Using the ∂¯-steepest descent method associated with the Riemann-Hilbert problem, we study the long-time asymptotics of solutions to the defocusing fifth-order modified Korteweg-de Vries (mKdV) equation with a non-vanishing background, which admits discrete spectra. Taking the self-similar variable ξ=x/(5t), we derive distinct long-time asymptotic behaviors in two solitonless regions with bounded ξ: ξ > 6 and ξ < 1.2, respectively. In the region ξ > 6, the phase function possesses four real and six purely imaginary phase points. Accordingly, the asymptotic expansion of the solution consists of the leading term from the non-vanishing background, an order term from the continuous spectrum and a residual error from the ∂¯-problem. In the region ξ < 1.2, ten stationary points of the phase function lie off the jump contour, and the corresponding asymptotic expansion contains the nonvanishing background plus a residual error of order O(t−1).

This paper considers the well-posedness and continuous property of a 4-parameter family of nonlocal evolution equations with (k + 1)-degree nonlinearities, known as k-abc equation. It was introduced by Himonas and Mantzavinos (2016) [21] as a generalization of the ab-equation for studying the peakon traveling waves and includes the well-known Camassa-Holm, Novikov and Fokas-Olver-Rosenau-Qiao equations as special case. It is shown that the k-abc equation has a unique solution in the Besov space B addition, we show that this dependence is sharp by proving that the data-to-solution map is not uniformly continuous. Our results cover some works on well-posedness and non-uniform continuity of the Camassa-Holm type equations (Fu et al. (2013) [14], Li et al. (2021) [26], Wu et al. (2021) [32], etc.). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. 5 2,1(R) with solutions depending continuously on initial data. In 2

In this paper, we prove that the continuity of the solution for the generalized Camassa-Holm equation cannot be improved to the Hölder continuity. To be precise, the solution of the generalized Camassa–Holm equation belongs to 𝒞([0,T];B^s_p,r) but not to 𝒞^α ([0,T];B^s_p,r) with any α∈ (0,1) .

The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $B^{s}_{p,r}$ or $\dot{B}^{s}_{p,r}$ norms can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together with the structure of the Boussinesq equations help us to obtain the global smooth solutions.

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Our aim of this paper is two fold. Firstly, we construct a new initial data and present a simple proof of the ill-posedness of B^s_p,∞ solution of the Euler equations and the surface quasi-geostrophic equation, which covers the result obtained by Cheskidov and Shvydkoy in and Misiołek and Yoneda in . Secondly, we prove the failure of the B^s_p,∞-convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.

For the Fornberg–Whitham equation, the local well-posedness in the critical Besov space B_p, 1^1+1/p(ℝ) with 1≤ p <∞ has been studied in Guo (Nonlinear Anal RWA 70:103791, 2023). However, for the endpoint case p=∞ , whether it is locally well-posed or ill-posed in B_∞ , 1^1(ℝ) is still open. In this paper, we prove that the Fornberg–Whitham equation is well-posed in the critical Besov space B_∞ , 1^1(ℝ) with solutions depending continuously on initial data, which is different from that of the Camassa–Holm equation (Guo et al. in J Differ Equ 327:127, 2022). In addition, we show that this dependence is sharp by showing that the solution map is not uniformly continuous on the initial data.

For the Fornberg-Whitham equation, the local well-posedness in the critical Besov space B 1+( 1)/ p( p,1) (R) with 1 <= p < infinity has been studied in [7](Guo, Nonlinear Anal. RWA., 2023). However, for the endpoint case p = infinity, whether it is locally well-posed or ill-posed in B-1 (infinity,1) (R) is still unknown. In this paper, we prove that the Fornberg-Whitham equation is well-posed in the critical Besov space B-infinity,1(1 ) (R) with solutions depending continuously on initial data, which is different from that of the Camassa-Holm equation [6](Guo et al., J. Differ. Equ., 2022). In addition, we show that this dependence is sharp by showing that the solution map is not uniformly continuous on the initial data.

In this paper, we consider the Cauchy problem for a generalized rotation b-family system on the real line and prove that the data-to-solution map of this problem is not uniformly continuous in B(p,r)(s )x B-p,r(s-1). We have removed the restriction of mu(b + 1) = A sigma in Holmes et al. [Nonuniform dependence of the R-b-family system in Besov spaces. Z Angew Math Mech. 2021;101(8):18] and Yang [Non-uniform continuity of the solution map to the rotation-two-component Camassa-Holm system. J Differ Equ. 2020;268:4423-4463].

In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in L^∞ (0,T;B^s_2,r(ℝ)) to the inviscid Burgers equation as the filter parameter α tends to zero with the given initial data u_0∈ B^s_2,r(ℝ) . Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in B^s_2,r(ℝ) .

