The existence of global weak solution for a generalized two-component shallow water system with higher-order inertia operators is proved. A special kind of global weak solution but not strong solution—traveling wave solution, is then presented.

In this paper, we prove the existence of the global conservative solutions of the two-component b-family equations by using the method in [3]. It is worth noting that we propose a new transformation method when transforming the original equation into a semilinear system. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise {[ du+(u∂_xu+∂_xP[u]) dt=βu dW, u(0,x)=u_0(x), P[u]=(1-∂_x^2)^-1(u^2+1/2(∂_x u)^2) ]. depends almost surely continuously on the deterministic initial data in H^s for s>3/2. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise.

In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space H^β with 1/2<β<1. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the L^2 energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when 1/2<3/4. We find a phenomenon that (a,u)_L^2 still decays even if the initial data does not possess L^2 smallness. Notably, the low-frequency smallness assumption is removed in the case with 1/2<3/4.

In this paper, we consider the hyperdissipative Navier-Stokes equations with fractional dissipation (-Δ)^β with β>1. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in B^-β-α_∞,1(𝕋^d) for any α>0. Moreover, we show the existence of a solution with arbitrarily small initial data in B^-β-α_∞,1(𝕋^d) (α>0) that grows arbitrarily large in Ḃ^-s_∞,∞(𝕋^d) for all s∈ℝ in arbitrarily small time. It is worth pointing out that B^-β-α_∞,1(𝕋^d) lies in the subcritical regime when 0<α<β-1. To the best of our knowledge, this is the first non-uniqueness result of the Navier-Stokes equations with initial data at the subcritical regularity. To show the sharpness of the above results, we establish the local well-posedness of the hyperdissipative Navier-Stokes equations with initial data in Ḃ^-β-α_∞,∞(𝕋^d) with α< 0.

In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in $H^s$ framework with $s>1$. Furthermore, by virtue of improved Fourier splitting method and the Littlewood-Paley decomposition theory, we then establish the optimal time decay rate for low regularity data.

In this paper, we consider the Cauchy problem of the Geng-Xue system with cubic nonlinearity. Firstly, we prove a blow-up criteria in the low besov space. Secondly, we prove the blow-up phenomenon by using the method which does not require any conservation law. Finally, we extend our results to the b-family of two-component system with cubic nonlinearity.

In this paper, we primarily establish ill-posedness results for the N-abc family of Camassa-Holm type equations, which incorporate both dissipation and dispersion effects, in the Besov space B1 infinity,1 by demonstrating norm inflation phenomena. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we study the generalized solution of Fractional Singular Burgers equation driving by | D|^1/2ξ. We establish a framework to describe the equations satisfied by generalized solutions, termed the Generalized Fractional Singular Burgers equation(GFSB), and prove its local well-posedness. Finally, we prove that the solution of GFSB can be the generalized solution of Fractional Singular Burgers equation for γ>3/2.

We consider the Cauchy problem for the Euler-Poincar & eacute; equations in Rd with a varying dispersion parameter alpha. Based on the convex entropy structure and the modified commutator estimates, we prove that the Euler-Poincar & eacute; equations have a uniform existence time with respect to alpha in Sobolev spaces Hs. Combining this with the Bona-Smith method, we obtain convergence of the solutions to the Euler-Poincar & eacute; equations as alpha -> 0 in the same space as the initial data. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations ut−uxxt+(f(u))x−(f(u))xxx+(g(u)+f″(u)2ux2)x+λ(u−uxx)=0. This includes the weakly dissipative Camassa-Holm equation and the weakly dissipative hyperelastic rod wave equation as special cases. Specifically, we establish three global existence results: one concerning the energy conservative weak solutions in a time-weighted H1 space, and the other two concerning strong solutions, which include the cases of small initial data and sign-changing initial data. Our results recover and extend many known results for several classical models.

In this paper, we investigate the global conservative solutions to the generalized Camassa-Holm equation with dual-power nonlinearities. By introducing a new set of variables, we transform the original equation into an equivalent semi-linear system, which allows us to establish the global existence of conservative solutions. Furthermore, for a given global conservative solution, we construct some auxiliary variables tailored to its specific structure and demonstrate that they satisfy a semi-linear system with a unique solution, thereby deriving the uniqueness of conservative solutions to the original equation.

