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In this paper, we consider the Littlewood–Paley characterization for mixed Morrey spaces and its predual spaces. The topology to converge the Littlewood–Paley decomposition for the element of mixed Morrey spaces is the weak‐∗$*$ topology. If we consider the topology of mixed Morrey spaces, we must give other characterization by using the heat semigroup. As an application, we show the wavelet characterization for mixed Morrey spaces. In particular, this characterization can be shown without the Peetre maximal operator. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with global well-posedness and optimal decay rate for strong solutions of the compressible finite extensible nonlinear elastic (FENE) dumbbell model. For d ≥ 2 , we firstly prove that the compressible FENE dumbbell model admits the unique global strong solutions provided initial data are close to equilibrium state. Then, by the Littlewood-Paley decomposition theory and the Fourier splitting method, we show optimal L 2 decay rate of global strong solutions for d ≥ 3. Finally, we mainly study optimal decay rate to the 2-D FENE dumbbell model. The improved Fourier splitting method yields that the L 2 decay rate is ln − l ⁡ (e + t) for any l ∈ N. By virtue of the time weighted energy estimate, we can improve the decay rate to (1 + t) − 1 4 . Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov spaces with negative index and obtain optimal L 2 decay rate without the smallness restriction of low frequencies. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation (alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory. Then, the Hölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored. [ABSTRACT FROM AUTHOR]

In this paper, we obtain global small solutions and decay estimates for the two-dimensional MHD boundary layer in Gevrey space without any structural assumptions, generalizing the results of [20] in analytic space. The proof method is mainly inspired by [14] and [26] , using new auxiliary functions and finer structural analysis to overcome the difficulty of the loss of derivatives and then we obtain the global well-posedness of the MHD boundary layer in the Gevrey 3 2 space. [ABSTRACT FROM AUTHOR]

Consider the following time-dependent stable-like operator with drift: L t φ (x) = ∫ R d [ φ (x + z) − φ (x) − z (α) ⋅ ∇ φ (x) ] σ (t , x , z) ν α (d z) + b (t , x) ⋅ ∇ φ (x) , where d ⩾ 1 , ν α is an α -stable type Lévy measure with α ∈ (0 , 1 ] and z (α) = 1 α = 1 1 | z | ⩽ 1 z , σ is a real-valued Borel function on R + × R d × R d and b is an R d -valued Borel function on R + × R d. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with L t under the sharp balance condition α + β ⩾ 1 , where β is the Hölder index of b with respect to x. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form L t. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces. [ABSTRACT FROM AUTHOR]

We prove the asymptotic properties of the solutions to the 3D Navier-Stokes system with singular external force, by making use of Fourier localization method, the Littlewood-Paley theory and some subtle estimates in Fourier-Herz space. The main idea of the proof is motivated by that of Cannone et al. [J. Differential Equations, 314, 316–339 (2022)]. We deal either with the non-stationary problem or with the stationary problem where solution may be singular due to singular external force. In this paper, the Fourier-Herz space includes the function space of pseudomeasure type used in Cannone et al. [J. Differential Equations, 314, 316–339 (2022)] [ABSTRACT FROM AUTHOR]

Considered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approximate and actual solution is negligible, thus succeeding in proving the nonuniform dependence result in both supercritical Besov spaces B p , r s (T) × B p , r s (T) with s > max { 3 2 , 1 + 1 p } , 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and critical Besov space B 2 , 1 3 2 (T) × B 2 , 1 3 2 (T) . In the non-periodic case, we constructed two sequences of initial data with high and low-frequency terms by analyzing the inner structure of the system under investigation in detail, and we proved that the distance between the two corresponding solution sequences is lower-bounded by time t, but converges to zero at initial time. This implies that the solution map is not uniformly continuous both in supercritical Besov spaces B p , r s (R) × B p , r s (R) with s > max { 3 2 , 1 + 1 p } , 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and critical Besov spaces B p , 1 1 + 1 p (R) × B p , 1 1 + 1 p (R) with 1 ≤ p ≤ 2 . The proof of nonuniform dependence is based on approximate solutions and Littlewood–Paley decomposition theory. These approaches are widely applicable in the study of continuous properties for shallow water equations. [ABSTRACT FROM AUTHOR]

