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/).
This paper is a continuation of our previous work (Zhong and Zhou (2024) [40]), where the global well-posedness for the Vaigant-Kazhikhov model of compressible nematic liquid crystal flows in the whole plane was derived. We extend such a result to the case of general two-dimensional bounded domains with Navier-slip boundary condition for the velocity and Neumann boundary condition for the director field. To overcome the difficulties brought by boundary, some new estimates based on the effective viscous flux are established. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We investigate the global bifurcation structure of the radial nodal solutions to the coupled elliptic equations {-Delta u + u = u3 + /3uv2 in B1, -Delta v + v = v3 + /3u2vin B1, u, v is an element of H10,r(B1). Here B1 is a unit ball in R3 and /3 is an element of R the coupling constant is used as bifurcation parameter. For each k, the unique pair of nodal solutions +/- wk with exactly k-1 zeroes to the scalar field equation-Delta w+w = w3 generate exactly four synchronized solution curves and exactly four semi-trivial solution curves to the above system. We obtain a fairly complete global bifurcation structure of all bifurcating branches emanating from these eight solution curves of the system, and show that for different k these bifurcation structures are disjoint. We obtain exact and distinct nodal information for each of the bifurcating branches, thus providing a fairly complete characterization of nodal solutions of the system in terms of the coupling. To this end, new a-priori estimate and Liouville type results for non-cooperative elliptic systems are developed. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper focuses on the open problems proposed by Gan and Zhang (2008) [14], and the sharp criteria of blow-up and the global existence of solutions to the Davey-Stewartson system with a defocusing perturbation are first solved. Moreover, the dynamic properties of the blow-up solutions are obtained, including the lower bound rate and the sharp criteria for the existence of stable standing waves with any positive frequency. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This work investigates the boundary stabilization of flows in star-shaped and tree-shaped networks of open channels governed by the Saint-Venant equations with a friction term. Due to the existence of the friction term, the steady states are nonuniform. We show that any such network can be stabilized with only controls at the terminal nodes of the network, even when there are no controls at the nodes inside the network. The number of controls is optimal. The main tool we use is the Lyapunov approach, and the main challenge is that the state-of-the-art Lyapunov functions developed for the Saint-Venant equations with source terms cannot be used. In this work, we manage to construct a new efficient and explicit Lyapunov function and, in turn, we give explicit ranges of the control tuning parameters that depend only on the values of the given non-uniform steady states at the ends of the branches. Moreover, this Lyapunov function also improves the existing conditions found in the last decade for a single channel modeled by the Saint-Venant equations. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper investigates a supersonic reacting jet issuing from a two-dimensional, finite-length divergent nozzle. To model the jet, a free boundary value problem for the steady two-dimensional Zeldovdovich-von Neumann-During (ZND) combustion equations is formulated. The flow is assumed to satisfy a slip condition along the nozzle walls, with its state prescribed at the nozzle inlet. Two external conditions are examined. First, when the nozzle exhausts into a vacuum, the global existence of a supersonic reacting jet is established using the method of characteristics. Furthermore, a sufficient condition on the nozzle wall geometry is provided that leads to the formation of a vacuum region inside the nozzle; it is shown that any such vacuum region must be adjacent to one wall. Second, when the nozzle is surrounded by a static atmosphere whose pressure is lower than the jet pressure at the exit, the analysis demonstrates the formation of singularities (i.e., shock waves) in the jet. This result provides a rigorous mathematical confirmation of the physical phenomenon described in Sec. 148 of the classical text Supersonic Flow and Shock Waves. