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In this paper, we investigate singularity formation for the high-order generalized Fokas–Olver–Rosenau–Qiao equation which include the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation), and we are devoted to understanding how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we show the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, we investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces B p , r s (R) with 1 ≤ p , r ≤ ∞ and s > m a x { 1 + 1 p , 3 2 } , which improve the previous work in Sobolev spaces H s (R) = B 2 , 2 s (R) with s > 3 2 (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces B p , r s (R) with 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and s > m a x { 1 + 1 p , 3 2 } . [ABSTRACT FROM AUTHOR]

This paper investigates a noteworthy phenomenon within the framework of Majda-Biello systems, wherein the inclusion of lower-order terms can enhance the well-posedness of the system. Specifically, we investigate the initial value problem (IVP) of the following system: u t + u xxx = - v v x , v t + α v xxx + β v x = - (u v) x , (u , v) | t = 0 = (u 0 , v 0) ∈ H s (R) × H s (R) , x ∈ R , t ∈ R , where α ∈ R \ { 0 } and β ∈ R . Let s ∗ (α , β) be the smallest value for which the IVP is locally analytically well-posed in H s (R) × H s (R) when s > s ∗ (α , β) . Two interesting facts have already been known in literature: s ∗ (α , 0) = 0 for α ∈ (0 , 4) \ { 1 } and s ∗ (4 , 0) = 3 4 . Our key findings include the following: For s ∗ (4 , β) , a significant reduction is observed, reaching 1 2 for β > 0 and 1 4 for β < 0 . Conversely, when α ≠ 4 , we demonstrate that the value of β exerts no influence on s ∗ (α , β) . These results shed light on the intriguing behavior of Majda-Biello systems when lower-order terms are introduced and provide valuable insights into the role of α and β in the well-posedness of the system. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a new weakly dissipative quasilinear shallow-water waves equation, which can be formally derived from a model with the effect of underlying shear flow from the incompressible rotational two-dimensional shallow water in the moderately nonlinear regime by Wang, Kang and Liu (Appl Math Lett 124:107607, 2022). Considering the dissipative effect, the local well-posedness of the solution to this equation is first obtained by using Kato's semigroup theory. We then establish the precise blow-up criterion by using the transport equation theory and Moser-type estimates. Moreover, some sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied according to the different real-valued intervals in which the dispersive parameter θ being located. It is noteworthy that we need to overcome the difficulty induced by complicated nonlocal nonlinear structure and different dispersive parameter ranges to get corresponding convolution estimates. [ABSTRACT FROM AUTHOR]

We study the initial-boundary vaule problem of the long and short wave equations posed on the half line with initial datas (u 0 , n 0) ∈ H s 0 (R +) × H s 1 (R +) and boundary datas (h , f) ∈ H 2 s 0 + 1 4 (R +) × H s 1 (R +) . We show the local well-posedness by giving the bilinear estimates of the coupling terms for the equations in suitable spaces of Bourgain type. Moreover, we consider results concerning ill-posedness for the system. Finally, the system is proved to be globally well-posed in Sobolev spaces H 1 (R +) × L 2 (R +) . [ABSTRACT FROM AUTHOR]

This work is devoted to study the relation between regularity and decay of solutions of some dissipative perturbations of the Korteweg-de Vries equation in weighted and asymmetrically weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we delve into an analysis of the fifth-order Fokas-Olver-Rosenau-Qiao (FORQ) equation. We demonstrate local well-posedness for this equation under the condition that an initial data u 0 ∈ H s (R) for some s > 7 / 2 . We examine global existence and blow-up scenario. Additionally, we study the persistence properties of the equation in weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m , n) ∈ C k (R) ∩ W k , 1 (R) with k ∈ N ∪ { 0 } . This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂ t m (t , x) = ∂ x [ m (t , x) (u (t , x) - ∂ x u (t , x)) (u (- t , - x) + ∂ x (u (- t , - x))) ] , where m (t , x) = 1 - ∂ x 2 u (t , x) . Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class C k ∩ W k , 1 . Moreover, we derive criteria for blow-up of the local solution in this class. [ABSTRACT FROM AUTHOR]

