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In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k , then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ -almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T -invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space. [ABSTRACT FROM AUTHOR]

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by κ > 0 whose solution converges to that of the MHD equations as κ → 0. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure. [ABSTRACT FROM AUTHOR]

In this paper, we study the nearly critical Lane-Emden equations (⁎) { − Δ u = u p − ε in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N with N ≥ 3 , p = N + 2 N − 2 and ε > 0 is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions. In general, the solutions of (⁎) may blow-up at multiple points a 1 , ⋯ , a k of Ω as ε → 0. In particular, when Ω is convex, there must be a unique blow-up point (i.e., k = 1). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎). [ABSTRACT FROM AUTHOR]

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by T g f (z) = ∫ 0 z f (ζ) g ′ (ζ) d ζ , S g f (z) = ∫ 0 z f ′ (ζ) g (ζ) d ζ , and M g f (z) = g (z) f (z). We are concerned with the study of the boundedness of operators in the algebra A g generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in A g are finite linear combinations of finite products (words) of T g , S g , M g which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of A g can be characterized by one of the conditions g ∈ H ∞ , or g n belongs to B M O A or the Bloch space, for some integer n > 0. However, it is also proved that there are many operators, even single words in A g whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in A g in terms of a "fractional power" of the symbol g , that only depends on the number of appearances of each of the letters T g , S g , M g in the given word. [ABSTRACT FROM AUTHOR]

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous paper (Kaneko-Matsuzawa-Yamada, Discrete Contin. Dyn. Syst., 2022), where we have classified all large-time behaviors of radially symmetric solutions to a free boundary problem of reaction diffusion equation u t = Δ u + f (u) with positive bistable nonlinearity f in high space dimensions. The positive bistable nonlinearity means that f (u) = 0 has exactly two positive stable equilibria. Among the classified solutions, we are interested in a spreading solution, that is a solution (u (t , | x |) , h (t)) for x ∈ R N with free boundary | x | = h (t) such that, as t → ∞ , | x | ≤ h (t) expands to the whole space R N and u (t , ⋅) converges to a positive stable equilibrium for f (u) uniformly in any compact set of R N. When we discuss whole asymptotic profiles of spreading solutions, it has been known that they are generally described with use of a semi-wave obtained from the corresponding semi-wave problem. Our main purpose is to study precise asymptotic estimates for any spreading solution whose profile accompanies a propagating terrace with two different types of propagating speeds. Such a spreading phenomenon occurs when u (t , ⋅) converges to the largest equilibrium of f and the related semi-wave problem does not have a solution. We will prove that the propagating terrace consists of two functions; one is a semi-wave corresponding to a smaller positive equilibrium of f and the other is a traveling wave connecting two positive equilibria of f. In order to give sharp estimates for a radial terrace solution with the above properties, we need so called logarithmic shiftings, which are revealed by Uchiyama (1983) and Du-Matsuzawa-Zhou (2015) in higher dimensional cases. Moreover, we will see that two types of the logarithmic shiftings appear in the estimates. The proof for the higher dimensional case is very different from the one-dimensional case established by Kaneko-Matsuzawa-Yamada (2020). To derive exact logarithmic terms, we have to construct a series of upper and lower solutions to make estimates more and more accurate. [ABSTRACT FROM AUTHOR]

Stability results for the Helmholtz equations in both deterministic and random periodic structures are proved in this paper. Under the assumption of excluding resonances, by a variational method and Fourier analysis in the energy space, the stability estimate for the Helmholtz equation in a deterministic periodic structure is established. For the stochastic case, by introducing a variable transform, the variational formulation of the scattering problem in a random domain is reduced to that in a definite domain with random medium. Combining the stability result for the deterministic case with regularity and stochastic regularity of the scattering surface, Pettis measurability theorem and Bochner's Theorem further yield the stability result for the scattering problem by random periodic structures. Both stability estimates are explicit with respect to the wavenumber. [ABSTRACT FROM AUTHOR]

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed "odd viscosity", becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of a priori estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected. In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of good unknowns , which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established. [ABSTRACT FROM AUTHOR]

