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In this paper, we establish the global existence and uniqueness of solution to 2-D inhomogeneous incompressible Navier-Stokes equations (1.1) with initial data in the critical spaces. Precisely, under the assumption that the initial velocity u 0 in L 2 ∩ B ˙ p , 1 − 1 + 2 p and the initial density ρ 0 in L ∞ and having a positive lower bound, which satisfies 1 − ρ 0 − 1 ∈ B ˙ λ , 2 2 λ ∩ L ∞ , for p ∈ [ 2 , ∞ [ and λ ∈ [ 1 , ∞ [ with 1 2 < 1 p + 1 λ ≤ 1 , the system (1.1) has a global solution. The solution is unique if p = 2. With additional assumptions on the initial density in case p > 2 , we can also prove the uniqueness of such solution. In particular, this result improves the previous work in [Abidi-Gui, ARMA(2021)] where u 0 belongs to B ˙ 2 , 1 0 and ρ 0 − 1 − 1 belongs to B ˙ 2 ε , 1 ε , and we also remove the assumption that the initial density is close enough to a positive constant in [Danchin-Wang, CMP(2023)] yet with additional regularities on the initial density here. [ABSTRACT FROM AUTHOR]

An algebraic upper bound for the decay rate of solutions to the Navier–Stokes and Navier–Stokes–Coriolis equations in the critical space H ˙ 1 2 (R 3) is derived using the Fourier splitting method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts. The proof is carried on purely in the critical space, as no L 2 (R 3) estimates are available for the solution. This is the first instance in which such a method is used for obtaining decay bounds in a critical space for a nonlinear equation. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the well-posedness problem of the threedimensional incompressible viscous and resistive Hall-magnetohydrodynamics system (Hall-MHD) with variable density. We mainly prove the existence and uniqueness issues of the density-dependent incompressible Hall-magnetohydrodynamic system in critical spaces on R3. [ABSTRACT FROM AUTHOR]

The primitive equations for geophysical flows are studied under the influence of stochastic wind driven boundary conditions modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic L t q - H z − 1 , p L x y p -setting. This solution is constructed in critical spaces. [ABSTRACT FROM AUTHOR]

Abstract In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained.

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are L p (L q) -theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8] , the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space R 3 , with initial value u → 0 ∈ BMO - 1 (as in Koch and Tataru's theorem) and with force f → = div F where smallness of F ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin's solutions (in L 2 F - 1 L 1 ) or solutions in the multiplier space M (H ˙ t , x 1 / 2 , 1 ↦ L t , x 2) . [ABSTRACT FROM AUTHOR]

Consider the bidomain equations from electrophysiology with FitzHugh--Nagumo transport subject to Current noise, i.e., subject to stochastic forcing modeled by a cylindrical Wiener process. It is shown that this set of equations admits in two- and three-dimensional situations a unique, global, strong pathwise solution within the setting of critical spaces. The proof is based on combining methods from stochastic and deterministic maximal regularity. In addition, the method of extrapolation spaces from deterministic evolution equations is transferred to the stochastic setting. [ABSTRACT FROM AUTHOR]

This work establishes a global well-posedness result for the three dimensional axisymmetric Boussinesq system with critical rough initial data. Specifically, we aim at extending the results in our previous paper [22] to the case where the initial vorticity and density are finite Radon measures. The roadmap towards proving our claim is based upon the strategy in [22, 17]. Namely, performing a fixed point argument to build up a unique local solution, then, showing that the constructed solution is in fact global in time. Nevertheless, crucial arguments from [22], required for the construction of the local solution, need to be carefully analyzed when the datum, and more precisely when the initial density, is not a Lebesgue-integrable function. This is where several techniques of measure theory come into play. Accordingly, we first develop numerous notions on axisymmetric measures, in a general context. These techniques are employed thereafter to understand the coupling between some important quantities of the system and, eventually, to construct the desired solution whenever only the atomic parts of the initial measures are small enough. [ABSTRACT FROM AUTHOR]

We are concerned with the uniqueness of mild solutions in the critical Lebesgue space $ L^{\frac{n}{2}}(\mathbb{R}^{n}) $ for the parabolic-elliptic Keller-Segel system, $ n\geq4 $. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak-$ L^{\frac{n}{2}} $ space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e., $ L^{q}((0, T);L^{p}) $-norms with $ q\neq\infty $), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical $ C([0, T);L^{\frac{n}{2}}) $, our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate $ \gamma $ of the chemoattractant is also analyzed.

