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In this paper, we consider the Cauchy problem for the Burgers equation in the line. We shall prove that this problem is ill-posed in the Sobolev space H s (R) with 1 ≤ s < 3 2 in the sense of "norm inflation" by constructing an explicit example of initial data. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problem of the incompressible Hall-MHD system and prove the global well-posedness in the scaling critical class B ˙ p , ∞ - 1 + 3 p (R 3) × (B ˙ p , ∞ - 1 + 3 p (R 3) ∩ L ∞ (R 3)) for 3 < p < ∞ . Moreover, we also refine the smallness conditions and show that our global well-posedness holds for initial data whose B ˙ p , ∞ - 1 + 3 p (R 3) -norm is large, provided that some weaker norm is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we study the following chemotaxis system with signal-dependent motility arising from tumor invasion u t = Δ (φ (v) u) + ρ u - μ u l , x ∈ Ω , t > 0 , v t = Δ v + w z , x ∈ Ω , t > 0 , w t = - w z , x ∈ Ω , t > 0 , z t = Δ z - z + u , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 1) , where the parameters ρ ≥ 0 , μ ≥ 0 and l > 1 are constants, the motility function φ (v) satisfies φ (v) ∈ C 3 ([ 0 , + ∞)) , φ 1 ≤ φ (v) ≤ φ 2 and | φ ′ (v) | ≤ φ 3 with φ 1 , φ 2 , φ 3 > 0. The purpose of this paper is to prove that the existence of global bounded solution for 1 ≤ n ≤ 3 . For n ≥ 4 , we prove that the nonnegative classical solution (u, v, w, z) is globally bounded if l > n 2 . In addition, we also show that all the global bounded solution will converge to the non-trivial constant steady state exponentially. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for Cahn–Hilliard equation in R n ( n ∈ Z + , n ≥ 3 ). First, based on the higher-order norm estimates of solutions and the mollifier technique, we obtain the local well-posedness of strong solutions. Then, by using pure energy method, standard continuity argument together with negative Sobolev norm estimates, one proves the global well-posedness and time decay estimates provided that the H n 2 + 1 -norm of initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for a system of nonlinear Schrödinger equations with quadratic interactions and initial data belonging to a class of analytic Gevrey functions. Here, we present a local well-posedness result in the analytic Gevrey class G σ , s × G σ , s by proving some bilinear estimates in Bourgain's space with exponential weight. Furthermore, we prove that the obtained solution can be extended to any time T > 0 , as long as the radius of the spatial analyticity σ is bounded below by c T - 2 , if 0 < a < 1 / 2 , or c T - 4 , if a > 1 / 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the possible singular points of suitable weak solutions to the 3D co-rotational Beris-Edwards system. Inspired by the work of He et al. (J. Nonlinear Sci. 29:2681–2698, 2019) and Wang et al. (Nonlinearity 32:4817–4833, 2019) for Navier–Stokes equations, we established a new partial regularity criteria for co-rotational Beris-Edwards system. As an application, we prove the known Minkowski dimension of the potential interior singular set of suitable weak solutions of the co-rotational Beris-Edwards system is 7 6 (≈ 1.167) . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem of Keller–Segel–Navier–Stokes system with fractional diffusion. Making use of Fourier localization technique and Littlewood-Paley theory, we establish the global well-posedness of mild solution for small initial data in mixed time-space Besov spaces. Furthermore, we obtain the well-posedness of mild solution in time-weighted Besov spaces by Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the maximum norm of velocity divergence controls the breakdown of smooth (strong) solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier–Stokes equations with zero heat conduction. The results indicate that the nature of the blowup for the full compressible Navier–Stokes equations with zero heat conduction of viscous flow is similar to the barotropic compressible Navier–Stokes equations and does not depend on the temperature field. The main ingredient of the proof is a priori estimate to the pressure field instead of the temperature field and weighted energy estimates under the assumption that velocity divergence remains bounded. Furthermore, the initial vacuum states are allowed, and the viscosity coefficients are only restricted by the physical conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the energy equality for axisymmetric weak solutions of the 3D Navier–Stokes equations. The classical Shinbrot condition says that if the weak solution u of the Navier–Stokes equations belongs L q (0 , T ; L p (R 3)) with 1 q + 1 p = 1 2 and p ≥ 4 , then u must satisfy the energy equality. For the axisymmetric Navier–Stokes equations, in our previous paper, we found that it is enough to impose the Shinbrot condition to u ~ = u r e r + u z e z . The recent papers (Chiun-Chuan et al., Commun PDE 34(1–3):203–232, 2009; Koch et al., Acta Math 203(1):83–105, 2009) tell us if 0.1 | u ~ | ≤ 1 r , 0 < r ≤ 1 , then u is smooth , therefore the energy equality holds. It is natural to ask the relation between a priori bound on the velocity and the energy conservation. The aim of this paper is to investigate this problem. We shall prove that if 0.2 | u ~ | ≤ 1 r d , 0 < r ≤ 1 , d > 1 , and 0.3 ∇ u ~ ∈ L 6 d - 4 2 d - 1 (0 , T ; L 2 (R 3)) , then the energy equality holds. [ABSTRACT FROM AUTHOR]