In the present paper, we study the generalized Cauchy problem of the incompressible micropolar fluid system in R 3. By using the Fourier localization argument and the Littlewood-Paley theory, we establish the local existence for large initial data and the global existence for small initial data in critical Fourier-Besov-Morrey spaces. [ABSTRACT FROM AUTHOR]

As Fourier transformations of L p functions are the mathematical basis of various applications, it is necessary to develop L p theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this L p theory for 1 ≤ p < ∞ . By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on L 1 (R 2) . Together with the Plancherel identity and Hausdorff–Young inequality, we establish L p (R 2) multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the L 1 (R 2) signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images. [ABSTRACT FROM AUTHOR]

The global existence of solutions to a damped MHD system in Besov spaces is obtained. This research is motivated by the open question [Wu et al., J. Nonlinear Sci. 25 (2015), 157–192, Remark 5.1] on the possibility of extension to the existence of global small solutions from a potential Banach space to a Besov space. The present result positively confirms the open question. What is more, the initial data are not required to be uniformly small. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous incompressible magnetohydrodynamic (MHD) system. By the assumption of the smallness of initial velocity and magnetic fluids in the critical Besov space, the local and global well‐posedness of 3D inhomogeneous incompressible equations is obtained. It improves some previous results of MHD equations by generalizing the range of exponent p$$ p $$ in Besov spaces B˙p,13/p−1$$ {\dot{B}}_{p,1}^{3/p-1} $$ with 1

In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u 0 is discontinuous at t = 0 in the metric of B 2 , ∞ s (R) , s > 3 2. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces B p , 1 1 + d p (R d) , 1 ≤ p < ∞ , which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ. , 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵ Δ u as ϵ → 0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory. [ABSTRACT FROM AUTHOR]

Given initial data u0 ∈ Lp (ℝ3) for some p in , the auhtors first prove that 3D incompressible Navier-Stokes system has a unique solution u = uL+v with and for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631–1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in Ḣs(ℝ3) for . [ABSTRACT FROM AUTHOR]

The stability of smooth solutions of an anisotropic surface quasi-geostrophic equation with horizontal dissipation remains an open problem. In this work, we present a partial answer to this problem in a rougher function space. More precisely, if the initial data θ0 belong to anisotropic Sobolev space H0,s with 1 2 < s < 1 , then there exists a global small solution, provided that the initial data are small in H0,s. The stability problem in the space H0,1 or more regular spaces remains open. The main tools used are the Littlewood–Paley theory and standard techniques. [ABSTRACT FROM AUTHOR]

The Menchov-Rademacher inequality is an inequality in harmonic analysis that bounds the L_2 norm of a certain maximal operator. It was first established in order to prove almost everywhere convergence of a one-parameter series of orthogonal functions. When two-parameter series of orthogonal functions is considered, the exact way the series is grouped becomes essential. We will consider grouping of a two-parameter series, generated by a sequence of right triangles with a vertex at the origin, who might be non-equilateral, and prove almost everywhere convergence when the eccentricity of those triangles is bounded. In order to carry out the proof, we will derive an analogue of the Menchov-Rademacher inequality for right triangles. [ABSTRACT FROM AUTHOR]

We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calderón and Torchinsky, and Grafakos, He, Honzík, and Nguyen, and an improvement of the result of Triebel. For 0 < p < ∞ and 0 < q ≤ ∞ we obtain that if r > d s - (d / min (1 , p , q) - d) , then ‖ { (m k f k ^ ) ∨ } k ∈ Z ‖ L p (ℓ q) ≲ p , q sup l ∈ Z ‖ m l (2 l ·) ‖ L s r (R d) ‖ { f k } k ∈ Z ‖ L p (ℓ q) , f k ∈ E (A 2 k) , under the condition max (| d / p - d / 2 | , | d / q - d / 2 |) < s < d / min (1 , p , q) . An extension to p = ∞ will be additionally considered in the scale of Triebel–Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by Sobolev spaces L s r with r ≤ d s - (d / min (1 , p , q) - d) . [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result. [ABSTRACT FROM AUTHOR]

