior works, our results hold in the critical regularity setting without any smallness assumption. Our approach uses only elementary tools and applies indistinctly to the cases where the fluid domain is the whole plane, a smooth two-dimensional bounded domain, or the torus.

An L1-maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard (J. Math. Pures Appl. (9) 54 (1975), no. 3, 305-387) is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property which is of course decisive for free boundary problems is, firstly, illustrated by the analysis of the free boundary value problem describing the motion of viscous, incompressible Newtonian fluids without surface tension and, secondly, the motion of compressible, pressureless gases. To this end, an endpoint maximal L1-regularity approach to the Stokes and Lame systems is developed. It is applied then to establish global, strong wellposedness results for the free boundary problems described above in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm.

We consider the inhomogeneous incompressible Navier-Stokes system in a smooth two or three dimensional bounded domain, in the case where the initial density is only bounded. Existence and uniqueness for such initial data was shown recently in [10], but the stability issue was left open. After observing that the solutions constructed in [10] have exponential decay, a result of independent interest, we prove the stability with respect to initial data, first in Lagrangian coordinates, and then in the Eulerian frame. We actually obtain stability in $L_2({\mathbb R}_+;H^1(\Omega))$ for the velocity and in a negative Sobolev space for the density. Let us underline that, as opposed to prior works, in case of vacuum, our stability estimates are not weighted by the initial densities. Hence, our result applies in particular to the classical density patches problem, where the density is a characteristic function.

We consider the system governing the evolution of pressureless viscous gases in dimension two in the case where the initial density is just bounded and bounded away from zero. Assuming that the initial velocity is sufficiently small compared to the viscosity in the critical Lorentz space L2,1 (a large subspace of the natural energy space L2), we prove the global existence and uniqueness of a solution with Lipschitz flow. This improves our recent work (2021), which, in a different functional framework, established a global result under the assumption that the density variations are small. The main difficulty to get a global result lies in the fact that the density is just transported by the flow, with no diffusion, and does not decay to the reference density for large time. Our approach consists in proving time weighted energy estimates for the velocity (in the spirit of the work by Hoff (1995) on the compressible Navier-Stokes equations), then in taking advantage of a "dynamic" interpolation argument so as to establish that the gradient of the velocity field belongs to L1( +; L infinity). This latter property ensures the uniqueness of the solution, and the control of the lower and upper bounds of the density. To the best of our knowledge, this is the first global existence and uniqueness result for the system of pressureless gases with large density variations. The strategy is valid indistinctly in 2 or in smooth bounded domains of 2 and might be extendable to other models of nonhomogeneous viscous flows.

We are concerned with the isentropic compressible Navier–Stokes system in the two-dimensional torus, with rough data and vacuum; the initial velocity belongs to the Sobolev space H^1 and the initial density is only bounded and nonnegative. Arbitrary regions of vacuum are admissible, and no compatibility condition is required. Under these assumptions and for large enough bulk viscosity, global solutions have been constructed in Danchin and Mucha (Commun Pure Appl Math 76:3437–3492, 2023). The main goal of the paper is to establish that these solutions converge exponentially fast to a constant state, and to specify the convergence rate in terms of the viscosity coefficients. We prove similar exponential decay results for the solutions to the inhomogeneous incompressible Navier–Stokes equations, thereby extending to the torus the recent paper (Danchin et al. in Ann Anal Non Lin IHP, 2023) where bounded domains are considered.

We consider the evolution of two-dimensional incompressible flows with variable density, only bounded and bounded away from zero. Assuming that the initial velocity belongs to a suitable critical subspace of L^2 , we prove a global-in-time existence and stability result for the initial (boundary) value problem. Our proof relies on new time decay estimates for finite energy weak solutions and on a 'dynamic interpolation' argument. We show that the constructed solutions have a uniformly C^1 flow, which ensures the propagation of geometrical structures in the fluid and guarantees that the Eulerian and Lagrangian formulations of the equations are equivalent. By adopting this latter formulation, we establish the uniqueness of the solutions for prescribed data, and the continuity of the flow map in an energy-like functional framework. In contrast with prior works, our results hold true in the critical regularity setting without any smallness assumption. Our approach uses only elementary tools and applies indistinctly to the cases where the fluid domain is the whole plane, a smooth two-dimensional bounded domain or the torus.

We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by ε > 0 and formally tends to the classical pressure when ε approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter ε , which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates to use the L^1(ℝ_+; Ḃ^σ _2,1(ℝ^d)) spaces framework.

We are concerned with the isentropic compressible Navier-Stokes system in the two-dimensional torus, with rough data and vacuum : the initial velocity is in the Sobolev space H^1 and the initial density is only bounded and nonnegative. Arbitrary regions of vacuum are admissible, and no compatibility condition is required. Under these assumptions and for large enough bulk viscosity, global solutions have been constructed in [7]. The main goal of the paper is to establish that these solutions converge exponentially fast to a constant state, and to specify the convergence rate in terms of the viscosity coefficients. We also prove exponential decay estimates for the solutions to the inhomogeneous incompressible Navier-Stokes equations. This latter result extends to the torus case the recent paper [9] dedicated to this system in smooth bounded domains.

In this article, we prove the existence of global solutions to the inhomogeneous incompressible Navier--Stokes equations, whenever the initial velocity belongs to some subspace of $\mathrm{BMO}^{-1}$, and the initial density is sufficiently close to $1$ in the uniform metric. This is a natural extension to the variable density case of the celebrated result by H. Koch and D. Tataru concerning the classical Navier-Stokes equations.

We are concerned with the Cauchy problem for the two-dimensional compressible Navier-Stokes equations supplemented with general H-1 initial velocity and bounded initial density not necessarily strictly positive: it may be the characteristic function of any set, for instance. In the perfect gas case, we establish global-in-time existence and uniqueness, provided the volume (bulk) viscosity coefficient is large enough. For more general pressure laws (like e.g., P=& rho;& gamma;$P=\rho <^>\gamma$ with & gamma;>1$\gamma >1$), we still get global existence, but uniqueness remains an open question. As a by-product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier-Stokes equations when the bulk viscosity tends to infinity. In the three-dimensional case, similar results are proved for short time without restriction on the viscosity, and for large time if the initial velocity field is small enough.

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2[Formula: see text]D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron [Formula: see text] and as regards [Formula: see text]D flows, of the auxiliary vector-field [Formula: see text] that comes into play in the total magneto-helicity balance for the Hall-MHD system.