Search

This article investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a rigid body. It is shown that this problem is locally strongly well-posed, and that it also admits a unique, global strong solution for initial data close to constant equilibria. The proof of the global strong solution relies on a new splitting method for the director in a mean value zero and average part. [ABSTRACT FROM AUTHOR]

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic W 2 , p -regularity is imposed as an assumption. Also, we give a finite time blow-up criterion. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the Wong–Zakai approximation results for two dimensional stochastic Cahn–Hilliard–Navier–Stokes model. The model consists of a Navier–Stokes system coupled with convective Cahn–Hilliard equations. It describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids under the influence of multiplicative noise. Our main result describes the support of the distribution of solutions. As in [2], both inclusions are proved by means of a general Wong–Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert‐valued Brownian motion and some adapted finite‐dimensional approximation of this process. Note that the coupling between the Navier–Stokes system and the Cahn–Hilliard equations makes the analysis more involved. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to establish the nonlocal-to-local convergence of strong solutions to a Navier–Stokes–Cahn–Hilliard model with singular potential describing immiscible, viscous two-phase flows with matched densities, which is referred to as the Model H. This means that we show that the strong solutions to the nonlocal Model H converge to the strong solution to the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. Compared to previous results in the literature, our main novelty is to further establish corresponding convergence rates. Before investigating the nonlocal-to-local convergence, we first need to ensure the strong well-posedness of the nonlocal Model H. In two dimensions, this result can already be found in the literature, whereas in three dimensions, it will be shown in the present paper. Moreover, in both two and three dimensions, we establish suitable uniform bounds on the strong solutions of the nonlocal Model H, which are essential to prove the nonlocal-to-local convergence results. [ABSTRACT FROM AUTHOR]

In this study, we establish a regular criterion for the 2D compressible micropolar viscous fluids with vacuum that is independent of the velocity of rotation of the microscopic particles. Specifically, we show that if the density verifies ||ρ||L∞(0,T;L∞) < ∞, then the strong solution will exist globally on R² x(0, T). Consequently, we generalize the results of Zhong (Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), no. 12, 4603–4615) to the compressible case. In particular, we don’t need the additional compatibility condition. [ABSTRACT FROM AUTHOR]

We study a free boundary value problem modelling the motion of a piston in a viscous compressible fluid. The fluid is modelled by 1D compressible Navier–Stokes equations with possibly degenerate viscosity coefficient, and the motion of the piston is described by Newton’s second law. We show that the initial boundary value problem has a unique global in time solution, and we also determine the large time behaviour of the system. Finally, we show how our methodology may be adapted to the motion of several pistons. [ABSTRACT FROM AUTHOR]

We consider a class of bulk-surface reaction-adsorption-diffusion systems, i.e. a coupled system of reaction-diffusion systems on a bounded domain Ω ⊆ R d (bulk phase) and its boundary Σ = ∂ Ω (surface phase), which are coupled via nonlinear normal flux boundary conditions. In particular, this class includes a heterogeneous catalysis model with Fickian bulk and surface diffusion and nonlinear adsorption of Langmuir type, i.e. transport from the bulk phase to the active surface, and desorption. For this model, we obtain well-posedness, positivity and global-in-time existence of solutions under some realistic structural conditions on the chemical reaction network and the sorption model. We work in appropriate Sobolev-Slobodetskii settings, where we aim for a wide range for the integrability index, including in particular values p < d. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the maximum norm of velocity divergence controls the breakdown of smooth (strong) solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier–Stokes equations with zero heat conduction. The results indicate that the nature of the blowup for the full compressible Navier–Stokes equations with zero heat conduction of viscous flow is similar to the barotropic compressible Navier–Stokes equations and does not depend on the temperature field. The main ingredient of the proof is a priori estimate to the pressure field instead of the temperature field and weighted energy estimates under the assumption that velocity divergence remains bounded. Furthermore, the initial vacuum states are allowed, and the viscosity coefficients are only restricted by the physical conditions. [ABSTRACT FROM AUTHOR]

We study the dynamics of a coupled system, formed by a rigid body with a cavity entirely filled with magnetohydrodynamic compressible fluid. Our aim is to derive the global existence of the unique classical solutions and weak solutions to this system. Moreover, we show the weak-strong uniqueness principle which means that a weak solution coincides with a strong solution on the time existence of a strong solution, provided they emanate from the same initial data. [ABSTRACT FROM AUTHOR]

In this paper we study an optimal control problem associated to a parabolic–elliptic chemo-repulsion system with a linear production term in a two-dimensional domain. Under the injection/extract chemical substance on a control subdomain Ω c , we prove the existence and uniqueness of global-in-time strong solutions. Afterwards, for the optimal control problem, we prove the existence of at least one global optimal solution, and derive an optimality system via using a Lagrange multipliers theorem. [ABSTRACT FROM AUTHOR]