We consider the global solvability of the incompressible Chemotaxis-Navier–Stokes system in R d , d = 2 , 3. We first investigate the global well-posedness of Chemotaxis-Navier–Stokes equations in R 2 by involving a double exponential function of the initial velocity field u 0 and the initial chemical concentration c 0. By only giving the smallness assumption of initial cell density n 0 and the horizontal components of velocity field u 0 h = (u 0 1 , u 0 2) in critical Besov spaces, the global existence and uniqueness of the solution for Chemotaxis-Navier–Stokes equations in R 3 is also established. The main contribution of this paper is to supply an improved assumption on initial data and remove the smallness condition on initial chemical concentration c 0 in both dimensional two and three. [ABSTRACT FROM AUTHOR]

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G = def u + y 2 〈 t 〉 φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ , which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data. [ABSTRACT FROM AUTHOR]

In this paper, we study a hyperbolic version of the Navier–Stokes equations, obtained by using the approximation by relaxation of the Euler system, evolving in a thin strip domain. The formal limit of these equations is a hyperbolic Prandtl type equation, our goal is to prove the existence and uniqueness of a global solution to these equations for analytic initial data in the tangential variable, under a uniform smallness assumption. Then we justify the limit from the anisotropic hyperbolic Navier–Stokes system to the hydrostatic hyperbolic Navier–Stokes system with small analytic data. [ABSTRACT FROM AUTHOR]

Motivated by [8] , we prove the global existence and large time behavior of small solutions to 2-D Prandtl system for data with Gevrey 2 regularity in the x variable and Sobolev regularity in the y variable. In particular, we extend the global well-posedness result in [26] for 2-D Prandtl system with analytic data to data with optimal Gevery regularity in the sense of [11]. [ABSTRACT FROM AUTHOR]

Rubio de Francia proved the one-sided version of Littlewood–Paley inequality for arbitrary intervals. In this paper, we prove the similar inequality in the context of arbitrary Vilenkin systems (that is, for functions on infinite products of cyclic groups). There are no assumptions on the orders of these groups. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous incompressible asymmetric fluids system with density-dependent viscosity. By the assumption of the smallness of initial velocity in the critical Besov space, B ˙ p , 1 3 / p − 1 for 1 < p < 6 and the initial density in the critical Besov space and bounded away from vacuum, the local and global well-posedness of 3D inhomogeneous incompressible asymmetric fluids is obtained. By giving the different estimates for pressure in the system for 1 < p < 3 2 and 3 2 ≤ p < 6 , it shows the a priori estimate for the corresponding linearized equation. This not only improves the previous results for 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity, but also obtains the new results on the micropolar system without smallness for density in critical Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitable decay rates. The whole analysis is divided into two parts: When the initial far-field states are not small, we reformulate the original problem into a small perturbation problem by extracting a suitable background profile which is governed by a heat equation, then we prove a long-time existence of solutions, and the lower bound of lifespan of solutions is given in term of the small parameter of initial perturbation; After establishing the long-time existence, combining the decay properties of far-field state and the properties of solutions to the heat equation, it is shown that both the solution obtained in the first part and the background profile indeed satisfy some smallness requirements when time goes to the lower bound of lifespan of solutions. Then we can show the global existence of solutions after this moment. [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the existence of global large solutions and incompressible limit for the compressible flow of liquid crystals under the assumption on almost constant density and large bulk viscosity. The result is based on the Fourier analysis and involves the so-called critical homogeneous Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized rotation b‐family system (R‐b‐family system) which models the evolution of equatorial water waves. Based on local well‐posedness results and lifespan estimates, we establish sharpness of continuity on the data‐to‐solution map by showing that it is not uniformly continuous from Bp,rs×Bp,rs−1 to C([0,T];Bp,rs×Bp,rs−1). The proof of nonuniform dependence is based upon approximate solutions and Littlewood‐Paley decomposition theory. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the global wellposedness and large time behavior results of strong solutions for the compressible Oldroyd-B model derived by Barrett, Lu, Süli (Commun. Math. Sci., 15, 1265–1323, 2017). Exploiting the harmonic analysis tools (especially Littlewood-Paley theory), we first study the global well-posedness of the model with small initial data in spaces with low regularity. Then, under some suitable conditions involving only the low frequency of the initial data, we also obtain the optimal decay rates of the solutions. Compared with the result by Wang and Wen (Math. Models Methods Appl. Sci., 30, 139–179, 2020), the polymer number density is allowed to vanish and the stress tensor is near zero equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global well-posedness of the 3-D generalized incompressible Navier–Stokes equations in critical Besov spaces under a polynomial smallness assumption on the initial data. Moreover, we construct a class of initial data with large vertical component, which satisfies that polynomial condition but cannot satisfy the exponential condition in Liu (2020). [ABSTRACT FROM AUTHOR]