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm. [ABSTRACT FROM AUTHOR]

This paper investigates the Cauchy problem for the ab-family of equations with cubic nonlinearities, which contains the integrable modified Camassa–Holm equation ( a = 1 3 , b = 2 ) and the Novikov equation ( a = 0 , b = 3 ) as two special cases. When 3 a + b ≠ 3 , the ab-family of equations does not possess the H 1 -norm conservation law. We give the local well-posedness results of this Cauchy problem in Besov spaces and Sobolev spaces. Furthermore, we provide a blow-up criterion, the precise blow-up scenario and a sufficient condition on the initial data for the blow-up of strong solutions to the ab-family of equations. Our blow-up analysis does not rely on the use of the conservation laws. [ABSTRACT FROM AUTHOR]

We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device are reformulated as a transmission problem related to the interaction between waves and a fixed partially immersed structure. By imposing the conservation of the total fluid-OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in Bocchi et al. (ESAIM Proc Surv 70:68–83, 2021). We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE. [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]

This paper is concerned with the local well-posedness, wave breaking, blow-up rate and global existence of solutions for a weakly dissipative coupled Camassa–Holm system. First, by using Kato's theory, we obtain the local well-posedness of solutions in the Sobolev space. We then derive the wave breaking mechanism of solutions, and obtain the blow-up rate of blow-up solutions. Finally, we establish the global existence of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain T d , for space dimensions d = 2 , 3 . We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case k ≥ 0 ; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces H s , for any s > 1 + d / 2 . We expect this regularity to be optimal, due to the degeneracy of the system when k ≈ 0 . We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the local well-posednesss for the Cauchy problem of a generalized Camassa–Holm (gCH) equation in Besov spaces B p , 1 1 + 1 p with 1 ≤ p < + ∞. Then we gain two blow-up criterions, and present two new blow-up results. Finally, we prove the ill-posedness of the gCH equation in critical Besov spaces B 2 , r 3 2 , r ∈ (1 , + ∞ ]. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in B ∞ , 1 1 (R) . Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly continuous in B ∞ , 1 1 (R) . [ABSTRACT FROM AUTHOR]

This paper is concerned with the local well-posedness, wave breaking, blow-up rate, global existence and uniqueness of solutions for a class of integrable equations, including the classical and modified Camassa–Holm equation, the classical and modified Degasperis–Procesi equation, and Novikov equation, etc. Firstly, we obtain the local well-posedness of solutions by using Kato's theorem. Then we derive the wave breaking mechanism of solutions. Particularly, when k is even, we give two sufficient conditions on the initial datum for the occurrence of wave breaking, and obtain the upper bound of the maximal existence time and the blow-up rate. Finally, we establish the global existence and uniqueness of solutions. [ABSTRACT FROM AUTHOR]

We study a class of nonlinear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion. We first establish the local well-posedness (existence, uniqueness and continuous dependence) with initial data (u0, b0) in critical Besov spaces B p , 1 d p + 1 × B p , 1 d p with 1 ≤ p ≤ ∞, and give a lifespan T of the solution which depends on the norm of the Littlewood—Paley decomposition (profile) of the initial data. Then, we prove the global existence in critical Besov spaces. In particular, the results of global existence also hold in Sobolev space C ([ 0 , ∞) ; H s (S 2)) × (C ([ 0 , ∞) ; H s − 1 (S 2)) ∩ L 2 ([ 0 , ∞) ; H s (S 2))) with s> 2, when the initial data satisfies ∫ S 2 b 0 d x = 0 and ‖ u 0 ‖ B ∞ , 1 1 (S 2) + ‖ b 0 ‖ B ∞ , 1 0 (S 2) ≤ ϵ . It's worth noting that our results imply some large and low regularity initial data for the global existence. [ABSTRACT FROM AUTHOR]

