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In this work, we investigate the Cauchy problem for the pressureless Euler–Navier–Stokes system in R3. We first establish the global small solutions of this system with critical regularity and then obtain the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). The proof crucially depends on non-standard product estimates and interpolations. In comparison with previous studies about time-decay by Choi and Jung [J. Math. Fluid Mech. 23, 99 (2021); arXiv:2112.14449], the smallness requirement of the low frequencies of initial data could be removed.

The present paper is dedicated to the Cauchy problem of the two dimensional chemotaxis-shallow water system in the critical Besov spaces. We first obtain the global solutions to this system with small initial data and then investigate the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). Moreover, the bacterial density is allowed to vanish here.

We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

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 condit...

In this paper, we investigate the Cauchy problem for the shallow water type equation u t + ∂ x 2 n + 1 u + 1 2 ∂ x ( u 2 ) + ∂ x ( 1 − ∂ x 2 ) − 1 u 2 + 1 2 u x 2 = 0 with low regularity data in the periodic settings. Firstly, we proved that the bilinear estimate related to the nonlinear term of the equation in space W s (defined in page 5) is invalid with s − n 2 + 1 . Then, the locally well-posed of the Cauchy problem for the periodic shallow water-type equation is obtained in H s ( T ) with s > − n + 3 2 , n ≥ 2 for arbitrary initial data. Thus, our result improves the result of Himonas and Misiolek (Commun. Partial Differ. Equ, 23(1998), 123–139.), where they have proved that the problem is locally well-posed for small initial data in H s ( T ) with s ≥ − n 2 + 1 , n ∈ N + with the aid of the standard Fourier restriction norm method.

An acoustic sensor packaged with polydimethylsiloxane film sandwich structure is proposed in this paper. The sensor uses a flat glass with a square hole in the middle as the substrate structure. The cavity length of the microfiber knot resonator is significantly changed by the applied acoustic pressure, which is ultimately manifested as a change in the strength of the sensor signal. The sensor has a good response at frequencies from 0.1 to 10 kHz, and has a sensitivity of 0.92 mV Pa−1 at 1 kHz acoustic frequency. The sensor studied in this paper has the advantages of wide frequency band, small size and low cost, and has a good application prospect in the field of acoustic signal detection.

We obtain the global large solutions to the compressible Navier–Stokes equations in $${\mathbb {R}}^2$$ . The solution is large in the sense that there is no smallness assumption applied to one component of the initial incompressible velocity.

The present paper is dedicated to the global well-posedness issue for the Boussinesq system with the temperature-dependent viscosity in R 2 . Motivated by the work [Abidi and Zhang, Adv. Math. 305 (2017) 1202–1249] on the critical dissipation κ | D | α θ with α = 1 in the temperature equation of the system, we consider the extension of their work to the supercritical dissipation for 2 / 3 α ≤ 1 .

In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in ${\mathbb R}^3$. We establish the global well-posedness of the MHD system if the initial data satisfies \begin{document}$(ρ_0, u_0, H_0)∈ L^{∞}({\mathbb R}^3)× H^s({\mathbb R}^3)× H^s({\mathbb R}^3)$\end{document} with \begin{document}$\frac{1}{2} and \begin{document}$0 for some small \begin{document}$c>0$\end{document} which only depends on \begin{document}$\underline{ρ}, \overline{ρ}$\end{document} . As a byproduct, we also get the decay estimate of the solution.

In this paper, we study the Cauchy problem for the MHD-Boussinesq system with the temperature-dependent viscosity. We first establish the local well-posedness for this system in R d ( d = 2 , 3 ) by fully using the advantage of weighted function generated by heat kernel and Fourier localization technique. As a byproduct, we present a Beale–Kato–Majda type blow-up criterion. Then, under the assumptions on the viscosity coefficient is sufficiently close to some positive constant in L ∞ norm, we can also get a global solution for this system in R 2 .

Applying similar arguments as introduced in Bona–Smith [4] , we mainly study the inviscid limit of the inhomogeneous incompressible Navier–Stokes equations in R d . Some results of local uniform estimates on solutions of the Navier–Stokes equations, independent of the viscosity, are also obtained.

In this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in Rn with initial data in some critical Besov spaces. Under the assumptions that ...

The local existence of smooth solutions to the inviscid Hall-MHD system has been obtained since Chae, Degond and Liu [Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, {31} (2014), 555--565]. However, as we known, how to construct the global small solutions to the inviscid Hall-MHD system is still an open problem. In the present paper, we give a positive answer in $ \mathbb{T}^3$ when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition., Comment: 9pages

We obtain the global small solutions to the generalized Oldroyd-B model without damping on the stress tensor in R n . Our result give positive answers partially to the question proposed by Elgindi and Liu (Remark 2 in Elgindi and Liu (2015)). The proof relies heavily on the trick of transferring dissipation from u to τ , and a new commutator estimate which may be of interest for future works. Moreover, we prove a global result of inviscid limit of two dimensional Oldroyd type models in the Sobolev spaces. The convergence rate is also obtained simultaneously.

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.

In this paper, we investigate the Cauchy problem for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations in critical Besov spaces. We aim at proving the local and global well-posedness for respectively large and small initial data having critical Besov regularity, without assumptions of small density variation.

In the present paper, we consider the global well-posedness of the density-dependent incompressible flow of liquid crystals in $\mathbb{R}^{2}$ . The local existence and uniqueness of the system are obtained without the assumption of small density variation. The global well-posedness is proved when the initial density and liquid crystal orientation are small. However, the initial velocity field is allowed to be arbitrarily large.