According to Patil (2012), a course book should be used as a source book rather than a forced book (p. 67). Students' learning needs to be enriched by supplementing the text book with the help of authentic materials. One of the readily available and easily accessible authentic materials is the newspaper, which is generally read casually and discarded as a waste material. The use of this educationally rich resource of newspapers is not given a serious attention in the pedagogy of English as an International Language (EIL). Incidentally, EIL is a wide-ranging field meant to examine how English language teaching and learning is structured, how meanings and ideas are formed and encoded, how they are communicated and represented, decoded and interpreted, and also taught and learnt across the Globe. Drawing on the wide-ranging educational possibilities that the use of newspapers can offer, we propose to demonstrate, in this paper, how EIL teachers can fruitfully exploit the neglected resource of newspapers to promote language skills, sharpen critical thinking skills, and develop research skills of EIL learners. [ABSTRACT FROM AUTHOR]

The low-frequency L 1 assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible Navier-Stokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical L p framework. Precisely, it is proved that the Besov space B ˙ 2 , ∞ σ 1 -boundedness condition (with d 2 − 2 d p ≤ σ 1 < d 2 − 1) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of B ˙ 2 , ∞ σ 1 . To the best of our knowledge, our work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids. [ABSTRACT FROM AUTHOR]

We consider a smooth, compact and embedded hypersurface Σ without boundary and show that the corresponding (shifted) surface Stokes operator admits a bounded H ∞ -calculus with angle smaller than π / 2 . As an application, we consider critical spaces for the Navier–Stokes equations on the surface Σ . In case Σ is two-dimensional, we show that any solution with a divergence-free initial value in L 2 (Σ , T Σ) exists globally and converges exponentially fast to an equilibrium, that is, to a Killing field. [ABSTRACT FROM AUTHOR]

In this paper, we study the long time decay of global solution to the 3D incompressible Navier-Stokes equations. We prove that if u∈C(R+,X−1,σ(R3))u\in {\mathcal{C}}\left({{\mathbb{R}}}^{+},{{\mathcal{X}}}^{-1,\sigma }\left({{\mathbb{R}}}^{3})) is a global solution to the considered equation, where Xi,σ(R3){{\mathcal{X}}}^{i,\sigma }\left({{\mathbb{R}}}^{3}) is the Fourier-Lei-Lin space with parameters i=−1i=-1 and σ≥−1\sigma \ge -1, then ‖u(t)‖X−1,σ\Vert u\left(t){\Vert }_{{{\mathcal{X}}}^{-1,\sigma }} decays to zero as time goes to infinity. The used techniques are based on Fourier analysis.

In this paper, we prove the global existence of smooth solutions to axisymmetric Navier-Stokes equations with initial data in some critical spaces, provided the swirl part of the initial velocity is sufficiently small. [ABSTRACT FROM AUTHOR]

We prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular in (0 , T ] {(0,T]} if the L 1 ⁢ (0 , T ; B ˙ ∞ , ∞ - 1) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ (0 , T ; B ˙ ∞ , ∞ - 1) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , the mid frequency part of Fourier modes k 2 ≤ | ξ | < k {\frac{k}{2}\leq|\xi|

We prove that a Leray–Hopf weak solution u to 3D Navier–Stokes equations is regular in (0 , T ] {(0,T]} if the L 1 ⁢ (0 , T ; B ˙ ∞ , ∞ - 1) {L^{1}(0,T;\dot{B}^{-1}_{\infty,\infty})} - or L 2 ⁢ (0 , T ; B ˙ ∞ , ∞ - 1) {L^{2}(0,T;\dot{B}^{-1}_{\infty,\infty})} -norm of u k / 2 , k {u_{k/2,k}} , the mid frequency part of Fourier modes k 2 ≤ | ξ | < k {\frac{k}{2}\leq|\xi|

We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of L ∞ L 3 Leray–Hopf weak solutions of Navier–Stokes equation. We modify the argument, and extend the uniqueness result to the scaling critical anisotropic Lebesgue space with mixed-norms. As a consequence, our results cover the class of initial data and solutions which may be singular or decay with different rates along different spatial variables. The result relies on the establishment of several refined properties of solutions of the Stokes and Navier–Stokes equations in mixed-norm Lebesgue spaces which seem to be of independent interest. [ABSTRACT FROM AUTHOR]

We study global well-posedness for the inhomogeneous Navier–Stokes equations on R n , n ≥ 2 , with initial velocity in endpoint critical Besov spaces B q , ∞ - 1 + n / q (R n) , n ≤ q < 2 n , and merely bounded initial density with a positive lower bound. First, we consider a multiplication property of L ∞ -functions in some Bessel potential and Besov spaces. Based on it and on maximal L γ ∞ -regularity of the Stokes operator in little Nicolskii spaces, we show solvability for the momentum equations with fixed bounded density. Finally, proof for existence of a solution to the inhomogeneous Navier–Stokes equations is done via an iterative scheme when B q , ∞ - 1 + n / q -norm of initial velocity and relative variation of initial density are small, while uniqueness of a solution is proved via a Lagrangian approach when initial velocity belongs to B q , ∞ - 1 + n / q (R n) ∩ B r , ∞ - 1 + n / q (R n) for slightly larger r > q . [ABSTRACT FROM AUTHOR]