We obtain local well-posedness result for the generalized Hall-magneto-hydrodynamics system in Besov spaces B ˙ ∞ , ∞ - (2 α 1 - γ) × B ˙ ∞ , ∞ - (2 α 2 - β) (R 3) with suitable indexes α 1 , α 2 , β and γ. As a corollary, the hyperdissipative electron magneto-hydrodynamics system is globally well-posed in B ˙ - (2 α 2 - 2) ∞ , ∞ (R 3) for small initial data. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Cauchy problem and wave-breaking phenomenon for a sine-type modified Camassa–Holm (alias sine-FORQ/mCH) equation. Employing the transport equations theory and the Littlewood–Paley theory, we first establish the local well-posedness for the strong solutions of the sine-FORQ/mCH equation in Besov spaces. In light of the Moser-type estimates, we are able to derive the blow-up criterion and the precise blow-up quantity of this equation in Sobolev spaces. We then give a sufficient condition with respect to the initial data to ensure the occurance of the wave-breaking phenomenon by trace the precise blow-up quantity along the characteristics associated with this equation. [ABSTRACT FROM AUTHOR]

We are concerned with the well-posedness of the nonlinear wave system, which is a first-order hyperbolic system, in the vicinity of a right-angled spatial corner. The problem can be expressed as an initial boundary value problem (IBVP) involving a second-order hyperbolic equation in a spatial domain with a corner. The main difficulty in establishing the local well-posedness of the problem arises from the lack of smoothness in the spatial domain due to the presence of the corner point. Additionally, the Neumann-type boundary conditions on both edges of the corner angle do not satisfy the linear stability condition, posing challenges in obtaining higher-order a priori estimates for the boundary terms in the analysis. To address the corner singularity, modified extension methods will be employed in this paper. Furthermore, new techniques will be developed to control the boundary terms, leveraging the observation that the boundary operators are co-normal. [ABSTRACT FROM AUTHOR]

In this paper we first investigate the local well-posedness and global existence with large damping coefficient for the Cauchy problem of a Camassa-Holm type equation with cubic and quartic nonlinearities in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present norm inflation which implies ill-posedness for the equation in critical Besov spaces. [ABSTRACT FROM AUTHOR]

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in R N , i ∂ t u + Δ 2 u - | x | - b | u | 2 σ u = 0 , where σ > 0 and b > 0 . We first study the local well-posedness in H ˙ s c ∩ H ˙ 2 , for N ≥ 5 and 0 < s c < 2 , where s c = N 2 - 4 - b 2 σ . Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in H ˙ s c ∩ H ˙ 2 with 0 ≤ s c < 2 . Finally, we study the phenomenon of L σ c -norm concentration for finite time blow up solutions with bounded H ˙ s c -norm, where σ c = 2 N σ 4 - b . Our main tool is the compact embedding of L ˙ p ∩ H ˙ 2 into a weighted L 2 σ + 2 space, which may be seen of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, a modified Green-Naghdi system with the effect of the Coriolis force is derived, which is a model in the equatorial oceanography to describe the propagation of large amplitude surface waves. The effects of the Coriolis force caused by the Earth’s rotation and nonlinearities on local well-posedness and traveling wave solutions are then investigated. Employing Kato’s theory, the local well-posedness in Sobolev space H s with s > 5 2 is established. Based on the qualitative method combined with the bifurcation method of dynamical systems, the classification of all traveling wave solutions, all possible phase portraits of bifurcations and exact traveling wave solutions to this system are obtained under various conditions about the parameters depending on the value of the rotation Ω [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation in some special spaces E p , r s , which mix Lebesgue spaces and homogeneous Besov spaces. Then we present a global existence result and provide a sufficient condition for strong solutions to blow up in finite time for the equation. The global existence is a new result comparing to the circle case in Yin (SIAM J Math Anal 36:272–283, 2004), since Yin (SIAM J Math Anal 36:272–283, 2004) proved that all solutions of (HS) equation with initial data that are not constant functions blow up in finite time. Finally, we give the ill-posedness and the unique continuation of the (HS) equation. [ABSTRACT FROM AUTHOR]