The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper we consider the vector-valued Schrödinger operator − Δ + V , where the potential term V is a matrix-valued function whose entries belong to L loc 1 (R d) and, for every x ∈ R d , V (x) is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in L 1 (R d , R m). Assuming further that the minimal eigenvalue of V belongs to some reverse Hölder class of order q ∈ (1 , ∞) ∪ { ∞ } , we obtain maximal inequality in L p (R d , R m) , for p in between 1 and some q , and generation results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the local well-posedness of a scaled anisotropic Navier-Stokes-Maxwell system in a 2-D striped domain with initial data around some nonzero background magnetic field in Gevrey-2 class. Then we rigorously justify the limit from the scaled anisotropic equations to the associated hydrostatic system and provide with the precise convergence rate. Finally, with small initial data in Gevrey- 3 2 class, we also extend the lifespan of thus obtained solutions to a longer time interval. [ABSTRACT FROM AUTHOR]

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones. [ABSTRACT FROM AUTHOR]

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions ( λ ¯ γ , u γ) of the equation F (D 2 u γ) + λ ¯ γ u γ r γ = 0 in B (0 , 1) ∖ { 0 } , u γ = 0 on ∂ B (0 , 1) where u γ > 0 in B (0 , 1) ∖ { 0 } and γ > 0. We prove existence of radial solutions which are continuous on B (0 , 1) ‾ in the case γ < 2 , existence of unbounded solutions in the case γ = 2 and a non existence result for γ > 2. We also give, in the case of Pucci's operators, the explicit value of λ ¯ 2 , which generalizes the Hardy–Sobolev constant for the Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of spatially quasi-periodic solutions to the p -generalized KdV equation on the real line with quasi-periodic initial data whose Fourier coefficients are exponentially decaying. In order to solve for the Fourier coefficients of the solution, we first reduce the nonlinear dispersive partial differential equation to a nonlinear infinite system of coupled ordinary differential equations, and then construct the Picard sequence to approximate them. However, we meet, and have to deal with, the difficulty of studying the higher dimensional discrete convolution operation for several functions : c × ⋯ × c ︸ p (total distance) : = ∑ ♣ 1 , ⋯ , ♣ p ∈ Z ν ♣ 1 + ⋯ + ♣ p = total distance ∏ j = 1 p c (♣ j). In order to overcome it, we apply a combinatorial method to reformulate the Picard sequence as a tree. Based on this form, we prove that the Picard sequence is exponentially decaying and fundamental (i.e., a Cauchy sequence). The result has been known for p = 2 [11] , and the combinatorics become harder for larger values of p. For the sake of clarity, we first give a detailed discussion of the proof of the existence and uniqueness result in the simplest case not covered by previous results, p = 3. Next, we prove existence and uniqueness in the general case p ≥ 2 , which then covers the remaining cases p ≥ 4. As a byproduct, we recover the local result from [11]. In the process of proof, we give a combinatorial structure of tensor (multi-linear operator), exhibit the most important combinatorial index σ (it's related to the degree or multiplicity of the power-law nonlinearity), and obtain a relationship with other indices, which is essential to our proofs in the case of general p. [ABSTRACT FROM AUTHOR]

Frisch-Parisi conjecture claims the existence of Baire function spaces in which Baire typical functions share the same multifractal behavior, prescribed in advance, and obey a multifractal formalism. In this paper, we introduce a family B of heterogeneous Besov spaces, which generalize the standard Besov spaces - they are obtained by replacing the Lebesgue measure (which plays a key role in the definition of the standard Besov spaces) by multifractal Radon measures belonging to some class constructed in the companion paper [1]. We find a characterization of the elements of B in terms of wavelet coefficients, and then describe the multifractal properties (singularity spectrum, validity of the multifractal formalism) of their Baire typical functions. This allows us to fully solve the Frisch-Parisi conjecture inside B . [ABSTRACT FROM AUTHOR]

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O (ν 1 / 4) order terms in L ∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when ν = 0 , are nonlinearly unstable when ν > 0 , provided ν is small enough, up to O (ν 1 / 4) terms in L ∞ norm. [ABSTRACT FROM AUTHOR]