This paper considers the large time properties of global solutions to the 3D incompressible magnetohydrodynamic system in critical spaces H ˙ 1 2 (R 3) and L 3 (R 3) . More precisely, we obtain the temporal decay rate (1 + t) - 1 4 for a small global solution in the space H ˙ 1 2 (R 3) , and a small global solution in the space L 3 (R 3) is non-increasing and decays to zero at infinity. [ABSTRACT FROM AUTHOR]

We prove the global well-posedness of the Cauchy problem to the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial data in critical Besov spaces that allowing for different integrability indices for the velocity field u and magnetic field b (and its current J), which generalize the result in [13]. Meanwhile, we analyze the long-time behavior of the solutions and get some decay estimates. Finally, a stability theorem for global solutions is established. [ABSTRACT FROM AUTHOR]

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are C1+α–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are C1+α–close to a sphere, and we prove that these solutions become spherical as time goes to infinity. [ABSTRACT FROM AUTHOR]

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in [19,20]. Roughly speaking, we show essentially that if the initial data (v0,ρ0) is axisymmetric and (ω0,ρ0) belongs to the critical space L1(Ω) × L1(R3), with ω0 is the initial vorticity associated to v0 and Ω = {(r,z)∈ R2 : r > 0}, then the viscous Boussinesq system has a unique global solution. [ABSTRACT FROM AUTHOR]

In this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space Za, σ−1 (ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity. [ABSTRACT FROM AUTHOR]

For arbitrary initial data in Lei‐Lin‐Gevrey spaces, we investigate the blow‐up phenomena in finite time to the local unique solution of the three‐dimensional Boussinesq system. We determine the blow‐up profile explicitly as a function of time, and we identify the low frequencies part as a solely responsible of this phenomena. Frequencies decomposition, functional spaces interpolation, and Leray theory are used. [ABSTRACT FROM AUTHOR]

We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings. [ABSTRACT FROM AUTHOR]

In this paper, we prove two results about the blow‐up criterion of the three‐dimensional incompressible Navier‐Stokes equation in the Sobolev space Ḣ5/2. The first one improves the result of Cortissoz et al. The second deals with the relationship of the blow up in Ḣ5/2 and some critical spaces. Fourier analysis and standard techniques are used. [ABSTRACT FROM AUTHOR]

We consider the inertial motion of a system constituted by a rigid body with an interior cavity entirely filled with a viscous incompressible fluid. Navier boundary conditions are imposed on the cavity surface. We prove the existence of weak solutions and determine the critical spaces for the governing evolution equation. Using parabolic regularization in time-weighted spaces, we establish regularity of solutions and their long-time behavior. We show that every weak solution à la Leray–Hopf to the equations of motion converges to an equilibrium at an exponential rate in the Lq-topology for every fluid-solid configuration. A nonlinear stability analysis shows that equilibria associated with the largest moment of inertia are asymptotically (exponentially) stable, whereas all other equilibria are normally hyperbolic and unstable in an appropriate topology. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the optimal time decay rate of solutions for the compressible flow of nematic liquid crystals in Lp type critical framework and any dimensionN≥2. Concretely, if the low frequencies of the initial data are in e.g. Lp/2(RN), we could obtain that the Lp norm of the critical global solution (ρ-1,u) decays like t-Np+N4, while the Lp norm of d-d^ decays like t-N2p for t→∞, exactly as shown in the work by Gao et al. (J Differ Equ 261:2334-2383, 2016) when p=2 and N=3, for solutions with high Sobolev regularity. As a by-product, we derive an accurate description of the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The main tools used here are the Littlewood-Paley theory and refined time weighted inequalities in Fourier space. [ABSTRACT FROM AUTHOR]

The global existence issue in critical regularity spaces for the full Navier-Stokes equations satisfied by compressible viscous and heat-conductive gases has been first addressed in Danchin (Arch Ration Mech Anal 160:1-39, 2001), then recently extended to the general Lp framework in Danchin and He (Math Ann 64:1-38, 2016). In the present work, we establish decay estimates for the global solutions constructed in [13], under an additional mild integrability assumption that is satisfied if the low frequencies of the initial data are in Lr(Rd) with p2≤r≤min{2,d2}. As a by-product in the case d=3, we recover the classical decay rate t-34 for t→+∞ that has been observed by Matsumura and Nishida (J Math Kyoto Univ 20:67-104, 1980) for solutions with high Sobolev regularity. Compared to a recent paper of us (Danchin and Xu in Arch Ration Mech Anal 224:53-90, 2017) dedicated to the barotropic case, not only we are able to treat the full system, but we also weaken the low frequency assumption and improve the decay exponents for the high frequencies of the solution. [ABSTRACT FROM AUTHOR]