This paper proposes a higher-order μ -Camassa–Holm equation, which is regarded as a higher-order extension of the μ -Camassa–Holm equation, and preserves some properties of the μ -Camassa–Holm equation. We first show that the equation admits the peaked traveling wave solution, which is given by a Green function of the momentum operator. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established. Finally, the blow-up criterion and wave breaking mechanism for solutions with certain initial profiles are studied. It turns out that all the nonlinearities even the first-order nonlinearity may have the effect on the blow up. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem associated with a fractional Korteweg–de Vries-type equation ∂ t u + D β ∂ x u + | u | α ∂ x u = 0 , 0 < β ≤ 2. We prove the existence and uniqueness of solution for initial data in H s (R) where 3 2 - 3 β 8 < s < α when β ∈ (0 , 1 ] . We follow the lines of the compactness method combined with refined Strichartz estimates for the linear equation regularized by the viscous term ε ∂ x 2 u , ε > 0 . Since the existence of sufficiently smooth solutions for the equation is not obvious, we can not apply the classical compactness argument directly. [ABSTRACT FROM AUTHOR]

In this paper, we study a hydrodynamic system that models smectic-A liquid crystal flow in R 3 . This model consists of the Navier–Stokes equations for fluid velocity coupled with a fourth-order equation for the layer variable. The main purpose is to analyze the well-posedness and asymptotic behavior of strong solutions. We first prove the local well-posedness through the higher-order a prior estimates of the solution and Galerkin method. Then, we establish the existence of global strong solution provided that ‖ u 0 ‖ H ˙ 1 2 + ‖ φ 0 ‖ H ˙ 3 2 + ‖ φ 0 ‖ H ˙ 7 2 is sufficiently small. Finally, we show the temporary decay estimates for the higher-order derivatives of strong solution by using the negative Sobolev norm estimates. [ABSTRACT FROM AUTHOR]

In this paper we consider a Cauchy problem on the self-dual relativistic non-abelian Chern–Simons–Higgs model, which is the system of equations of sl (n , C) (n ≥ 2) -valued matter field ϕ and gauge field A. Based on the frequency localization as well as the null structure we show local well-posedness in the Sobolev space H s + 1 2 × H s for s > 1 4 . We also prove that the solution flow map (ϕ (0) , A (0)) ↦ (ϕ (t) , A (t)) fails to be C 2 at the origin of H s × H σ when σ < max (0 , s - 1) regardless of s ∈ R . [ABSTRACT FROM AUTHOR]

We consider the Euler–Korteweg system with space periodic boundary conditions x ∈ T d . We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data. [ABSTRACT FROM AUTHOR]

This paper proves the local well-posedness of strong solutions to a two-phase model with magnetic field and vacuum in a bounded domain Ω ⊂ ℝ3 without the standard compatibility conditions. [ABSTRACT FROM AUTHOR]

We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in R N i u t - L a u + λ | x | - b | u | α u = 0 , L a = - Δ + a | x | 2 , where λ = ± 1 , α , b > 0 and a > - (N - 2) 2 4 . We first establish sufficient conditions for global existence and blow-up in H a 1 (R N) for λ = 1 , using a Gagliardo–Nirenberg-type estimate. In the sequel, we study local and global well-posedness in H a 1 (R N) in the H 1 -subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in H a 1 (R N) , for the mass-supercritical and energy-subcritical case. [ABSTRACT FROM AUTHOR]

Consideration in this paper is the generalized fractional Camassa–Holm equation. The local well-posedness is established in Besov space B 2 , 1 s 0 with s 0 = 2 ν - 1 2 for ν > 3 2 and s 0 = 5 2 for 1 < ν ≤ 3 2 . Then, with a given analytic initial data, the analyticity of the solutions in both variables, globally in space and locally in time, is established. Finally, a blow-up criterion is presented. [ABSTRACT FROM AUTHOR]

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity u α - 1 u for α ∈ [ 1 , 5 ] and initial data u 0 ∈ H 1 (T) + L 2 (R) . We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ( α = 2 ) we obtain global well-posedness in the space C (R , H 1 (T) + L 2 (R)) via Gronwall’s inequality. [ABSTRACT FROM AUTHOR]