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The objective of this paper is to investigate the uniqueness theorems for solutions of the following mixed-order semi-linear elliptic system in R-4: { (-Delta)(1/2)u(1)(x) = f(1) (u(1), u(2), u(3,) u(4)), (-Delta)u(2)(x) = f(2) (u(1), u(2), u(3,) u(4)), (-Delta)(3/2)u(3)(x) = f(3) (u(1), u(2), u(3,) u(4)) (-Delta)(2)u(4)(x) = f(4) (u(1), u(2), u(3,) u(4)) where u(j) >= 0(j = 1, 2, 3), u(4) may change signs, f(i)(u(1), u(2), u(3), u(4))(i = 1, 2, 3, 4) are monotone continuous functions (which can be either increasing or decreasing), and f(4) satisfies the condition of finite total curvature, i.e., f(R)(4) f(4)(u(1), u(2), u(3), u(4))(x)dx < +infinity. Initially, we establish an integral representation formula equivalent to (0.1) and demonstrate a crucial asymptotic property concerning u4 under specific assumptions. Finally, under certain assumptions, we prove the uniqueness theorem of the classical solutions of the above equations under two different conditions by using the moving method in conjunction with integral inequalities. Our work establishes uniqueness results for semi-linear elliptic system without the assumption that the nonlinearity is a nondecreasing function (only assuming that the nonlinearity is monotone function). We also extend the mixed-order elliptic equations with general nonlinear terms in R-4 to the higher-dimensional case of R-n. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The damped Euler equations model compressible fluid flow in a porous medium or under the influence of a frictional background. This work provides the first rigorous justification of this model as the compressible fluid limit of the frictional Boltzmann equation (FBE) in the whole space, uniformly in the Knudsen number and globally in time. For well-prepared initial data, we prove that the solution F epsilon of the FBE remains close to a local Maxwellian whose hydrodynamic parameters satisfy the damped Euler system, and we establish the explicit convergence rate sup x is an element of R-3 ||F-epsilon (t, x, v)-M-[rho,M-& ufr;,M-theta](t, x, v) /root mu|| L-2 R-v(3) less than or similar to (1 + t)(-3/8) epsilon(1/2) , for any Knudsen number epsilon is an element of (0, 1] and for all t >= 0. The proof relies on a delicate micro-macro decomposition involving Burnett functions, the construction of a singular-in- high-order energy functional, and the use of the rapid time-decay of solutions to the damped Euler equations in three dimensions. The result illustrates how friction can prevent shock formation and allow a global-in-time hydrodynamic limit, in contrast to the pure Euler limit which is only local in time.
Existence and non-existence for weighted fourth order elliptic problems in exterior domains are studied. The Liouville type results are obtained for the subcritical case and a unique nontrivial nonnegative radial fast-decay solution is obtained for the supercritical case. It is also seen that the nontrivial nonnegative radial solution does not exist for the critical case.
This paper establishes several weak KAM type results for Hamiltonian systems with infinite dimensional normal directions, which we call infinite co-dimensional Hamiltonian systems. We first study the Mather's minimization problem for a class of infinite co-dimensional Hamiltonians. Then we prove the existence of weak solutions of the corresponding Hamilton-Jacobi equation which satisfy the equation a-almost everywhere, where a is a projected minimal measure satisfying a divergence equation. The abovementioned weak solution transforms the infinite co-dimensional Hamiltonian system into a system with an integrable structure in the weak sense. Finally, we apply our results to a class of one-dimensional nonlinear Schr & ouml;dinger equations. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper is concerned with the existence, the multiplicity and the precise spatial decay of forced waves to the following heterogeneous Fisher-KPP equation with a degenerate shifting environmentut=uxx+u(a(x−ct)−u),t>0,x∈R, where c>0 is a shifting speed of the resource and a(z) can be any given positive function satisfying a(−∞)=α>0=a(+∞),00 the existence/nonexistence of forced waves decaying non-exponentially at z=+∞ are solely determined by the convergence/divergence of the integral ∫z0+∞σ˜2(z)dz. Precisely speaking, we prove that there exists no forced wave decaying exponentially when c≥2α; and there exists a forced wave solution decaying exponentially for each c∈(0,2α), which is also the unique or the minimal forced wave. For the case ∫z0+∞σ˜2(z)dz<+∞, we prove that for any c>0 there exist infinitely many forced waves decaying non-exponentially and only the maximal forced wave is not in L1([z0,+∞)). We also obtained the precise decaying rates of all the forced waves at z=+∞.