This paper proposes a geometrisation of N -manifolds of degree n as n -fold vector bundles equipped with a (signed) S n -symmetry. More precisely, it proves an equivalence between the categories of [ n ] -manifolds and the category of (signed) symmetric n -fold vector bundles, by finding that symmetric n -fold vector bundle cocycles and [ n ] -manifold cocycles are identical. This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an S 2 -action. [ABSTRACT FROM AUTHOR]

In an extended mean field game the vector field governing the flow of the population can be different from that of the individual player at some mean field equilibrium. This new class strictly includes the standard mean field games. It is well known that, without any monotonicity conditions, mean field games typically contain multiple mean field equilibria and the wellposedness of their corresponding master equations fails. In this paper, a partial order for the set of probability measure flows is proposed to compare different mean field equilibria. The minimal and maximal mean field equilibria under this partial order are constructed and satisfy the flow property. The corresponding value functions, however, are in general discontinuous. We thus introduce a notion of weak-viscosity solutions for the master equation and verify that the value functions are indeed weak-viscosity solutions. Moreover, a comparison principle for weak-viscosity semi-solutions is established and thus these two value functions serve as the minimal and maximal weak-viscosity solutions in appropriate sense. In particular, when these two value functions coincide, the value function becomes the unique weak-viscosity solution to the master equation. The novelties of the work persist even when restricted to the standard mean field games. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets–Plateau's problem in the setting of sets. In particular, a direct application of the above result leads to various types of new singularities for Almgren minimal sets, e.g. the product of any paired calibrated cone (such as the cone over the d − 2 skeleton of the unit cube in R d , d ≥ 4) with homogeneous area minimizing hypercones (such as the Simons cone). [ABSTRACT FROM AUTHOR]

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDE s) associated with the quantum cohomology of P 1 -bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P 1 -bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17] , called Borel (α , β) -multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P 1 -bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDE s. As an example, we show how to integrate the qDE of blow-up of P 2 at one point via Borel multitransforms of solutions of the qDE of P 1. [ABSTRACT FROM AUTHOR]

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in [19] , we prove the existence of a nontrivial solution for the problem under consideration. The results we obtain here are the nonlocal counterparts of the ones obtained in [19] in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest. We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in [27]. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear boundary diffusion equation of porous medium type arising from a boundary control problem. We give a complete and sharp characterization of the asymptotic behavior of its solutions, and prove the stability of its separable solutions. [ABSTRACT FROM AUTHOR]

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in R d for d = 2 or d = 3. Using ideas from transformation optics, we construct an approximate cloak by "blowing up" a small ball of radius ϵ > 0 to one of radius 1. In the anisotropic cloaking layer resulting from the "blow-up" change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed ϵ) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as ϵ approaches 0, the L 2 -norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves. [ABSTRACT FROM AUTHOR]

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behaviors of a plate with non-planar top surface in the framework of linear elasticity. For this plate, we give a decomposition of the displacements. We show that every displacement of the plate is the sum of a Kirchhoff-Love displacement and a residual displacement that takes into account the deformations of the fibers of the plate and shear. We also prove Korn's type inequalities. [ABSTRACT FROM AUTHOR]

This paper is concerned with a controllability problem of blowup points for the heat equation. It can be described as follows: In the absence of control, the solution to the linear heat equation globally exists in a bounded domain Ω. We wonder whether for a given time T > 0 and a point a in this domain, we can find a feedback control, acting on a given internal subset ω of this domain, such that the corresponding solution to the heat equation blows up at time T and holds a as a unique blowup point. In this paper, we positively answer the question, when a ∈ ω. On the contrary, when a ∈ Ω ∖ ω ‾ , we show this is not possible, for any open-loop or feedback control which guarantees that the solution is L ∞ (Ω) whenever it exists. [ABSTRACT FROM AUTHOR]

We study the insulated conductivity problem with inclusions embedded in a bounded domain in R n , for n ≥ 3. The gradient of solutions may blow up as ε , the distance between the inclusions, approaches to 0. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on S n − 2. [ABSTRACT FROM AUTHOR]