In this paper, we establish the local well-posedness of strong solutions to the three dimensional viscous non-resistive MHD system with internal surface wave in low regularity Sobolev spaces. Due to the absence of magnetic diffusion, the proof will be based on the new Stokes estimates involving in optimal surface regularity, the weighted energy estimates and the higher order horizontal regularity estimates. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for nonlinear Schrödinger equations with repulsive inverse-power potentials i ∂ t u + Δ u − c | x | − σ u = ± | u | α u , c > 0. We study the local and global well-posedness, finite time blow-up and scattering in the energy space for the equation. These results extend a recent work of Miao-Zhang-Zheng (2018, ) to a general class of inverse-power potentials and higher dimensions. [ABSTRACT FROM AUTHOR]

The article is devoted to the study of Cauchy problem to the Rayleigh–Bénard convection model for the micropolar fluid in two dimensions. We first prove the unique local solvability of smooth solution to the system when the system has only velocity dissipation, and then establish a criterion for the breakdown of smooth solutions imposed only the maximum norm of the gradient of scalar temperature field. Finally, we show the global regularity of the system with zero angular viscosity. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrödinger equation ut = iαuxx + βu2ūx + γ∣u∣2ux + i∣u∣2u on the half-line with inhomogeneous boundary condition. We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces. Moreover, we show that the nonlinear part of the solution on the half-line is smoother than the initial data. [ABSTRACT FROM AUTHOR]

This paper is devoted to understanding how higher-order nonlinearities affect the dispersive dynamics. As a prototype, a class of higher-order Camassa–Holm equations which can be viewed as a generalization of the Camassa–Holm equation is studied. The local well-posedness of the Cauchy problem in Besov spaces and Sobolev spaces is established. Furthermore, a delicate analysis is employed to investigate the formation of singularities, and some sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solutions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to a new integrable two-component Novikov equation with Lax pairs and bi-Hamiltonian structures. Ons the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. On the other hand, we prove that the strong solutions maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has ( N + 1 )-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported. [ABSTRACT FROM AUTHOR]

This paper discusses the Cauchy problem for the fractional Navier–Stokes–Coriolis equation (FNSC). The FNSC equation refers to that obtained by replacing the Laplacian in the Navier–Stokes–Coriolis equation by the more general operator (- Δ) α with α > 0 . We prove the time-local existence and uniqueness of the mild solution for every Ω ∈ R \ { 0 } and u 0 ∈ H ˙ s (R 3) 3 with 1 / 4 < α ⩽ 3 / 2 , 3 / 2 - α < s < 5 / 4 . Furthermore, we give a lower bound for the time interval of its local existence in terms of | Ω | and ‖ u 0 ‖ H ˙ s . It follows from our characterization that the existence time T of the solution can be arbitrarily large provided the speed of rotation | Ω | is sufficiently fast. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem in n-dimensional space for the system of Navier–Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong, and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper demonstrate the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Young's inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz–Leray projection, and the boundedness of the Riesz transform are developed in mixed-norm Lebesgue spaces. These analysis results are topics of independent interests, and they are potentially useful in other problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to a new integrable two-component Novikov equation, lax pairs and bi-Hamiltonian structures. Firstly, the local well-posedness in nonhomogeneous Besov spaces is established by using the Littlewood–Paley theory and transport equations theory. Then, we verify the blow-up that occurs for this system only in the form of breaking waves. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

The Vlasov–Poisson–Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We consider the system in a convex domain with the Cercignani–Lampis boundary condition. We construct a uniqueness local-in-time solution based on an L ∞ -estimate and W 1 , p -estimate. In particular, we develop a new iteration scheme along the characteristic with the Cercignani–Lampis boundary for the L ∞ -estimate, and an intrinsic decomposition of boundary integral for W 1 , p -estimate. [ABSTRACT FROM AUTHOR]

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood's lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space B 2 , 1 1 2 . Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion. [ABSTRACT FROM AUTHOR]

In this paper, we establish the local well-posedness for the quasi-geostrophic equations and obtain a blow-up criterion of smooth solutions in the Besov–Lorentz spaces. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Cauchy problem of the multi-component Novikov equation. We establish the local well-posedness in a range of the Besov spaces by using Littlewood–Paley decomposition and transport equation theory. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]