Archive for Rational Mechanics and Analysis最新文献

We prove the existence of steady space quasi-periodic stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel ({{mathbb {R}}}times [-1,1]). These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash–Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin’s cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.

We establish (textrm{C}^{infty })-partial regularity results for relaxed minimizers of strongly quasiconvex functionals

subject to a q-growth condition (|F(z)|leqq c(1+|z|^{q})), (zin mathbb {R}^{Ntimes n}), and natural p-mean coercivity conditions on (Fin textrm{C}^{infty }(mathbb {R}^{Ntimes n})) for the basically optimal exponent range (1leqq pleqq q<min {frac{np}{n-1},p+1}). With the p-mean coercivity condition being stated in terms of a strong quasiconvexity condition on F, our results include pointwise (p, q)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (p, q)-growth conditions, our results extend the previously known exponent range from Schmidt’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for (p=1). We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la Fonseca and Malý (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997).

This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As has already been pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, concerns the properties of the projected image and its higher regularity, which we show to be at most of class (C^{2,1}). In arbitrary dimensions, we prove that the image map is globally of class (W^{3,BMO}), and locally of class (C^{2,1}) around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.

Consider the elliptic operator given by

for some smooth vector field (varvec{b}:{mathbb R}^drightarrow {mathbb R}^d) and a small parameter (varepsilon >0). Consider the initial-valued problem

for some bounded continuous function (u_0). Denote by (mathcal {M}_0) the set of critical points of (varvec{b}) which are stable stationary points for the ODE (dot{varvec{x}} (t) = varvec{b} (varvec{x}(t))). Under the hypothesis that (mathcal {M}_0) is finite and (varvec{b} = -(nabla U + varvec{ell })), where (varvec{ell }) is a divergence-free field orthogonal to (nabla U), the main result of this article states that there exist a time-scale (theta ^{(1)}_varepsilon ), (theta ^{(1)}_varepsilon rightarrow infty ) as (varepsilon rightarrow 0), and a Markov semigroup ({p_t: tge 0}) defined on (mathcal {M}_0) such that

for all (t>0) and (varvec{x}) in the domain of attraction of (varvec{m}) [for the ODE (dot{varvec{x}}(t) = varvec{b}(varvec{x}(t)))]. The time scale (theta ^{(1)}) is critical in the sense that, for all time scales (varrho _varepsilon ) such that (varrho _varepsilon rightarrow infty ), (varrho _varepsilon /theta ^{(1)}_varepsilon rightarrow 0),

for all (varvec{x} in mathcal {D}(varvec{m})). Namely, (theta _varepsilon ^{(1)}) is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [20] we extend this result finding all critical time-scales at which the solution of the initial-valued problem (0.2) evolves smoothly in time and we show that the solution (u_varepsilon ) is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of (varvec{b}).

We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility (m_varepsilon >0) in the Allen–Cahn equation tends to zero in a subcritical way, i.e., (m_varepsilon = m_0 varepsilon ^beta ) for some (beta in (0,2)) and (m_0>0). The proof proceeds by showing via a relative entropy argument that the solution to the Navier–Stokes/Allen–Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term (m_varepsilon H_{Gamma _t}) in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.

This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation spheres, we interpret the equilibrium shape of the drop as a global minimizer of the sum of its surface energy and the electrostatic repulsive energy between the charges under fixed drop volume. For all model parameters, we establish the existence of generalized minimizers that consist of at most a finite number of components “at infinity”. We also give several existence and non-existence results for classical minimizers consisting of only a single component. In particular, we identify an asymptotically sharp threshold for the number of charges to yield existence of minimizers in a regime corresponding to macroscopically large drops containing a large number of charges. The obtained non-trivial threshold is significantly below the corresponding threshold for the Rayleigh model, consistently with the ill-posedness of the latter and demonstrating a particular regularizing effect of the charge discreteness. However, when a minimizer does exist in this regime, it approaches a ball with the charge uniformly distributed on the surface as the number of charges goes to infinity, just as in the Rayleigh model. Finally, we provide an explicit solution for the problem with two charges and a macroscopically large drop.

This paper is concerned with the 2-dim two-phase interface Euler equation linearized at a pair of monotone shear flows in both fluids. We extend the Howard’s Semicircle Theorem and study the eigenvalue distribution of the linearized Euler system. Under certain conditions, there are exactly two eigenvalues for each fixed wave number (kin mathbb {R}) in the whole complex plane. We provide sufficient conditions for spectral instability arising from some boundary values of the shear flow velocity. A typical mode is the ocean-air system in which the density ratio of the fluids is sufficiently small. We give a complete picture of eigenvalue distribution for a certain class of shear flows in the ocean-air system.

A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schrödinger equations. Matrix displacement convexity is stronger than the classical notions of displacement convexity, and its verification (formal and rigorous) relies on matrix differential inequalities along the density flows. The matrical nature of these differential inequalities upgrades dimensional functional inequalities to their intrinsic dimensional counterparts, thus improving on many classical results. Applications include turnpike properties, evolution variational inequalities, and entropy growth bounds, which capture the behavior of the density flows along different directions in space.

We consider the two-dimensional capillary-gravity water waves problem where the free surface (Gamma _t) intersects the bottom (Gamma _b) at two contact points. In our previous works (Ming and Wang in SIAM J Math Anal 52(5):4861–4899; Commun Pure Appl Math 74(2), 225–285, 2021), the local well-posedness for this problem has been proved with the contact angles less than (pi /16). In this paper, we study the case where the contact angles belong to ((0, pi /2)). It involves much worse singularities generated from corresponding elliptic systems, which have this strong influence on the regularities for the free surface and the velocity field. Combining the theory of singularity decompositions for elliptic problems with the structure of the water waves system, we obtain a priori energy estimates. Based on these estimates, we also prove the local well-posedness of the solutions in a geometric formulation.

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side (f in L^q) for (q > n+2). In the case of the heat equation, we also show the optimal (C^{1-varepsilon }) regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are (C^{1,alpha }) in the parabolic obstacle problem and in the parabolic Signorini problem.