We present an L q (L p) -theory for the equation ∂ t α u = ϕ (Δ) u + f , t > 0 , x ∈ R d ; u (0 , ⋅) = u 0. Here p , q > 1 , α ∈ (0 , 1) , ∂ t α is the Caputo fractional derivative of order α , and ϕ is a Bernstein function satisfying the following: ∃ δ 0 ∈ (0 , 1 ] and c > 0 such that (0.1) c (R r) δ 0 ≤ ϕ (R) ϕ (r) , 0 < r < R < ∞. We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove ‖ | ∂ t α u | + | u | + | ϕ (Δ) u | ‖ L q ([ 0 , T ] ; L p) ≤ N (‖ f ‖ L q ([ 0 , T ] ; L p) + ‖ u 0 ‖ B p , q ϕ , 2 − 2 / α q ) , where B p , q ϕ , 2 − 2 / α q is a modified Besov space on R d related to ϕ. Our approach is based on BMO estimate for p = q and vector-valued Calderón-Zygmund theorem for p ≠ q. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence and time decay rates of the solutions to Boltzmann equation for hard potential in the critical Chemin-Lerner type space. The evolution of the energy in negative Besov space is derived to be preserved. Our results are based on the modified trilinear estimates and some new observations. We use the Lyapunov-type inequality of the energy functionals to show the time decay rates without linear decay analysis. Compared with the previous work, the decay rates of the global solution are established for lower-regularity initial data. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global existence and the large time decay estimate of solutions to the two-dimensional MHD boundary layer equations with small initial data, which is analytical in the tangential variable. The main idea of the proof is motivated by that of [28]. The additional difficulties are: 1. There appears the magnetic field; 2. The far field here depends on the tangential variable; 3. The Reynolds number is different from magnetic Reynolds number. In particular, we solved an open question in [33] concerning the large time existence of analytical solutions to the MHD boundary layer equations. [ABSTRACT FROM AUTHOR]

We obtain the global large solutions to the compressible Navier–Stokes equations in R 2 . The solution is large in the sense that there is no smallness assumption applied to one component of the initial incompressible velocity. [ABSTRACT FROM AUTHOR]

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from [1] , we prove that if one component of the velocity remains small enough in a sub-space of H ˙ 1 2 "almost" scaling invariant, then the 3D Navier-Stokes equations are globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form L T p ( B ˙ 2 , ∞ α , 2 p − 1 2 − α) or L T p ( B ˙ q , ∞ 2 p + 3 q − 2). [ABSTRACT FROM AUTHOR]

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space B 2 , 1 1 / 2 (T). Our results show that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L 1 -integrability of the variable coefficients α (t) and γ (t) , but nothing to do with the shape of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α (t) an γ (t). We believe that the method in this paper can be applied to the study of well-posedness for other transport type equations. [ABSTRACT FROM AUTHOR]

The Littlewood–Paley theory of homogeneous product Besov and Triebel–Lizorkin spaces is developed in the spirit of the φ -transform of Frazier and Jawerth. This includes the frame characterization of the product Besov and Triebel–Lizorkin spaces and the development of almost diagonal operators on these spaces. The almost diagonal operators are used to obtain product wavelet decomposition of the product Besov and Triebel–Lizorkin spaces. The main application of this theory is to nonlinear m-term approximation from product wavelets in L p and Hardy spaces. Sharp Jackson and Bernstein estimates are obtained in terms of product Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove the almost global existence of classical solutions to the 3D Prandtl system with the initial data which lie within ε of a stable shear flow. Using anisotropic Littlewood-Paley energy estimates in tangentially analytic norms and introducing new linearly-good unknowns, we prove that the 3D Prandtl system has a unique solution with the lifespan of which is greater than exp (ε − 1 / log (ε − 1)) . This result extends the work obtained by Ignatova and Vicol (Arch. Ration. Mech. Anal. 2:809–848, 2016) on the 2D Prandtl equations to the three-dimensional setting. [ABSTRACT FROM AUTHOR]