We study the symmetric stochastic p-Stokes system, p ∈ (1 , ∞) , in a bounded domain. The results are two-fold: First, we show that in the context of analytically weak solutions, the stochastic pressure—related to non-divergence free stochastic forces—enjoys almost - 1 / 2 temporal derivatives on a Besov scale. Second, we verify that the velocity u of strong solutions obeys 1/2 temporal derivatives in an exponential Nikolskii space. Moreover, we prove that the non-linear symmetric gradient V (ϵ u) = (κ + ϵ u) (p - 2) / 2 ϵ u , κ ≥ 0 , which measures the ellipticity of the p-Stokes system, has 1/2 temporal derivatives in a Nikolskii space. [ABSTRACT FROM AUTHOR]

We analyze a diffuse interface model for multi-phase flows of N incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space dimensions and a singular free energy density is shown. Moreover, in two space dimensions global existence for sufficiently regular initial data is proven. In three space dimension, existence of strong solutions locally in time is shown as well as regularization for large times in the absence of exterior forces. Moreover, in both two and three dimensions, convergence to stationary solutions as time tends to infinity is proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider the strong solutions under three different types of boundary problems to a compressible nematic liquid crystals system in three dimensions. We establish the blow‐up criterion in terms of the density and the gradient of direction field, provided the shear viscosity and bulk viscosity coefficients of the fluid satisfy an additional relation (i.e., μ>λ$$ \mu >\lambda $$). In particular, the initial vacuum is allowed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the first boundary value problem for a class of steady non-Newtonian micropolar fluid equations with heat convection in the three-dimensional smooth and bounded domain $ \Omega $. By using the fixed-point theorem and introducing a family of penalized problems, under the condition that the external force term and the vortex viscosity coefficient are appropriately small, the existence and uniqueness of strong solutions of the problem are obtained.

In this paper, we consider the first boundary value problem for a class of steady non-Newtonian micropolar fluid equations with heat convection in the three-dimensional smooth and bounded domain Ω. By using the fixed-point theorem and introducing a family of penalized problems, under the condition that the external force term and the vortex viscosity coefficient are appropriately small, the existence and uniqueness of strong solutions of the problem are obtained. [ABSTRACT FROM AUTHOR]

Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro-macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry. [ABSTRACT FROM AUTHOR]

In this paper, we concern the Cauchy problem of two‐dimensional (2D) compressible nematic liquid crystal flows with vacuum as far‐field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem. [ABSTRACT FROM AUTHOR]

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

In general, the study of inverse problems is realizable only in the case when the corresponding direct problems have the unique solution with some necessary properties such as continuity and regularity. In this paper, we study initial-boundary value problems for the system of 2D-3D nonlinear Kelvin-Voigt equations with memory, which describes a motion of an incompressible homogeneous non-Newtonian fluids with viscoelastic and relaxation properties. The investigation of these direct problems is related to the study of inverse problems for this system, which requires the continuity and regularity of solutions to these direct problems and their derivatives. The system, in addition to the initial condition, is supplemented with one of the boundary conditions: stick and slip boundary conditions. In both cases of these boundary conditions, the global in time existence and uniqueness of strong solutions to these initial-boundary value problems were proved. Moreover, under suitable assumptions on the data, the regularity of solutions and their derivatives were established.

In this paper, we study a free-boundary problem of the liquid-gas two-phase drift-flux model with constant viscosity, where the initial liquid and gas masses are assumed to be connected with vacuum continuously, and the pressure function takes a singular form. We utilize the flow map to reformulate the problem and prove the global existence and decay rates of strong solutions with general initial data that involve transition to pure single-phase points or regions. In particular, both the optimal decay rates of the mass functions and the decay estimates of the velocity towards its asymptotic profile as time goes to infinity are studied. As far as we are concerned, this is the first result on the global existence of strong solutions to the two-phase drift-flux model with singular pressure function and with general initial data that allow for transition to single-phase flow. It is also worth noting that our method in recovering the non-weighted estimates of the solution by utilizing the Poincaré type inequality can apply to the single-phase problem studied in Zeng (2015) [37] , so as to derive the decay estimates for the velocity and relax the constraints on the initial data. [ABSTRACT FROM AUTHOR]

We consider a stochastic 2D liquid crystal model with a multiplicative noise of Lévy type, which models the dynamic of nematic liquid crystals under the influence of stochastic external forces of jump type. We derive a large deviation principle for the model. The proof is based on the weak convergence method introduced in [Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincar Probab Stat. 2011;47(3):725–747]. [ABSTRACT FROM AUTHOR]

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L 4 (Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t → ∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities. [ABSTRACT FROM AUTHOR]

