Archive for Rational Mechanics and Analysis最新文献

The liquid drop model was introduced by Gamow in 1928 and Bohr–Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface (Sigma =partial Omega ) in ({mathbb {R}}^3) that is critical for the energy

under the volume constraint (|Omega | = m). The term (mathrm{Per,} (Omega ) ) corresponds to the surface area of (Sigma ). The associated Euler–Lagrange equation is

where (H_Sigma ) stands for the mean curvature of the surface, and where (lambda in {mathbb {R}}) is the Lagrange multiplier associated to the constraint (|Omega |=m). Round spheres enclosing balls of volume m are always solutions; they are minimizers for sufficiently small m. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a “pearl necklace” with an axis located on a large circle, with a shape close to a Delaunay’s unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since, for the closely related constant mean curvature problem (H_Sigma = lambda ), the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

In this pages, we consider the p order nonlinear half wave Schrödinger equations

on the plane (mathbb {R}^2) with (1<ple 2). We prove the global well-posedness of this equation in (L_x^2 H_y^s(mathbb {R}^2) cap H_x^1 L_y^2(mathbb {R}^2))((frac{1}{2}le s le 1)), which is the first global well-posedness result of nonlinear half wave Schrödinger equations. With the global well-posedness in the energy space for the focusing equation and the study on the solitary wave in Bahri et al. (Commun Contemp Math 23(05), 2020), we complete the proof of the stability of the set of ground states. Moreover, we consider the half wave Schrödinger equations on (mathbb {R}_{x}times mathbb {T}_{y}), which can also be called the wave guide Schrödinger equations on (mathbb {R}_{x}times mathbb {T}_{y}). Using a similar approach in the analysis of the Cauchy problem of half wave Schrödinger equations on (mathbb {R}^2), we can also deduce the global well-posedness of p ((1<ple 2)) order wave guide Schrödinger equations in (L_x^2 H_y^s(mathbb {R}times mathbb {T}) cap H_x^1 L_y^2(mathbb {R}times mathbb {T})) with (frac{1}{2}le s le 1). With the global well-posedness in the energy space for the focusing wave guide Schrödinger equations and the study on the ground states in Bahri et al. J Dyn Differ Equ 1–43, 2021), we complete the proof of the orbital stability of the ground states with small frequencies.

In this paper, we study the interaction between two closely spaced rigid inclusions suspended in a Stokes flow. It is well known that the stress significantly amplifies in the narrow region between the inclusions as the distance between them approaches zero. To gain deeper insight into these interactions, we derive high-order derivative estimates for the Stokes equation in the presence of two rigid inclusions in two dimensions. Our approach resonates with the method used to handle the incompressibility constraint in the standard convex integration scheme. Under certain symmetric assumptions on the domain, these estimates are shown to be optimal. As a result, we establish the precise blow-up rates of the Cauchy stress and its higher-order derivatives in the narrow region.

For a genuinely nonlinear (2times 2) hyperbolic system of conservation laws, assuming that the initial data have a small (textbf{L}^infty ) norm but a possibly unbounded total variation, the existence of global solutions was proven in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like (t^{-1}). Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with a faster decay rate: (hbox {Tot.Var.}bigl {u(t,cdot )bigr }le C t^{alpha -1}). For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with (Vert {bar{u}}Vert _{textbf{L}^infty } le varepsilon _1) small enough, solutions with a fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to 1 by further shrinking the value of (varepsilon _1>0). An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.

We prove the conjectured first order expansion of the Levy–Lieb functional in the semiclassical limit, arising from Density Functional Theory (DFT). In particular, we prove a general asymptotic first order lower bound in terms of the zero point oscillation functional and the corresponding asymptotic upper bound in the case of two electrons in one dimension. This is accomplished by interpreting the problem as the singular perturbation of an Optimal Transport problem via a Dirichlet penalization.

We prove a conjecture regarding the asymptotic behavior at infinity of the Kantorovich potential for the Multimarginal Optimal Transport with Coulomb and Riesz costs.

We establish a new sharp estimate of the order of the vanishing of solutions to parabolic equations with variable coefficients. For real-analytic leading coefficients, we prove a localised estimate of the nodal set, at a given time-level, that generalises the celebrated one of Donnelly and Fefferman. We also establish Landis type results for global solutions.

We consider the problem of the asymptotic stability of constant flows in the Aw-Rascle-Zhang traffic model. By using a perturbative approach, we are able to compute where the perturbation is mainly localised in space for a given time, based on the localisation of the perturbation initially. These new ideas can be applied to various other one dimensional models of hyperbolic conservation laws with relaxations.

Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. Using the Itô–Schrödinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime.

A fundamental result in global analysis and nonlinear elasticity asserts that given a solution (mathfrak {S}) to the Gauss–Codazzi–Ricci equations over a simply-connected closed manifold ((mathcal {M}^n,g)), one may find an isometric immersion (iota ) of ((mathcal {M}^n,g)) into the Euclidean space (mathbb {R}^{n+k}) whose extrinsic geometry coincides with (mathfrak {S}). Here the dimension n and the codimension k are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on (mathfrak {S}) and (iota ). The best result up to date is (mathfrak {S} in L^p) and (iota in W^{2,p}) for (p>n ge 3) or (p=n=2). In this paper, we extend the above result to (iota in mathcal {X}) the topology of which is strictly weaker than (W^{2,n}) for (n ge 3). Indeed, (mathcal {X}) can be taken as the Morrey space (L^{p, n-p}_{2}) with arbitrary (p in ]2,n]). This appears to be the first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss–Codazzi–Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges—in particular, Rivière–Struwe’s work (Rivière and Struwe in Comm Pure Appl Math 61:451–463, 2008) on harmonic maps in arbitrary dimensions and codimensions—and the theory of compensated compactness.