In this note we investigate some conditions of Hörmander-Mihlin type in order to assure the L∞-BMO boundedness for pseudo-multipliers of the harmonic oscillator. The H¹-L¹ continuity for Hermite multipliers also is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and uniqueness of strong solutions to the 2D anisotropic Boussinesq system for rough initial data with striated regularity. We prove the global well-posedness of the Boussinesq system with anisotropic thermal diffusion with initial vorticity being discontinuous across some smooth interface. In the case of an anisotropic horizontal viscosity, we study the propagation of the striated regularity for the smooth temperature patches problem. The proofs rely on the idea of Chemin to solve the 2-D vortex patch problem for ideal fluids, namely the striated regularity can help to bound the gradient of the velocity. [ABSTRACT FROM AUTHOR]

Let (X , ρ , μ) be Ahlfors-1 regular metric spaces. By developing the Littlewood–Paley characterization of Lipschitz spaces over (X , ρ , μ) and establishing a density argument in the weak sense, the authors give a necessary and sufficient condition for the boundedness of Calderón–Zygmund operators on the Lipschitz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the well‐posedness of the thermal boundary layer equation in two‐dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two‐dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood‐Paley theory, we obtain the local‐in‐time existence and uniqueness of solution to this thermal boundary layer problem. [ABSTRACT FROM AUTHOR]

Suppose that g(f) are bi-parameter Littlewood-Paley square functions which were introduced by H. Martikainen. It is known that the L2(ℝn × ℝm) boundedness and the H1 (ℝn × ℝm)-L1(ℝn × ℝm) boundedness of g(f) have been proved by H. Martikainen and by Z. Li and Q. Xue, respectively. In this paper, we apply the vector-valued theory, the atomic decomposition of product Hardy spaces, and Journé's covering lemma to show that g(f) are bounded from Hp(ℝn × ℝm) to Lp(ℝn × ℝm) with p smaller than 1. [ABSTRACT FROM AUTHOR]

This paper is devoted to the global large solutions to the 3 dimensional compressible Navier-Stokes equations in the critical Besov spaces with initial data satisfying a nonlinear smallness condition. Here the "large solutions" mean that the any component of the initial velocity could be arbitrarily large. Moreover, we give an example of initial data satisfying the nonlinear smallness condition, while the norms of each component are arbitrarily large. Our approach is inspired by the weighted Chemin-Lerner technique used for the incompressible Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L (1 , φ ; κ) ⁢ (G) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system which includes the Fokas-Olver-Rosenau-Qiao (FORQ) or modified Camassa-Holm (MCH) equation and the two-component MCH system as two special cases. The system is integrable in the sense of possessing a non-isospectral Lax pair with the spectrum depending on time t , and admits multi-peakon solutions in an explicit form. Furthermore, we establish the local well-posedness for the system in the Besov space B 2 , r s (T) with s > 3 / 2 , 1 ≤ r ≤ ∞ , where the key ingredients include the Friedrichs regularization method, the Littlewood-Paley decomposition theory, and the transport theory in Besov spaces. Then we derive a precise blow-up criteria, which is dependent of the parameters α (t) and γ (t). Moreover, by the intrinsic structure of the system, we obtain a new blow-up result for strong solutions with sufficient conditions on the initial data and parameters. The entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves. [ABSTRACT FROM AUTHOR]

By developing the Littlewood-Paley characterization for product homogeneous Lipschitz spaces Lip(#945;1, α2) (ℝn x ℝm) and Lipb (α1, α2) (ℝn x ℝm), and establishing a density argument for Lipb (α1, α2) (ℝn x ℝm in the weak sense, we give a Tb theorem for the generalized singular integral operator on Lipb(α1; α2) (ℝn x ℝm), where b(x, y) = b1(x)b2(y), b1, b2 are para-accretive functions on ℝn and ℝm, respectively. [ABSTRACT FROM AUTHOR]

In this work, we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant norms. In particular we prove that it is not possible for one component of the velocity field to tend to 0 too fast near blow up. We also introduce a space "almost" invariant under the action of the scaling such that if one component of the velocity field measured in this space remains small enough, then there is no blow up. [ABSTRACT FROM AUTHOR]