We develop a theory of invariant manifolds for phase-field system on some high-dimensional domains. The phase-field system is a generalized model describing the statics and dynamics of an interface between two phases of specific materials within the framework of Landau-Ginzburg theory. The underlying feature of the problem under consideration is the intrinsic hyperbolic nature. We prove the existence of a finite-dimensional Lipschitz manifold in the possible absence of the spectral gap condition. The manifold, which has a certain regularity, is locally forward invariant. Its basin of exponential attraction is a proper subspace of the phase space. Moreover, we prove that the manifold contains a global attractor with additional technical assumptions on the nonlinearities, which implies in particular that the basin of attraction coincides with the whole phase space. The manifold gives a geometric insight into the long-term asymptotic behaviors from the perspective of finite-dimensional reduction. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Local well-posedness of the compressible Navier-Stokes equations with a free boundary condition is considered in the scaling critical spaces. We prove local well-posedness for the Lagrange transformed compressible Navier-Stokes system in the homogeneous Besov space B-p,1(center dot)n/p (& Ropf;(n)(+)) & times; B-p,1(center dot)-1+n/p (& Ropf;(n)(+)) with the exponent n-1 < P < 2n-1 along Solonnikov's formulation. To show local well-posedness, we use end-point maximal L-1-regularity for the corresponding linear initial-boundary value problem of the Lam & eacute; equations associated with the velocity derived by the explicit Fourier symbols under the free boundary condition. (c) 2026 The Authors. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton–Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.