In this article, we study the long time decay of global solution to 3D incompressible Navier-Stokes equations. We prove that if $u\in{\mathcal C}([0,\infty),H^1_{a,\sigma}(\mathbb{R}^3))$ is a global solution, where $H^1_{a,\sigma}(\mathbb{R}^3)$ is the Sobolev-Gevrey spaces with parameters $a>0$ and $\sigma>1$, then $\|u(t)\|_{H^1_{a,\sigma}(\mathbb{R}^3)}$ decays to zero as time approaches infinity. Our technique is based on Fourier analysis.

In this paper, we obtain analyticity of the inhomogeneous Navier–Stokes equations. The main idea is to use the exponential operator e ϕ ( t ) | D | , where ϕ ( t ) = δ − θ ( t ) , δ > 0 is the analyticity radius of ( ρ 0 − 1 , u 0 ) , and | D | is the differential operator whose symbol is given by ‖ ξ ‖ l 1 . We will show that for sufficiently small initial data, solutions are analytic globally in time in critical Besov spaces. [ABSTRACT FROM AUTHOR]

The bidomain problem with FitzHugh-Nagumo transport is studied in the Lp-Lq-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension d≤4, by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria are the same as for the classical FitzHugh-Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028-2074, 2018), and qualitative theory of evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the axisymmetric MHD system with nearly critical initial data having the special structure: u0=u0rer+u0θeθ+u0zez, b0=b0θeθ. We prove that, this system is globally well-posed provided the scaling-invariant norms ∥ru0θ∥L∞, ∥r−1b0θ∥L32 are sufficiently small. [ABSTRACT FROM AUTHOR]

The abstract theory of critical spaces developed in Prüss and Wilke (J Evol Equ, 2017. doi:10.1007/s00028-017-0382-6), Prüss et al. (Critical spaces for quasilinear parabolic evolution equations and applications, 2017) is applied to the Navier-Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the Lp-Lq setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an H∞-calculus with H∞-angle 0, and the real and complex interpolation spaces of these operators are identified. [ABSTRACT FROM AUTHOR]

A model from electro-magneto-hydrodynamics describing a completely ionized gas, a plasma, is studied. Local well posedness of the problem in time-weighted Lp spaces is obtained by means of maximal regularity of the linearized problem, and the induced local semiflow in the proper state space is constructed. If the underlying domain is bounded and simply connected, based on the principle of linearized stability, it is shown that the trivial solution of the problem is exponentially stable. Moreover, if the orbit of a solution is relatively compact in the state space, the solution exists globally, and it converges to the trivial equilibrium. [ABSTRACT FROM AUTHOR]

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal L p -regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier–Stokes problem, convection–diffusion equations, the Nernst–Planck–Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the large time behavior of global solutions to the initial value problem for the compressible Navier–Stokes–Poisson system in the L p critical framework and in any dimension N ≥ 3 . We obtain the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents , which improves the decay results in high Sobolev regularity. The proof is mainly based on the Littlewood–Paley theory and refined time weighted inequalities in Fourier space. [ABSTRACT FROM AUTHOR]

This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper (LeCrone et al. J Evolut Equ 14:509-533 2014). The improvement is explained in 'The improvement' section and proved in 'Proof of the main result' section. The importance of the extension is demonstrated in 'Application to the Navier-Stokes equations' section with an application to the Navier-Stokes system in critical $$L_q$$ -spaces. [ABSTRACT FROM AUTHOR]

We prove global well-posedness for the Navier-Stokes-Coriolis system (NSC) in a critical space whose definition is based on Fourier transform, namely the Fourier-Besov-Morrey space $$\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$$ with $$0<\mu <3$$ . The smallness condition on the initial data is uniform with respect to the angular velocity $$\omega $$ . Our result provides a new class for the uniform global solvability of (NSC) and covers some previous ones. For $$\mu =0$$ , (NSC) is ill-posedness in $$\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$$ which shows the optimality of the results with respect to the space parameter $$\mu >0$$ . The lack of Hausdorff-Young inequality in Morrey spaces suggests that there are no inclusions between $$\mathcal {FN}_{1,\mu ,\infty }^{\mu -1}$$ and the largest previously known existence classes of Kozono-Yamazaki (Besov-Morrey space) and Koch-Tataru ( $$\textit{BMO}^{-1}$$ ) for Navier-Stokes equations (3DNS). So, taking in particular $$\omega =0$$ , we obtain a critical initial data class that seems to be new for global existence of solutions of (3DNS). [ABSTRACT FROM AUTHOR]