Archive for Rational Mechanics and Analysis最新文献

The Ericksen-Leslie model for nematic liquid crystal flows in case of an isothermal and incompressible fluid with general Leslie stress and anisotropic elasticity, i.e. with general Ericksen stress tensor, is shown for the first time to be strongly well-posed. Of central importance is a fully nonlinear boundary condition for the director field, which, in this generality, is necessary to guarantee that the system fulfills physical principles. The system is shown to be locally, strongly well-posed in the (L_p)-setting. More precisely, the existence and uniqueness of a local, strong (L_p)-solution to the general system is proved and it is shown that the director d satisfies (|d|_2equiv 1) provided this holds for its initial data (d_0). In addition, the solution is shown to depend continuously on the data. The results are proven without any structural assumptions on the Leslie coefficients and in particular without assuming Parodi’s relation.

We are concerned with inverse problems for supersonic potential flows past infinite axisymmetric Lipschitz cones. The supersonic flows under consideration are governed by the steady isentropic Euler equations for axisymmetric potential flows, which give rise to a singular geometric source term. We first study the inverse problem for the stability of an oblique conical shock as an initial-boundary value problem with both the generating curve of the cone surface and the leading conical shock front as free boundaries. We then establish the existence and asymptotic behavior of global entropy solutions with bounded BV norm of the inverse problem, under the condition that the Mach number of the incoming flow is sufficiently large and the total variation of the pressure distribution on the cone is sufficiently small. To this end, we first develop a modified Glimm-type scheme to construct approximate solutions by self-similar solutions as building blocks to balance the influence of the geometric source term. Then we define a Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions. Meanwhile, the approximate generating curves of the cone surface are also constructed. Next, when the Mach number of the incoming flow is sufficiently large, by asymptotic analysis of the reflection coefficients in those interaction estimates, we prove that appropriate weights can be chosen so that the corresponding Glimm-type functional decreases in the flow direction. Finally, we determine the generating curves of the cone surface and establish the existence of global entropy solutions containing a strong leading conical shock, besides weak waves. Moreover, the entropy solution is proved to approach asymptotically the self-similar solution determined by the incoming flow and the asymptotic pressure on the cone surface at infinity.

We study the quantitative pointwise behavior of solutions to the Boltzmann equation for hard potentials and Maxwellian molecules, which generalize the hard sphere case introduced by Liu and Yu (Commun Pure Appl Math 57:1543–1608, 2004). The large time behavior of the solution is dominated by fluid structures, similar to the hard sphere case (Liu and Yu in Commun Pure Appl Math 57:1543–1608, 2004; Liu and Yu in Bull Inst Math Acad Sin (N.S.) 6:151–243, 2011). However, unlike the hard sphere case, the spatial decay here depends on the potential power (gamma ) and the initial velocity weight. A key challenge in this problem is the loss of velocity weight in linear estimates, which makes standard nonlinear iteration infeasible. To address this, we develop an Enhanced Mixture Lemma, demonstrating that mixing the transport and gain part of the linearized collision operator can generate arbitrary order regularity and decay in both space and velocity variables. This allows us to decompose the linearized solution into fluid (arbitrary regularity and velocity decay) and particle (rapid space-time decay, but with loss of velocity decay) parts, making it possible to solve the nonlinear problem through this particle-fluid duality.

This paper analyzes weak solutions of the quantum hydrodynamics (QHD) system with a collisional term posed on the one-dimensional torus. The main goal of our analysis is to rigorously prove the time-relaxation limit towards solutions to the quantum drift-diffusion (QDD) equation. The existence of global in-time, finite energy weak solutions can be proved by straightforwardly exploiting the polar factorization and wave function lifting tools previously developed by the authors. However, the sole energy bounds are not sufficient to show compactness and then pass to the limit. For this reason, we consider a class of more regular weak solutions (termed GCP solutions), determined by the finiteness of a functional involving the chemical potential associated with the system. For solutions in this class and bounded away from vacuum, we prove the time-relaxation limit and provide an explicit convergence rate. Our analysis exploits compactness tools and does not require the existence (and smoothness) of solutions to the limiting equations or the well-preparedness of the initial data. As a by-product of our analysis, we also establish the existence of global in time (H^2) solutions to a nonlinear Schrödinger–Langevin equation and construct solutions to the QDD equation as strong limits of GCP solutions to the QHD system.

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.