In this paper, based on a perturbation argument and fully exploring the nonlinear structure of the considered system, we establish the global existence to the three-dimensional generalized incompressible Hall-MHD equations for a class of large initial data, whose L-p, 1 <= p <= infinity norms can be arbitrarily large. Our obtained result improves considerably the recent result in Li et al. [A class large solution of the3DHall-magnetohydrodynamic equations. J Differ Equ. 2020;268(10):5811-5822] to the generalized HallMHD system with 0 <= alpha <= 1 equipping with different viscosity and magnetic diffusion coefficients.

In this paper, we prove that the Cauchy problem for a generalized Camassa–Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space B^1_∞ ,1(ℝ) . It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa–Holm equation is ill-posed in B^1_∞ ,1(ℝ) , here we turn our attention to a higher-order nonlinear generalization of Camassa–Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space B^1_∞ ,1(ℝ) which leads to ill-posedness.

In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data (ρ0,u0) in Bp,∞s−1(R)×Bp,∞s(R) with s>max⁡{2+1p,52} and 1≤p≤∞, we show that any energy bounded solution starting from (ρ0,u0) does not converge back to (ρ0,u0) in the metric of Bp,∞s−1(R)×Bp,∞s(R) as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.

In this paper, we prove that the Cauchy problem for the Fornberg–Whitham equation is not locally well-posed in Bp,rs(R) with (s,p,r)∈(1,1+1p)×[2,∞)×[1,∞] or (s,p,r)∈{1}×[2,∞)×[1,2] by showing norm inflation phenomena of the solution for some special initial data.

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761–781, 2005). We prove that the solution map of the generalized Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces. Our result includes the present work Li et al. (Differ Equ 269:8686–8700, 2020) on Camassa–Holm equation with $$Q=1$$ and extends the previous non-uniform continuity in Sobolev spaces Mi and Mu (Monatsh Math 176:423–457, 2015) to Besov spaces.

This paper studies a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system as special case. It is shown that the solution map of this system is not uniformly continuous on the initial data in Besov spaces B_p, r^s-1(ℝ)× B_p, r^s(ℝ) with s>max{1+1/p, 3/2} , 1≤ p, r< ∞ . Our result covers and extends the previous non-uniform continuity in Sobolev spaces H^s-1(ℝ)× H^s(ℝ) for s>5/2 to Besov spaces (Nonlinear Anal., 2014, 111: 1-14). Compared with the generalized rotation b-family system considered by Holmes et al. (Z. Angew. Math. Mech., 2021), our non-uniform continuity is established in a broader range of Besov spaces.

In this paper, we consider the Cauchy problem to the tri‐dimensional compressible Navier‐Stokes‐Korteweg system with a specific choice on the Korteweg tensor in the whole space and establish the global solutions to the tri‐dimensional Navier‐Stokes‐Korteweg equations with a class of large initial data whose L2 norm can be arbitrarily large.

The aim is to present the inverse scattering transform (IST) with Riemann-Hilbert problem (RHP) for a higher-order extended modified Korteweg-de Vries (emKdV) equation with zero/nonzero boundary conditions (Z/NZBC) at infinity, where the emKdV equation contains third- and fifth-order dispersion coefficients matching with relevant nonlinearity terms, respectively. Under reflectionless condition, the RHP of emKdV equation is firstly solved when scattering data have two cases: multiple simple poles and higher-order poles, and corresponding formulae of multiple simple poles soliton and higher-order poles solution are shown in terms of determinants, respectively. One simple soliton and two simple soliton solutions are obtained in detail according to variable values of third- and fifth-order dispersion coefficients, the effect power of which are specially displayed in dynamic structures. In addition, as a very important application for solving higher-order solutions by RHP, we extend Laurent's series of residue conditions of idea to this higher-order emKdV equation.

In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data (ρ_0, u_0) in B_p, ∞^s-1(ℝ)× B_p, ∞^s(ℝ) with s>max{2+1/p, 5/2} and 1≤ p ≤∞, we show that any energy bounded solution starting from (ρ_0, u_0) does not converge back to (ρ_0, u_0) in the metric of B_p, ∞^s-1(ℝ)× B_p, ∞^s(ℝ) as time goes to zero, thus results in discontinuity of the data-to-solution map and ill-posedness.

In this paper, we show that the solution map of the two-component Novikov system is not uniformly continuous on the initial data in Besov spaces Bp,rs−1(R)×Bp,rs(R) with s>max{1+1p,32}, 1≤p≤∞, 1≤r<∞. Our result covers and extends the previous non-uniform continuity in Sobolev spaces Hs−1(R)×Hs(R) for s>52 to Besov spaces.

In this paper, we consider the solution map of the Cauchy problem to the Fokas–Olver–Rosenau–Qiao equation on the real line and prove that the solution map of this problem is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous results in Himonas and Mantzavinos (Nonlinear Anal 95:499–529, 2014) and Li et al. (J Math Fluid Mech 22:50, 2020).