Scaling Invariant Spaces Research Articles

On the global stability of large Fourier mode for 3-D Navier-Stokes equations

In this paper, we first prove the global existence of strong solutions to 3-D incompressible Navier-Stokes equations with solenoidal initial data, which writes in the cylindrical coordinates is of the form: A(r,z)cos⁡Nθ+B(r,z)sin⁡Nθ, provided that N is large enough. In particular, we prove that the corresponding solution has almost the same frequency N for any positive time. The main idea of the proof is first to write the solution in trigonometrical series in θ variable and estimate the coefficients separately in some scale-invariant spaces, then we handle a sort of weighted sum of these norms of the coefficients in order to close the a priori estimate of the solution. Furthermore, we shall extend the above well-posedness result for initial data which is a linear combination of axisymmetric data without swirl and infinitely many large mode trigonometric series in the angular variable.

Global existence of solutions for Boussinesq system with energy dissipation

The Boussinesq system coupled by an energy dissipation brings new challenges in the study of global existence of solutions, for instance, this system does not have scale invariance which makes it difficult to show existence of mild solutions when initial data belongs to scaling invariant function spaces. In this paper we are interested to show global existence of solutions [u,θ] for Boussinesq system coupled by bilinear energy dissipation Φ(u)=2μE(u)⋅E(u) on smooth bounded domain Ω⊆Rn or whole space Rn, n⩾3, when the initial data [u0,θ0]∈X0=Wσ1,n/2(Ω)×Ln/2(Ω) is sufficiently small and the external force F(θ)=ϱfθen has low regularity in the sense t1/2−b/2nf∈L∞((0,T):Lb(Ω)) or f∈Ls((0,T):Lb(Ω)), where 2s=1−nb and b∈[n,∞). It seems that our results on the global and local well-posedness are the first to provide an Lp-approach when the initial velocity data u0 belongs to the space W˙σ1,n/2(Rn) that is scaling invariant.

Global well-posedness of 3D inhomogeneous Navier–Stokes system with small unidirectional derivative

In Liu and Zhang (2020 Arch. Ration. Mech. Anal. 235 1405–44); Liu et al (2020 Arch. Ration. Mech. Anal. 238 805–43), the authors proved that as long as the one-directional derivative of the initial velocity is sufficiently small in some scaling invariant spaces, then the (anisotropic) Navier–Stokes (NS) system has a unique global solution. The goal of this paper is to extend this type of result to the 3D inhomogeneous (density-dependent) NS system. More precisely, given initial density such that and the initial velocity with belonging to , then the inhomogeneous NS system has a unique global solution provided that being sufficiently small for some bounded function f depending on and . This provide a more general result that of Chemin et al (2014 Commun. Math. Phys. 272 529–66); Chemin and Zhang (2015 Commun. PDE 40 878–96).

Critical exponent for global existence of solutions to the Schrödinger equation with a nonlinear boundary condition

We consider the nonlinear initial boundary value problem of one dimensional Schrödinger equation with the power nonlinearity of order q in boundary conditions. Our purpose in this paper is to prove that if q>2 and the initial data are small in suitable norms, then we have a global in time of solutions in a scale invariant space. Furthermore, we present a blow up result of small amplitude solutions when 1<q≤2 and derive an upper-bound of existence time Our results indicate that the quadratic nonlinearity is a critical exponent.

The two‐dimensional stationary Navier–Stokes equations in toroidal Besov spaces

AbstractWe consider the stationary Navier–Stokes equations in the two‐dimensional torus . For any , we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces for given small external forces in when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case , we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from to .

Large global solutions of the compressible Navier-Stokes equations in three dimensions

&lt;p style='text-indent:20px;'&gt;This work concerns the initial value problem for the three dimensional compressible Navier-Stokes equations (both isentropic and polytropic). By exploiting the famous Fujita-Kato theorem to the Classical incompressible Navier-Stokes equations, we prove the existence of global-in-time unique solutions under as weak as possible smallness conditions in the scaling invariant spaces. In particular, our results improve the classical theorems obtained by Danchin [Invent. Math., &lt;b&gt;141&lt;/b&gt;, 579–614, 2000] and Danchin [Arch. Ration. Mech. Anal., &lt;b&gt;160&lt;/b&gt;, 1–39, 2001].&lt;/p&gt;

Various regularity estimates for the Keller-Segel-Navier-Stokes system in Besov spaces

We show the local well-posedness for the Keller-Segel-Navier-Stokes system with initial data in the scaling invariant Besov spaces, where the solution exists globally in time if the initial data is sufficiently small. We also reveal that the solution belongs to the Lorentz spaces in time direction, while the solution is smooth in space and time. Moreover, we obtain the maximal regularity estimates of solutions under the certain conditions. We further show that the solution has the additional regularities if the initial data has higher regularities. This result implies that global solutions decay as the limit t→∞ in the same norm of the space of the initial data. Our results on the Lorentz regularity estimates are based on the strategy by Kozono-Shimizu (2019) [26].

An optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component

In this paper, we provide an optimal regularity criterion for 3D Navier-Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces. More precisely, employing the anisotropic Littlewood-Paley theory, we prove that a weak solution u is regular if ∇u3 belongs to scaling invariant space L2(0,T;Lv∞Lh2), where h and v denote the horizontal and vertical components, respectively. This result verifies the limiting case of a previous result established by Guo, Caggio and Skalák (2017).

Wellposedness of Keller-Segel systems in mixed norm spaces

We study the well-posedness of the Cauchy problem for the Keller-Segal system in the setting of mixed norm spaces. We prove existence of mild solutions n scaling invariant spaces and uniqueness in a special case. These results allow for existence and uniqueness when the initial data has anisotropic properties. In particular, persistence of anisotropic properties under the evolution is demonstrated which could be of biological interest. For more information see https://ejde.math.txstate.edu/conf-proc/26/r1/abstr.html

Long-time asymptotic behavior of the generalized two-dimensional quasi-geostrophic equation

In this paper, we are concerned with the long-time asymptotic behavior of the generalized two-dimensional quasi-geostrophic equation. More precisely, we obtain the sharp time-decay of the smooth solution of the generalized quasi-geostrophic equation with the sub-critical dissipation and the critical dissipation for the general initial data, and the sup-critical dissipation for the small initial data in the scale invariant Besov space. In addition, the sharp time-decay of any weak solution of the generalized quasi-geostrophic equation for the general initial data is also derived. Moreover, we also derive the decay estimates for the difference between the full solution and the solution to the corresponding linear part.

Global existence for the defocusing Sobolev critical Schrödinger equation under the finite variance condition of initial data

In unbounded domain RN,N≥1, we consider the defocusing nonlinear Schrödinger equation with power-type nonlinearity containing −|u|4N−2su. For 2s∗=2N−2s, with s<N2 and s≤1, the corresponding scaling invariant space is homogeneous Sobolev Ḣxs(RN) and in this case implies that the critical regularity is sc=N2−12s∗≡s. Under the finite variance condition of initial data and the solutions of the problem satisfy the pseudo-conformal conservation law, we investigate the global existence of the solutions in Ltq−Lxp spaces for some constants p,q≥2 depending on N,s. The new results of this study encompass the existing results on mass critical (sc=0) in Dodson (2012) and energy critical (sc=1) in Killip and Visan (2010).

Local and global existence of solutions of a Keller-Segel model coupled to the incompressible fluid equations

We consider a Keller-Segel model coupled to the incompressible fluid equations which describes the dynamics of swimming bacteria. We mainly take the incompressible Navier-Stokes equations for the fluid equation part. In this case, we first show the existence of unique local-in-time solutions for large data in scaling invariant Besov spaces. We then proceed to show that these solutions can be defined globally-in-time if some smallness conditions are imposed to initial data. We also show the existence of unique global-in-time self-similar solutions when initial data are sufficiently small in scaling invariant Besov spaces. But, these solutions do not exhibit (expected) temporal decay rates. So, we change the fluid part to the Stokes equations and we derive temporal decay rates of the bacteria density and the fluid velocity.

Global Well-Posedness and Analyticity for the Three-Dimensional Incompressible Nematic Liquid Crystal Flows in Scaling Invariant Spaces

The Cauchy problem for the three-dimensional incompressible flows of liquid crystals in scaling invariant spaces is considered. In this work, we exhibit three results. First, we prove the global well-posedness of mild solution for the system without the supercritical nonlinearity∇d2dwhen the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients. Our second result is a proof of the global existence of mild solution in the time dependent spaces for the system including the term∇d2dfor small initial data. Lastly, we also get analyticity of the solution.

Global solutions for the Muskat problem in the scaling invariant Besov space [formula omitted

The one-phase and two-phase Muskat problems with arbitrary viscosity contrast are studied in all dimensions. They are quasilinear parabolic equations for the graph free boundary. We prove that small data in the scaling invariant homogeneous Besov space B˙∞,11 lead to unique global solutions. The proof exploits a new structure of the Dirichlet-Neumann operator which allows us to implement a robust fixed-point argument. As a consequence of this method, the initial data is only assumed to be in B˙∞,11 and the solution map is Lipschitz continuous in the same topology. For the general Muskat problem, the only known scaling invariant result was obtained in the Wiener algebra (plus an L2 assumption) which is strictly contained in B˙∞,11.