Due to the high degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behaves singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It is shown in this paper that the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities of the velocity and temperature can be propagated for viscous and heat conductive gases in ℝ3, provided that the initial vacuum occurs only at far fields with suitably slow decay of the initial density. Precisely, it is proved that for any strong solution to the Cauchy problem of the heat conductive compressible Navier-Stokes equations, the corresponding entropy keeps uniformly bounded and the L2 regularities of the velocity and temperature can be propagated, up to the existing time of the solution, as long as the initial density vanishes only at far fields with a rate not faster than O (1 | x | 2 ) . The main tools are some singularly weighted energy estimates and an elaborate De Giorgi type iteration technique. We apply the De Giorgi type iterations to different equations in establishing the lower and upper bounds of the entropy. [ABSTRACT FROM AUTHOR]

In this paper, we show the important roles of sharp minima and strong minima for robust recovery. We also obtain several characterizations of sharp minima for convex regularized optimization problems. Our characterizations are quantitative and verifiable especially for the case of decomposable norm regularized problems including sparsity, group-sparsity and low-rank convex problems. For group-sparsity optimization problems, we show that a unique solution is a strong solution and obtains quantitative characterizations for solution uniqueness. [ABSTRACT FROM AUTHOR]

In this paper we study a stationary chemo-repulsion model and analyze a related optimal bilinear control problem. We prove the existence of strong solutions of the state equations with a non-smooth source term in the chemical concentration equation, in bounded domains of R N , N = 1 , 2 , 3 , for any given mass, which permit us to consider optimal bilinear control problems. We prove the existence of optimal solutions and, by using a Lagrange multipliers theorem in Banach spaces, we obtain some first-order optimality conditions. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of local strong solution for a perfect compressible viscous fluid, heat conductive and self-gravitating, coupled with a first-order kinetics used in astrophysical hydrodynamical models. In our setting, the vacuum is allowed and as a byproduct of the existence result we get a blow-up criterion for the local strong solution. Moreover we prove a blow-up criterion for the local strong solutions in terms of the velocity gradient, the mass fraction gradient and the temperature similar to the well-known Beale–Kato–Majda criterion for ideal incompressible flows. [ABSTRACT FROM AUTHOR]

In this article we investigate the temporal regularity of strong solutions to the stochastic p-Laplace system in the degenerate setting, p ∊ [2, ∞), driven by a multiplicative nonlinear stochastic forcing. We establish 1/2 time differentiability in an exponential Besov--Orlicz space for the solution process u. Furthermore, we prove 1/2 time differentiability of the nonlinear gradient V (∇ u) in a Nikolskii space. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space χ−1(ℝ2) with any initial data in χ−1(ℝ2) ∩ L2(ℝ2) is established. Furthermore, the uniqueness of the strong solution in χ−1(ℝ2) and the Leray-Hopf weak solution in L2(ℝ2) is proved. [ABSTRACT FROM AUTHOR]

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In our previous work [38] , we investigate the global existence of strong solutions to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity. However, it is unknown whether the entropy remains its boundedness or not? Thus, the main novelty of this paper is to give a positive response to this problem. In fact, we show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. Modified De Giorgi type iteration techniques and useful singularly weighted estimates are developed to deduce the lower and upper bounds on the entropy. [ABSTRACT FROM AUTHOR]

We analyze a three-dimensional rapidly rotating convection model of tall columnar structure in the limit of infinite Prandtl number, i.e., when the momentum diffusivity is much more dominant than the thermal diffusivity. Consequently, the dynamics of the velocity field takes place at a much faster time scale than the temperature fluctuation, and at the limit the velocity field formally adjusts instantaneously to the thermal fluctuation. We prove the global well-posedness of weak solutions and strong solutions to this model.

We use asymptotic analysis to study the existence of solutions of a one-dimensional nonlinear system of ordinary differential equations with different powers of a small parameter at higher derivatives. A specific feature of the problem is the presence of a discontinuity of the first kind in the right-hand side of the equation in the unknown variable at the level , while the right-hand side of the second equation is assumed to be smooth in all variables. We define a generalized solution of the problem is in terms of differential inclusions. Conditions under which generalized solutions turn into strong ones are proposed, and the possibility that the -component of the solution intersects zero only once is studied. The existence theorems are proved by using the asymptotic method of differential inequalities. [ABSTRACT FROM AUTHOR]

We examine the long-term behavior of strong solutions for the wave equation $ \varepsilon(t)u_{tt}-\Delta u_{t}-\Delta u+f(u) = g(x) $ and obtain the time-dependent attractor in $ \mathcal{H}_{t}^{1} = [H^{2}(\Omega)\cap H_{0}^{1}(\Omega)]\times H_{0}^{1}(\Omega) $. Moreover, the attractor is bounded in $ [H^{2}(\Omega)\cap H_{0}^{1}(\Omega)]\times[H^{2}(\Omega)\cap H_{0}^{1}(\Omega)] $. [ABSTRACT FROM AUTHOR]