In this paper, we prove the non-uniqueness of three-dimensional magneto-hydrodynamic (MHD) system in C ([ 0 , T ] ; L 2 (T 3)) for any initial data in H β ¯ (T 3) (β ¯ > 0), by exhibiting that the total energy and the cross helicity can be controlled in a given positive time interval. Our results extend the non-uniqueness results of the ideal MHD system to the viscous and resistive MHD system. Different from the ideal MHD system, the dissipative effect in the viscous and resistive MHD system prevents the nonlinear term from balancing the stress error ( R ˚ q , M ˚ q) as doing in [4]. We introduce the box flows and construct the perturbation consisting in seven different kinds of flows in convex integral scheme, which ensures that the iteration works and yields the non-uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of nontrivial contractible domains Ω ⊂ S d , d ≥ 2 , such that the overdetermined elliptic problem { − ε Δ g u + u − u p = 0 in Ω, u > 0 in Ω, u = 0 on ∂Ω, ∂ ν u = constant on ∂Ω, admits a positive solution. Here Δ g is the Laplace-Beltrami operator in the unit sphere S d with respect to the canonical round metric g , ε > 0 is a small real parameter and 1 < p < d + 2 d − 2 (p > 1 if d = 2). These domains are perturbations of S d ∖ D , where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains. [ABSTRACT FROM AUTHOR]

In this paper, we establish the locally diffeomorphic property of the solution to McKean-Vlasov stochastic differential equations defined in the Euclidean space. Our approach is built upon the insightful ideas put forth by Kunita. We observe that although the coefficients satisfy the global Lipschitz condition and some suitable regularity condition, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time, which sets McKean-Vlasov stochastic differential equations apart significantly from classical stochastic differential equations. Finally, we provide an example to complement our results. [ABSTRACT FROM AUTHOR]

This paper concerns the homogenization of Schrödinger equations for non-crystalline matter, that is to say the coefficients are given by the composition of stationary functions with stochastic deformations. Two rigorous results of so-called effective mass theorems in solid state physics are obtained: a general abstract result (beyond the classical stationary ergodic setting), and one for quasi-perfect materials (i.e. the disorder in the non-crystalline matter is limited). The former relies on the double-scale limits and the wave function is spanned on the Bloch basis. Therefore, we have extended the Bloch Theory which was restricted until now to crystals (periodic setting). The second result relies on the Perturbation Theory and a special case of stochastic deformations, namely stochastic perturbation of the identity. [ABSTRACT FROM AUTHOR]

In this paper, we study the global well-posedness problem for the 1d compressible Navier–Stokes systems (cNSE) in gas dynamics with rough initial data. First, Liu and Yu (2022) [30] established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d full cNSE with initial velocity and temperature data in BV space by Wang et al. (2022) [31]. We improve the global well-posedness result of Liu and Yu with initial velocity data in W 2 γ , 1 space; and of Wang–Yu–Zhang with initial velocity data in L 2 ∩ W 2 γ , 1 space and initial data of temperature in W ˙ − 2 3 , 6 5 ∩ W ˙ 2 γ − 1 , 1 for any γ > 0 arbitrarily small. Our essential ideas are based on establishing various "end-point" smoothing estimates for the 1d parabolic equation. [ABSTRACT FROM AUTHOR]

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a "projection" of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting. [ABSTRACT FROM AUTHOR]

In this paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power ((ρ ϵ) δ with δ > 1), for regular solutions to the corresponding Cauchy problem, via introducing one "quasi-symmetric hyperbolic"–"degenerate elliptic" coupled structure to control the behavior of the velocity near the vacuum, we establish the uniform energy estimates for the local sound speed in H 3 and (ρ ϵ) δ − 1 2 in H 2 with respect to the viscosity coefficients for arbitrarily large time under some smallness assumption on the initial density. Second, by making full use of this structure's quasi-symmetric property and the weak smooth effect on solutions, we prove the strong convergence of the regular solutions of the degenerate viscous flow to that of the inviscid flow with vacuum in H 2 for arbitrarily large time. It is worth pointing out that the result obtained here seems to be the first one on the global-in-time inviscid limit of solutions with large velocities and vacuum for compressible flow in 3D space without any symmetric assumption. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable – reversibility depending – Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures. [ABSTRACT FROM AUTHOR]