Existence, Gevrey regularity, and decay properties of solutions to Active models in critical spaces

In this paper, we provide some mathematical results to models of phase separation of self-propelled particles. More precisely, we show the existence of a unique global-in-time solution, Gevrey regularity, and decay properties with small initial data in scaling-invariant spaces.

Blow-up of radial solutions for the intercritical inhomogeneous NLS equation

We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in RNi∂tu+Δu+|x|−b|u|2σu=0, where N≥3, 0<b<min⁡{N2,2} and 2−bN<σ<2−bN−2. The scaling invariant Sobolev space is H˙sc with sc=N2−2−b2σ. The restriction on σ implies 0<sc<1 and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let u0∈H˙sc∩H˙1 be a radial initial data and u(t) the corresponding solution to the INLS equation. We first show that if E[u0]≤0, then the maximal time of existence of the solution u(t) is finite. Also, for all radially symmetric solution of the INLS equation with finite maximal time of existence T⁎>0, then limsupt→T⁎‖u(t)‖H˙sc=+∞. Moreover, under an additional assumption and recalling that H˙sc⊂Lσc with σc=2Nσ2−b, we can in fact deduce, for some γ=γ(N,σ,b)>0, the following lower bound for the blow-up ratec‖u(t)‖H˙sc≥‖u(t)‖Lσc≥|log⁡(T−t)|γ,ast→T⁎. The proof is based on the ideas introduced for the L2 super critical nonlinear Schrödinger equation in the work of Merle and Raphaël [14] and here we extend their results to the INLS setting.

The primitive equations in the scaling-invariant space L^{infty }(L^1)

Consider the primitive equations on mathbb {R}^2times (z_0,z_1) with initial data a of the form a=a_1+a_2, where a_1 in mathrm{BUC}_sigma (mathbb {R}^2;L^1(z_0,z_1)) and a_2 in L^infty _sigma (mathbb {R}^2;L^1(z_0,z_1)). These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for a_1 arbitrary large and a_2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the L^infty (L^1)-setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the L^infty (L^1)-setting.

Medical image segmentation algorithm based on positive scaling invariant-self encoding CCA

The quality of medical image segmentation results directly affect disease analysis and diagnosis. Although traditional medical image segmentation methods have achieved certain results, this type of method is not only computationally inefficient, but also difficult to achieve fully automatic segmentation. Deep learning has a good generalization ability, which provides a new technical approach to solve the above problems. However, the following problems exist in the application of deep learning in medical image segmentation: (1) Optimization of deep learning models. The parameters of the existing deep learning model have not been well optimized; it is easy to cause wrong medical image segmentation. (2) Overfitting problem. As the number of deep learning model layers increases, it makes the deep model fit the training data easily, but its generalization performance on the test set will be poor. In view of this, in order to solve the problem of deep learning model optimization, this paper first reveals that the weight space of deep learning is not equivalent to its parameter space, and explains in detail how to optimize the deep learning model in a positive scaling invariant space. The method of mapping the derivative of the base path back to the weight, and analyzed the stochastic optimization algorithm in F space. In addition, to solve the overfitting problem of deep learning models, this paper first gives a method of self-encoding Canonical Correlation Analysis (Self-encoding CCA) to optimize both self-coding loss and correlation loss in the fine-tuning stage. Based on the above ideas, this paper proposes a medical image segmentation algorithm based on positive scaling invariant-self encoding CCA. The experimental results show that the index obtained by the method proposed in this paper is not only improved by a large margin compared with traditional machine learning methods, but also improved to a certain extent compared with other deep learning methods.

On a repulsion Keller–Segel system with a logarithmic sensitivity

In this paper, we study the initial-boundary value problem of a repulsion Keller–Segel system with a logarithmic sensitivity modelling the reinforced random walk. By establishing an energy–dissipation identity, we prove the existence of classical solutions in two dimensions as well as existence of weak solutions in the three-dimensional setting. Moreover, it is shown that the weak solutions enjoy an eventual regularity property, i.e., it becomes regular after certain time T &gt; 0. An exponential convergence rate towards the spatially homogeneous steady states is obtained as well. We adopt a new approach developed recently by the author to study the eventual regularity. The argument is based on observation of the exponential stability of constant solutions in scaling-invariant spaces together with certain dissipative property of the global solutions in the same spaces.

Refined blow-up criteria for the full compressible Navier–Stokes equations involving temperature

In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier–Stokes equations, we improve almost all the blow-up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier–Stokes equations. Enlightening regular criteria via pressure \(\Pi =\frac{\text{ divdiv }}{-\Delta }(u_{i}u_{j})\) of the 3D incompressible Navier–Stokes equations on bounded domain, we generalize Beirao da Veiga’s result in (Chin Ann Math Ser B 16:407–412, 1995) from the incompressible Navier–Stokes equations to the isentropic compressible Navier–Stokes system in the case away from vacuum.