In recent paper, we study the compressible biaxial nematic liquid crystal flow in a simply connected and smooth bounded domain Ω⊂ℝ3$$ \Omega \subset {\mathbb{R}}^3 $$. An initial vacuum may exist in this flow. We firstly obtain a local existence of a unique strong solution by Galerkin's approximation method. Finally, we get a blow up criterion for such a local strong solution in terms of blow up of ρLt∞BMOx$$ {\left\Vert \rho \right\Vert}_{L_t^{\infty } BM{O}_x} $$ and ∇nLtrLx∞+∇mLtrLx∞$$ {\left\Vert \nabla n\right\Vert}_{L_t^r{L}_x^{\infty }}+{\left\Vert \nabla m\right\Vert}_{L_t^r{L}_x^{\infty }} $$ for any r>3$$ r>3 $$. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two dimensional PDE model describing the proliferation-invasion structure of hypoxic glial cells in the brain, in which the cell movement is determined not only by natural diffusion but also, by the gradient of the concentration of oxygen. This model is given by a second order nonlinear system involving the density of cancer cells and the oxygen concentration, which are coupled by a chemoattraction term, a source of logistic growth, and a reaction term of Michaelis-Menten type. The contribution of this paper is summarized in the following two main aspects: first, we prove the existence and uniqueness of strong solutions; and second, we propose a nonlinear fully discrete finite element (FE) approximation for the model. We prove its well-posedness, some uniform estimates, the positivity for the discrete oxygen concentration and approximated positivity for the discrete tumour cells, which are required in this biological model. The key point to develop the numerical analysis is to control properly the second order nonlinear chemotaxis term as well as to obtain a discrete energy estimate which, in particular, gives a bounded energy; this energy estimate is derived by using a regularization technique. Finally, we present some numerical simulations which allow us to validate the theoretical results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the continuation criteria to the 3D isentropic compressible Navier–Stokes equations in Lorentz spaces. We prove that there exists a positive constant ε$$ \varepsilon $$ such that no blowup occurs at time T$$ T $$ in this system provided that the supernorm of the density is bounded and the space‐time Lorentz spaces norm ‖ρu‖Lp,∞(0,T;Lq,∞)$$ {\left\Vert \sqrt{\rho }u\right\Vert}_{L&amp;#x0005E;{p,\infty}\left(0,T;{L}&amp;#x0005E;{q,\infty}\right)} $$ with 2/p+3/q=1$$ 2/p&amp;#x0002B;3/q&amp;#x0003D;1 $$(3≤q<∞)$$ \left(3\le q&lt;\infty \right) $$ is small. As a direct application, we established some Serrin's blow‐up criteria in the Lorentz spaces. [ABSTRACT FROM AUTHOR]

We prove an existence result for strong solutions $u\in W^{2,q}\left(\Omega\right) $ of singular semilinear elliptic problems of the form $-\Delta u=g\left( \cdot,u\right) $ in $\Omega,$ $u=\tau$ on $\partial\Omega,$ where $1

In this paper, we consider the conditional regularity for the 3D incompressible Navier-Stokes equations in Vishik spaces. These results will be regarded an improvement of the results given by Huang-Li-Xin, (SIAM J. Math. Anal., 2011) and Jiu-Wang-Ye, (J. Evol. Equ., 2021).

In this paper, we study the compressible barotropic Navier-Stokes equations in a general two-dimensional bounded domain with the Navier-slip boundary conditions. It is shown that the strong solution exists globally if the density satisfies ‖ ρ | | L ∞ (0 , t ; L q) < ∞ for some q < ∞ , which extends previous one ‖ ρ | | L ∞ (0 , t ; L ∞) < ∞ . The initial vacuum is permitted in this paper. [ABSTRACT FROM AUTHOR]

In this work, we study the very weak solution for the stationary exterior Stokes problem with Navier slip boundary condition. We try to investigate some results of the existence and the uniqueness of solutions related to this problem in an exterior domain in Lp$$ {L}^p $$‐theory where 1

In this paper we study a 3D non-stationary Boussinesq system with Navier-slip boundary conditions for the velocity field and Neumann boundary conditions for the temperature. Considering the viscosity and diffusion coefficient as explicit functions depending on the temperature, we prove the existence of weak solutions and present a regularity result that allow us obtain global-in-time strong solutions. [ABSTRACT FROM AUTHOR]

In this paper, we first establish well-posedness results for one-dimensional McKean–Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study Euler–Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean–Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order 1/2 known for the Lipschitz case cannot be recovered for all presented schemes. [ABSTRACT FROM AUTHOR]

The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space B ˙ p , r 4 - 2 α - 3 p . Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense. [ABSTRACT FROM AUTHOR]