In the celebrated 1933 paper [22] by J. Leray, the invading domains method was proposed to construct D -solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D -solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, as well as the ideas developed in the recent works [19,20]. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the existence and non-existence of isolated singularities of positive solutions for the following Choquard equation (1) { − Δ u + u = I α [ u p ] u q in R + N , − Δ u + u = 0 on ∂ R + N ∖ { 0 } , where p , q > 0 , N ≥ 2 , α ∈ (0 , N) and I α [ u p ] (x) = ∫ R + N u (y) p | x − y | N − α d y. Here we obtain the existence of singular solutions to (1) by considering the very weak solutions of { − Δ u + u = I α [ u p ] u q in R + N , − Δ u + u = k δ 0 on ∂ R + N , where k > 0 and δ 0 is Dirac mass concentrated at the origin. [ABSTRACT FROM AUTHOR]

This paper is dedicated to L p bounds on eigenfunctions of a Schrödinger-type operator (− Δ g) α / 2 + V on closed Riemannian manifolds for critically singular potentials V. The operator (− Δ g) α / 2 is defined spectrally in terms of the eigenfunctions of − Δ g. We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation (∂ t 2 + (− Δ g) α / 2 + V) u = 0. The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao [4] and Shao-Yao [27] play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the "uniform Sobolev estimates" by Kenig-Ruiz-Sogge [18]. These enable us to handle the critically singular potentials V and prove the quasimode estimates. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case. [ABSTRACT FROM AUTHOR]

We study the focusing energy-critical NLS (NLS) i ∂ t u + Δ x , y u = − | u | 4 d − 1 u on the waveguide manifold R x d × T y with d ≥ 2. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of (NLS) are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem m c defined on the semivirial-vanishing manifold with prescribed mass c. We prove that for all sufficiently large mass the variational problem m c has a unique optimizer u c satisfying ∂ y u c = 0 , while for all sufficiently small mass, any optimizer of m c must have non-trivial y -dependence. Afterwards, we prove that m c characterizes a sharp threshold for the bifurcation of finite time blow-up (d = 2 , 3) and globally scattering (d = 3) solutions of (NLS) in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting. [ABSTRACT FROM AUTHOR]

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential − log ⁡ u such that u is C ∞ -smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles S (L) of negative Hermitian line bundles (L , h) over Kähler manifolds (M , g). We prove that if (M , g) has constant Ricci eigenvalues, then S (L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M , g) is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S (L) when (M , g) is a Kähler surface (dim ⁡ M = 2) with constant scalar curvature. [ABSTRACT FROM AUTHOR]

We consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space H 1 × L 2. This model has solitary waves with speeds − 1 < c < 1. When | c | approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. In this paper we consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space. [ABSTRACT FROM AUTHOR]

Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the first non-vanishing higher order averaged function. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations. • We study higher order perturbed 2D non-autonomous periodic ODE. • We provide a continuation result for normally hyperbolic invariant tori. • Averaging theory provides asymptotic estimates by means of the Guiding System (GS). • The GS is a 2D autonomous ODE given by the first non-vanishing averaged function. • We prove that hyperbolic limit cycles of GS imply normally hyperbolic invariant tori. [ABSTRACT FROM AUTHOR]