We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness of the problem and regularity of the solution. Our main theorem is a criterion which roughly states that if the flow remains C 1 , α -regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., C ∞ in time and space, assuming that the initial data is smooth. Our main focus is on the regularity of the shape of the drop. Indeed, due to the appearance of Taylor cones, which are singularities with Lipschitz-regularity, we expect the C 1 , α -regularity assumption to be optimal. We also quantify the C ∞ -regularity via high order energy estimates which, in particular, implies the well-posedness of the problem. [ABSTRACT FROM AUTHOR]

This paper reports the analytical results on the non-isothermal stationary fluid flow inside thin vertical annular region formed by two co-axial cylinders. The annulus is packed with the fluid-saturated sparsely packed porous medium which is cooled through the side wall. The flow is governed by the prescribed pressure drop between the top and bottom walls which are maintained at uniform, but different temperatures. The main objective of this work is to propose the approximate model describing the effective flow using rigorous asymptotic analysis with respect to the thickness of the annular region. Starting from the dimensionless Darcy-Brinkman-Boussinesq system endowed with the appropriate boundary conditions, we derive the explicit asymptotic approximation clearly showing the effects of the porous structure and thermal transfer. We also provide the theoretical error analysis in order to indicate the order of accuracy of the proposed model and justify its usage. [ABSTRACT FROM AUTHOR]

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s > 1 2 , to those of BO in the deep-water limit (as δ → ∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s > 3 2 in the deep-water limit and s ⩾ 2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂ x (u k) for k ⩾ 2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s ⩾ 3 4 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments. [ABSTRACT FROM AUTHOR]

This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients where Du is the symmetric part of the gradient ∇u. We first establish an ε-regularity criterion involving both the excess functional of the symmetric gradient Du and Wolff potentials of the nonhomogeneous term f to guarantee the local vanishing mean oscillation (VMO)-regularity of Du in an open subset Ωu of Ω with full measure. Such an ε-regularity criterion leads to a pointwise Wolff potential estimate of Du, which immediately infers that Du is partially C0-regular under appropriate assumptions. Finally, we give a local continuous modulus estimate of Du. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness and large time behavior of solutions to the 3D incompressible magneto-micropolar equations. By virtue of the $ L_p-L_q $ L p − L q estimate obtained through the spectral decomposition of the linearized magneto-micropolar equations, we show the existence and uniqueness of small $ L_3 $ L 3 -strong solutions of the equations with small initial data. Then basing on this result, we derive sharp time decay estimates of the $ L_3 $ L 3 -strong solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of equations of viscous compressible and heat-conducting full magnetohydrodynamic (MHD) steady flows in a horizontal layer under the gravitational force and a large temperature gradient across the layer. We assume as boundary conditions, periodic conditions in the horizontal directions, while in the vertical directions, slip-boundary is assumed for the velocity, vertical conditions for the magnetic field, and fixed temperature or fixed heat flux are prescribed for the temperature. The existence of stationary solution in a small neighborhood of a stationary profile close to hydrostatic state is obtained in Sobolev spaces as a fixed point of some nonlinear operator. [ABSTRACT FROM AUTHOR]

In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented example consists of two disjoint, simply connected patches. The rates of growth for both curvature and perimeter in this example are at least algebraic. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for the three-dimensional electron magnetohydrodynamic (EMHD) equations. Firstly, we establish sufficient conditions to guarantee the energy (magnetic helicity) balance of weak solutions for the EMHD equations based on the magnetic field, which can be viewed as an analogue of famous Lions' energy balance criterion of the Navier–Stokes equations for the EMHD equations. Secondly, in the spirit of recent works due to Berselli and Chiodaroli (Nonlinear Anal 192: 111704, 2020), as reported by Berselli (Three-Dimensional Navier–Stokes Equations for Turbulence. Academic Press, London, 2021), Berselli (Mathematics 11(4): 1–16, 2023), Berselli (J Differ Equ 368: 350–375, 2023), Berselli and Georgiadis (Nonlinear Differ Equ Appl 31(33): 1–14, 2024), we present energy (magnetic helicity) preservation criteria in terms of the current density in this system for both the whole space and the torus cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global well-posedness for the 3D Boussinesq equations. Firstly, we prove the global existence and uniqueness of the solution in critical space for small initial data. Then, aimed at the large initial data giving rise to global solution, we obtain a certain class of large solutions with initial data slowly varying in one direction. The proof relies on the special structure of the equation and some new refined estimates for the velocity and temperature. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the well-posedness issue for the 3D incompressible Hall-MHD system obtained from kinetic models. Our analysis strongly relies on the use of the Fourier analysis. We establish the global existence of smooth solutions for a class of large initial data, this result implies the initial velocity and magnetic field can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution $ (u,b) $ (u , b) of the non-resistive MHD equations for the initial data $ u_0\in \dot B^{\frac {d}{p}-1}_{p,1}(\mathbb {R}^{d}) $ u 0 ∈ B ˙ p , 1 d p − 1 (R d) and $ b_0\in \dot B^{\frac {d}{p}}_{p,1}(\mathbb {R}^{d}) $ b 0 ∈ B ˙ p , 1 d p (R d) with $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d. Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to $ 1\le p \le \infty $ 1 ≤ p ≤ ∞ from $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires $ 1\le p\le 2d $ 1 ≤ p ≤ 2 d yet. [ABSTRACT FROM AUTHOR]