Let X be a compact nonsingular real algebraic set and let M be a compact C infinity submanifold of X, with dim.X = n and dimM = d. Under the assumption 2d + 1 <= n we prove that M can be approximated by nonsingular algebraic subsets of X if and only if certain mod 2 homology classes of X associated with the inclusion map M -> X are algebraic. This allows us to give a rather precise description of the approximation properties of k-regulous maps from X to the unit p-sphere S-p in the space of all C-k maps, where k is a nonnegative integer and 2p >= n + 1 A map : X -> S-p is called k-regulous if it is of class C(k )and its restriction to a Zariski open dense subset of X is a regular map. (c) 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
This paper is the second part of our series of works to establish L2 estimates and existence theorems for the 8 operators in infinite dimensions. In this part, we consider the most difficult case, i.e., the underlying space is a general pseudoconvex domain. In order to handle this longstanding unsolved problem, we introduce several new concepts and techniques, which have independent interest and may be applied in other places. (c) 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We consider networks of elastic strings with end masses, where the coupling is modeled via elastic springs. The model is representative of a network of nonlinear strings in which the strings are coupled to elastic bodies. The coupled system converges to the classical string-network model with Kirchhoff and continuity transmission conditions as the spring stiffness terms tend to infinity and the masses at the nodes vanish. Due to the presence of point masses at the nodes, the boundary conditions become dynamic and the corresponding first-order system of quasilinear balance laws exhibits nonlocal boundary conditions. We demonstrate well-posedness in the sense of semi-global classical solutions, namely on arbitrarily large time intervals provided that the initial and boundary data are sufficiently small, and we observe additional regularity at the masses. We prove local and global-local exact boundary controllability of a star-like network when controls are active at the endpoints of the string-spring-mass system except for one clamped end. At multiple nodes, a complex smoothing pattern appears, leading to asymmetric control spaces when springs and masses are present. Furthermore, the rank of the Laplacian matrix at the junction is crucial for controllability, particularly in models containing wave equations with degeneration at dynamic boundaries, which may be interpreted as damage in mechanical vibration systems where some springs are missing.
In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problemdiv(∇u1−|∇u|2)+f(u)=0,x∈RN in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality∫RN(1−1−|∇u|2)dx≥CN,p(∫RN|u|pdx)NN+p, for p>2⁎ as well as the characterization of the optimal constant CN,p in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Pohožaev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever N≥4; in particular, this settles a previously open problem in the case N=5.
Korn's inequalities show that the L2-norm of Vu can be controlled by the L2-norm of Sym(Vu), which only has d(d+ 1)/2 components. In (Chipot, 2021 [22]) Chipot posed the question of how many scalar measurements are needed to have a Korn-type control on Vu when u is in H01 (Q) and H1(Q), introducing the minimal numbers N(d, Q) and N '(d, Q) respectively. He proved general bounds and calculated several low-dimensional values of N, N '. We reframe Chipot's problem in the language of rank-one convexity and quasiconvexity and obtain a purely algebraic characterisation of when such inequalities hold, which yields the sharp bounds N(d, Q) = 2d(1-o(1)) N '(d, Q) = 2d-1. As a consequence, we recover and streamline several of Chipot's results, we obtain a dimension-optimal Korn inequality and several sharp estimates for the best constant for various Korn-type inequalities. Generalisations to the rectangular case and to general Lp estimates are also considered. The central new ingredient of our approach is a systematic connection between laminates and martingales which produces explicit families of laminates realising these bounds. This method is of independent interest in the calculus of variations: for instance, we use it to obtain a new quick and quantitative proof of Ornstein's non-inequality, valid for all first order homogeneous operators in R2 & times;2 and for a large class of operators in general dimensions (including Korn's del u+del ut 2 and del u+del ut 2-div(u)Id d ). (c) 2026 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
In this work, we study harmonic analysis in self-similar measures. A set A is called a spectral eigenvalue set of u if there exists ACR such that the family (alpha lambda: alpha Alpha} are spectra for mu. Given a Hadamard triple (q, D, L), & Lstrok;aba and Wang [33] proved that the associated self-similar measure Hq,D is spectral. We establish that the set T={t is an element of Z: (q, D, tL) forms a Hadamard triple} supseteq \{p \in Z / (gcd(p,q)) = 1\} constitutes a spectral eigenvalue set for HD. Furthermore, we demonstrate that for any prescribed Beurling dimension se [0, 1], the corresponding spectra log have the cardinality of the continuum. This result provides a complete answer to the question posed by Kong, Li and Wang [30]. As an application, we characterize the eigenvalue sets for N-Bernoulli convolutions, proving that A is an eigenvalue set if and only if ACT for some TET. 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, Al training, and similar technologies.