In this paper, we consider the global existence and properties of H^1 martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions. The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method. The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. In particular, by means of the Girsanov-type transform for regular viscous approximations and the convergence of Skorokhod-Jakubowski representations, we are able to establish the one-sided supernorm estimate and space-time higher regularity of the first-order spatial derivative, and large-time behavior of the weak martingale solution in the stochastic framework.

This paper is devoted to the existence of solution for logarithmic Schr & ouml;dinger equations. In contrast to most studies, we consider that the potential is indefinite. With the help of Morse theory, the existence of nontrivial solutions for the above problem is obtained.

This paper is concerned with the Cauchy problem for the 3D incompressible magnetohydrodynamic (MHD) equations in supercritical Sobolev spaces. It is well known that the system is locally well-posed in subcritical Sobolev spaces, whereas the supercritical regime remains largely open. In this work, we establish norm inflation for the incompressible MHD equations, both with and without Laplacian dissipation, in supercritical Sobolev spaces, thereby revealing strong ill-posedness of the system at this regularity level. A distinctive feature of our approach is the introduction of a novel geometric construction, termed the “Magnetic-solo ansatz”, through which, for the ideal MHD system, norm inflation occurs exclusively in the magnetic field b in H^s with 0<s<5/2, while the H^s-norm of the velocity field u remains uniformly bounded. This asymmetric behavior shows that supercritical ill-posedness can be driven exclusively by the magnetic field, highlighting its essential role in the breakdown of well-posedness. Our findings fill a significant gap in the supercritical regularity theory for incompressible MHD and shed light on the distinct mechanisms governing the fluid and magnetic dynamics.

We study the Cauchy problem for the short-pulse equation on the circle, addressing global well-posedness, finite-time singularity formation, and global weak solutions. For small initial data we prove global existence in H^2(𝕊) via a sharp criterion expressed through two conserved quantities arising from the geometric correspondence with the sine-Gordon equation. For large data we exhibit a new class of solutions developing finite-time singularities, derived through Lagrangian coordinates. Finally, for any zero-mean initial datum in L^4(𝕊) we construct a unique global entropy weak solution, and we show that any weak solution in L^∞ ((0,T);𝒞_b(𝕊)) —not necessarily satisfying an entropy condition-is unique.

In this paper, we consider the Cauchy problem of the 3-component Degasperis-Procesi equation. Firstly, we discuss a local well-posedness result and a blow-up criterion in the low besov space. Secondly, we study the blow-up phenomenon by using the method which does not require any conservation law. Finally, we investigate some persistence properties.

Recently, Coiculescu and Palasek shows the non-uniqueness of solutions for the 3D incompressible Navier-Stokes equations with initial data in BMO^-1. Inspired by their breakthrough work, we develop their schemes for the incompressible magnetohydrodynamic equations and obtain a similar result in 5 dimensional case. More precisely, we construct two distinct global solutions with a initial data, which has nonvanishing velocity and magnetic fields in BMO^-1(𝕋^5).

This paper focuses on the d-dimensional (d >= 2) Boussinesq equation with fractional dissipation (-Delta)(alpha) on the torus. We show that the uniqueness property breaks down within the function space (LtLx infinity)-L-p for any p < 2 alpha/2 alpha-1 when 1 <= alpha < d+1/2 when and the function space (Lt2 alpha/2 alpha-1Lxq) for any q < infinity when 1 < alpha < (d + 1)/2 More-weak solutions construct are smooth outside a set of singular times with Hausdorff dimension 1 <= alpha < (d + 1)/2 20 arbitrarily small. This result is sharp, as weak-strong uniqueness holds in the space (LT2 alpha/2 alpha-1Lx infinity) (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, Al training, and similar technologies.

We study the Cauchy problem for a nonlocal evolution model arising in equatorial ocean flows. We prove global existence of solutions for small initial data in critical Besov spaces and show that these solutions become instantly spatially analytic. Our approach introduces exponentially weighted unknowns to capture the interplay between the dissipative linear operator and the nonlocal Hilbert transform, allowing precise control of the bilinear nonlinearity. This provides a unified framework to extend small-data global existence and analyticity results from classical fluid equations to a nonlocal dispersive-dissipative setting.