In this paper, a graphical mean curvature flow with the velocity involving a positive power of the mean curvature and a driving force is studied. We first give a complete classification of the radially symmetric translating solutions, and select one of them so that it satisfies the prescribed boundary angle in a cylinder. For the initial boundary value problem in a cylinder, by showing the a priori estimates (especially the uniform-in-time bound for the mean curvature which guarantees the uniform parabolicity of the corresponding fully nonlinear equation), we can establish the global existence for the solution and prove its convergence to some corresponding translating solution finally. The aim is to provide an exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angle on the existence and stability of radially symmetric translating solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the hydrodynamic limit to contact discontinuity of the compressible Euler system for the Landau equation with the physical Coulomb interaction as the Knudsen number ε > 0 is vanishing. Our results are twofold. First, we construct the unique global-in-time solution to the Landau equation with suitably small initial data as ε > 0 is sufficiently small. Second, we prove that the solution of the Landau equation converges to a local Maxwellian whose fluid quantities are the contact discontinuity of the corresponding Euler system uniformly away from the discontinuity as ε → 0. Moreover, the uniform convergence rate is also obtained. The main idea is to introduce a new time-velocity weigh function to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of the exponential stability of density-velocity systems with boundary conditions. Density-velocity systems are typical hyperbolic systems that are omnipresent in physics as they encompass all systems that consist in a flux conservation and a momentum equation. In this paper we show that any such system can be stabilized exponentially quickly in the H 2 norm using simple local boundary feedbacks, provided a condition on the source term is valid. This condition holds for most physical systems, even when the source term is not dissipative. Besides, the feedback laws obtained only depend on the target values at the boundaries, which implies that they do not depend on the expression of the source term or the force applied on the system. This makes them both very easy to implement in practice and robust to model errors. For instance, for a river modeled by Saint-Venant equations this means that the feedback law does not require any information on the friction model, the slope or the shape of the channel considered. This feat is obtained by showing the existence of a basic H 2 Lyapunov function. We apply it to several systems: the general Saint-Venant equations, the isentropic Euler equations, the motion of water in rigid-pipe, the osmosis phenomenon, the traffic flow, etc. [ABSTRACT FROM AUTHOR]

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2]. One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n. The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Z̆itković [8]. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Z̆itković developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result. [ABSTRACT FROM AUTHOR]

In this paper, we study the diffusive limit of the steady state radiative heat transfer system for non-homogeneous Dirichlet boundary conditions in a bounded domain with flat boundaries. By taking account of the boundary layers, a composite approximate solution is constructed using asymptotic analysis. The convergence to the composite approximate solution in the diffusive limit is proved using a Banach fixed point theorem. The major difficulty lies in the nonlinear coupling between elliptic and kinetic transport equations. To overcome this difficulty, a spectral assumption is proposed to ensure the linear stability of boundary layers. Moreover, a combined L 2 - L ∞ estimate and the Banach fixed point theorem are used to obtain the convergence proof. This result extends our previous work [6] for the well-prepared boundary data case to the general boundary date. [ABSTRACT FROM AUTHOR]

Recent field and experimental studies show that mobility patterns for humans exhibit scale-free nonlocal dynamics with heavy-tailed distributions characterized by Lévy flights. To study the long-range geographical spread of infectious diseases, in this paper we propose a susceptible-infectious-susceptible epidemic model with Lévy flights in which the dispersal of susceptible and infectious individuals follows a heavy-tailed jump distribution. Owing to the fractional diffusion described by a spectral fractional Neumann Laplacian, the nonlocal diffusion model can be used to address the spatiotemporal dynamics driven by the nonlocal dispersal. The primary focuses are on the existence and stability of disease-free and endemic equilibria and the impact of dispersal rates and fractional powers on the spatial profiles of these equilibria. A variational characterization of the basic reproduction number R 0 is obtained and its dependence on dispersal rates and fractional powers is also examined. Then R 0 is utilized to investigate the effects of spatial heterogeneity on the transmission dynamics. It is shown that R 0 serves as a threshold for determining the existence and nonexistence of an epidemic equilibrium as well as the stability of the disease-free and endemic equilibria. In particular, in low-risk regions both dispersal rates and fractional powers play a critical role and are capable of altering the threshold value. Numerical simulations were performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the L 2 -gradient flow for the modified p -elastic energy defined on planar closed curves. We formulate a notion of weak solution for the flow and prove the existence of global-in-time weak solutions with p ≥ 2 for initial curves in the energy space via minimizing movements. Moreover, we prove the existence of unique global-in-time solutions to the flow with p = 2 and obtain their subconvergence to an elastica as t → ∞. [ABSTRACT FROM AUTHOR]