This paper is concerned with the uniqueness and energy conservation of weak solutions for Electron-MHD system. Under suitable assumptions, we first show that the Electron-MHD system has a unique weak solution. In addition, we show that weak solution conserves energy if ∇ × b ∈ L 2 (0 , T ; L 4 (R d)) (d ≥ 2) or ∇ × b ∈ L 4 d + 8 d + 4 0 , T ; L 4 d + 8 d + 4 (R d) (d = 2 , 3 , 4) . [ABSTRACT FROM AUTHOR]

This paper is devoted to the temporal decay of solutions of a coupled parabolic-elliptic equations in R 2 . It is proved that weak solutions of the equations decay to zero in L 2 , ∞ × L 2 without a uniform rate, and these decay estimates are optimal. Furthermore, the uniform logarithmic decay estimates of weak solutions are established when initial data are in L 1 ∩ L 2 . In addition, the temporal decay estimates of weak solutions at the sharp rate are also shown. The proofs are based on the Fourier splitting method and the generalized Ladyzhenskaya inequality for weak type spaces. [ABSTRACT FROM AUTHOR]

In this paper, we show global smoothness of solutions to the Navier–Stokes/Poisson–Nernst–Planck system for the transport and diffusion of ions in electrolyte solutions. For the multi-ionic species model, the key step to obtain global smoothness is to enhance the regularity of ions density and the velocity field by using the L ∞ L p -estimate of the charge density, which is from a clear energy-dissipation equality. As their direct consequence, utilizing Duhamel's principle, we obtain global smoothness for the multi-ionic species case in R 2 . Moreover, the decay rate of solutions for the coupled Navier–Stokes/Poisson–Nernst–Planck system with two-ionic species is given at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal control problem for the 2D MHD equations with memory in bounded domains with Dirichlet boundary conditions, where the time needed to reach the desired state plays an essential role. We first prove the existence of optimal solutions. Then we establish the first-order necessary and second-order sufficient optimality conditions. The second-order optimality ones obtained in the paper seem to be optimal in the sense that the gap between them is minimal. [ABSTRACT FROM AUTHOR]

In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed, subject to the boundary conditions (∇n − nS(x, n, c)∇c) · ν = ∇c · ν = 0 and u = 0, and suitably regular initial data (n0(x), c0(x), u0(x)), where Ω ⊂ ℝ3 is a bounded domain with smooth boundary ∂Ω. Here S is a chemotactic sensitivity satisfying ∣S(x, n, c)∣ ≤ CS(1 + n)−α with some CS > 0 and α > 0. The greatest contribution of this paper is to consider the large time behavior of solutions for the system (KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system (KSS) decays to ( 1 ∣ Ω ∣ ∫ Ω n 0 , 1 ∣ Ω ∣ ∫ Ω n 0 , 0) exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient CS of chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS. [ABSTRACT FROM AUTHOR]

A generalized incompressable magnetohydrodynamics system is considered in this paper. Furthermore, results of global well-posednenss are established with the aid of Littlewood–Paley decomposition and Fourier localization method in mentioned system with small initial condition in the variable exponent Fourier–Besov–Morrey spaces. Moreover, the Gevrey class regularity of the solution is also achieved in this paper. [ABSTRACT FROM AUTHOR]

This paper proves the continuation criteria of the strong solution for the 3D double-diffusive convection system involving the deformation tensor in Vishik spaces. As a bi-product, our theorem also improve some well-known results on conditional regularity for the particular case of classical Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time t. We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids. [ABSTRACT FROM AUTHOR]

The zero-relaxation limit of Euler–Poisson systems for ion dynamics in a slow time scale leads to drift-diffusion equations. This fact was justified in previous works. This paper concerns global error estimates between solutions of Euler–Poisson systems and that of drift-diffusion equations. In the proof, we use Sobolev inequalities and employ a multiplier related to the electric potential. [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]

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $ -\frac {\mu }{(1+t)^{\lambda }}n_iu_i(i=1, 2) $ − μ (1 + t) λ n i u i (i = 1 , 2) for $ -1 − 1 < λ < 1 , μ > 0. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions. [ABSTRACT FROM AUTHOR]