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3] , Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. 1 p 6 . The main novelty of this work is to apply the CRW theorem obtained by Coifman, Rochberg, Weiss in [11] to get a new a priori estimate for an elliptic equation with variable coefficients. The uniqueness of the solution also relies on a Lagrangian approach as in [16] , [17] , [18] .

In this paper, we consider the global wellposedness of the 3-D density-dependent incompressible flow of liquid crystals in the critical Besov spaces. We first get the local solution for this system without assumptions of small density variation and then extend this local solution to be global under the smallness assumptions on initial density, initial orientation field and the horizontal components of the initial velocity field. This result implies the global well-posedness of this system with large initial vertical velocity component.

In this paper, we consider the 2-D nonhomogeneous incompressible magnetohydrodynamic equations with variable viscosity and variable conductivity. We obtain the global existence of solutions for this system with initial data in the scaling invariant Besov spaces and without size restriction for the initial velocity and magnetic field.

The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier–Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (2012) [2] , and (2013) [3] to a lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as L 2 in R 3 . Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations.

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in \begin{document}$\mathbb{R}^d \, (d≥2)$\end{document} by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on \begin{document}$\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$\end{document} , \begin{document}$\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$\end{document} and \begin{document}$\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$\end{document} . This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in \begin{document}$\mathbb{R}^2$\end{document} provided the initial cell density \begin{document}$n_0$\end{document} and the initial chemical concentration \begin{document}$c_0$\end{document} are doubly exponential small compared with the initial velocity field \begin{document}$u_0$\end{document} .

In this paper, we mainly investigate the Cauchy problem for the Phan-Thein-Tanner (PTT) model. The PPT model can be viewed as a Navier-Stokes equations couple with a nonlinear transport system. This model is derived from network theory for the polymeric fluid. We study about the global well posedness of the PTT model in critical Besov spaces. When the initial data is a small perturbation over around the equilibrium, we prove that the strong solution in critical Besov spaces exists globally., Comment: arXiv admin note: substantial text overlap with arXiv:1811.07462

The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in $\mathbb{R}^n(n=2,3)$. This result allows to construct global solutions for a class of highly oscillating initial velocity. The proof uses the special structure of the system. Moreover, our theorem extends the previous result by Zhu [19] and covers the recent result by Chen and Hao [4]., Comment: We have made some small changes. arXiv admin note: text overlap with arXiv:1708.06172 by other authors

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier–Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for p ∈ ( 1 , 4 ) and a ∈ B ˙ p , 1 2 p ( R 2 ) that the solution mapping H a : F ↦ ∇ Π to the 2-D elliptic equation div ( ( 1 + a ) ∇ Π ) = div F is bounded on B ˙ p , 1 2 p − 1 ( R 2 ) . More precisely, we prove that ‖ ∇ Π ‖ B ˙ p , 1 2 p − 1 ≤ C ( 1 + ‖ a ‖ B ˙ p , 1 2 p ) 2 ‖ F ‖ B ˙ p , 1 2 p − 1 . The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15] , [16] , [17] . When the viscosity coefficient μ ( ρ ) is a positive constant, we prove that (1.2) is globally well-posed.

In this paper, we study the Cauchy problem for the incompressible inhomogeneous magnetohydrodynamic (MHD) system in R 3 . We obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field and magnetic field in the critical Besov spaces. Our methods mainly rely upon the Fourier frequency localization and Bony's paraproduct decomposition.

This paper is dedicated to the study of the Navier-Stokes-Landau-Lifshitz system. We obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field. Our methods mainly rely upon the Fourier frequency localization and Bony's paraproduct decomposition.

The present paper is dedicated to the global large solutions and incompressible limit for the compressible Navier-Stokes system in $\mathbb{R}^d$ with $d\ge 2$. We aim at extending the work by Danchin and Mucha (Adv. Math., 320, 904--925, 2017) in $L^2$ structure to that in a critical $L^p$ framework. The result implies the existence of global large solutions initially from large highly oscillating velocity fields., The final version for publish

The present paper is dedicated to the global large solutions and incompressible limit for the compressible flow of liquid crystals under the assumption on almost constant density and large volume viscosity. The result is based on Fourier analysis and involved so-called critical Besov norm., 19 pages

In this paper, we prove that as the viscosity and resistivity go to zero, the solution of the Cauchy problem for the homogeneous incompressible MHD system converges to the solution of the ideal MHD system in the L p type space. The convergence rate is also obtained simultaneously. To the author’s knowledge, this is the first instance in which considering the inviscid and non-resistive limit for MHD system in the L p ( p ≠ 2 ) type space.

In this paper, we consider the global well-posedness of the incompressible magnetohydrodynamic system with anisotropic Besov initial data in the critical Besov spaces. Under the nonlinear smallness condition on the critical anisotropic Besov norm of the horizontal components of the initial velocity field and magnetic field, we get the system has a unique global solution. Our result allows to construct global solutions for a class of highly oscillating initial data of Cannone's type.

In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{align*} u_{t}+\partial_{x}^{3}u + \tfrac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\tfrac{1}{2} u_{x}^{2}\right]=0, \ \ x\in {\mathbf T}={\mathbf R}/2\pi \lambda, \end{align*} with low regularity data and $\lambda\geq1$. By applying the bilinear estimate in $W^{s}$, Himonas and Misiołek (Commun. Partial Diff. Eqns., 23 (1998), 123-139) proved that the problem is locally well-posed in $H^{s}([0, 2\pi))$ with $s\geq {1}/{2}$ for small initial data. In this paper, we show that, when $s < {1}/{2}$, the bilinear estimate in $W^{s}$ is invalid. We also demonstrate that the bilinear estimate in $Z^{s}$ is indeed valid for ${1}/{6} < s < {1}/{2}$. This enables us to prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with ${1}/{6} < s < {1}/{2}$ for small initial data.