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In this paper, we study the prescribed Q -curvature flow equation on a arbitrary even dimensional closed Riemannian manifold (M , g) , which was introduced by S. Brendle in [3] , where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and ∫ M Q d μ < (n − 1) ! Vol (S n). In this paper we study the critical case that ∫ M Q d μ = (n − 1) ! Vol (S n) , we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence result in [21] in dimension 4 and extend the work of Li-Zhu [22] in dimension 2 to general even dimensions. In the proof, we give a explicit expression of the limit of the corresponding energy functional when the blow up occurs. [ABSTRACT FROM AUTHOR]

In this paper we provide the classification of positive solutions to the critical p -Laplace equation on R n , for 1 < p < n , possibly having infinite energy. If n = 2 , or if n = 3 and 3 2 < p < 2 we prove rigidity without any further assumptions. In the remaining cases we obtain the classification under energy growth conditions or suitable control of the solutions at infinity. Our assumptions are much weaker than those already appearing in the literature. We also discuss the extension of the results to the Riemannian setting. [ABSTRACT FROM AUTHOR]

Let (M , ω) be a compact connected Kähler manifold of complex dimension four and let [ χ ] ∈ H 1 , 1 (M ; R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑ i arctan ⁡ (λ i) = θ ˆ , where λ i are the eigenvalues of χ with respect to ω and θ ˆ is a topological constant. This conjecture was stated in [8] , wherein they proved that the existence of a supercritical C -subsolution or the existence of a C -subsolution when θ ˆ ∈ [ ((n − 2) + 2 / n) π / 2 , n π / 2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θ ˆ > (n − 2) π / 2 , where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θ ˆ is close to the supercritical phase π from the right, then the existence of a C -subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation. [ABSTRACT FROM AUTHOR]

In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel T × [ − 1 , 1 ] , and asymptotic behavior of Couette flow in Gevrey spaces on T × R for the β -plane equation. Let T > 0 be the horizontal period of the channel and α = 2 π T be the wave number. We obtain a sharp region O in the whole (α , β) half-plane such that non-shear traveling waves do not exist for (α , β) ∈ O and such traveling waves indeed exist for (α , β) in the remaining regions, near Couette flow for H ≥ 5 velocity perturbation. The borderlines between the region O and its remaining are determined by two curves of the principal eigenvalues of singular Rayleigh-Kuo operators. Our results reveal that there exists β ⁎ > 0 such that if | β | ≤ β ⁎ , then non-shear traveling waves do not exist for any T > 0 , while if | β | > β ⁎ , then there exists a critical period T β > 0 so that such traveling waves exist for T ∈ [ T β , ∞) and do not exist for T ∈ (0 , T β) , near Couette flow for H ≥ 5 velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any β ≠ 0 and T > 0 , we prove that there exist no non-shear traveling waves with traveling speeds converging in (− 1 , 1) near Couette flow for H ≥ 5 velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for H < 5 2 velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the Coriolis effects. Finally, we prove nonlinear inviscid damping for Couette flow in some Gevrey spaces by extending the method of [4] to the β -plane equation on T × R. [ABSTRACT FROM AUTHOR]

In this paper, we establish the continuous dependence for the non-resistive MHD equations in Sobolev spaces. Our obtained result fills considerably the recent result [5] for the continuous dependence on initial data. We also show that this result is optimal. More precisely, we proved that the data-to-solution map is not uniform continuous dependence in Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the soliton resolution and large time behavior of solutions to the Cauchy problem for the Novikov equation with a nonzero background u t − u t x x + 4 u x = 3 u u x u x x + u 2 u x x x , u (x , 0) = u 0 (x) , where u 0 (x) → κ > 0 , x → ± ∞ and u 0 (x) − κ is assumed in the Schwarz space. It is shown that the solution of the Cauchy problem can be characterized via a Riemann-Hilbert problem in a new scale (y , t) with y = x − ∫ x ∞ ((u − u x x + 1) 2 / 3 − 1) d s. According to the distribution of other four phase points on the jump contour, we divide the upper half-plane { (y , t) : y ∈ R , t > 0 } into four space-time regions: ξ = y / t ∈ (− ∞ , − 1 / 8) ∪ (1 , + ∞) and ξ ∈ (− 1 / 8 , 0) ∪ (0 , 1). Via ∂ ‾ -steepest descent analysis, we obtain the different long time asymptotic expansions of the solution u (y , t) in these different space-time solitonic regions. The corresponding residual error functions from singularities and a ∂ ‾ -equation respectively. Our result implies that soliton resolution can be characterized with an N (Λ) -soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the regions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinear parabolic equations with weights in the Muckenhoupt class A q p (R n + 1) for q ≥ p on non-smooth domains. Here the quasilinear nonlinearity is modeled after the standard p -Laplacian operator. Until now, all the weighted estimates for the gradient were obtained only for exponents q > p. The results for exponents q > p used the full complicated machinery of the Calderón-Zygmund theory developed over the past few decades, but the constants blow up as q → p (essentially because the Maximal function is not bounded on L 1). In order to prove the weighted estimates for the gradient at the natural exponent, i.e., q = p , we need to obtain improved a priori estimates below the natural exponent. To this end, we develop the technique of Lipschitz truncation based on [3,25] and obtain significantly improved estimates below the natural exponent. Along the way, we also obtain improved, unweighted Calderón-Zygmund type estimates below the natural exponent which is new even for the linear equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any C 1 -regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond. [ABSTRACT FROM AUTHOR]

It is known that the Maxwell–Klein–Gordon equations in R 3 + 1 admit global solutions with finite energy data. In this paper, we present a new approach to study the asymptotic behavior of these global solutions. We show the quantitative energy flux decay of the solutions with data merely bounded in some weighted energy space. We also establish an integrated local energy decay and a hierarchy of r -weighted energy decay. The results in particular hold in the presence of large total charge. This is the first result to give a complete and precise description of the global behavior of large nonlinear charged scalar fields. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to -unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton''s center and momentum) is close to Newton''s equation but with a dissipative term due to radiation of the energy to infinity. [Copyright &y& Elsevier]

We consider the Yang-Mills equations in (1 + d) -dimensional Minkowski spacetime. It is known that in the supercritical case, i.e., for d ≥ 5 , these equations admit closed form equivariant self-similar blowup solutions [2]. These solutions are furthermore conjectured to be the universal attractors for generic large equivariant data evolutions. In this paper we partially prove this conjecture. Namely, we show that for all odd d ≥ 5 the blowup mechanism exhibited by these solutions is stable. [ABSTRACT FROM AUTHOR]

In this paper, we develop a systematical approach in applying the method of moving planes to study qualitative properties of solutions for nonlocal parabolic equations. We first establish a series of key ingredients in the proofs, such as maximum principles for antisymmetric functions, narrow region principles, maximum principles near infinity, and the Hopf lemma, then we demonstrate how these new tools can be employed to derive the radial symmetry and monotonicity of ancient positive solutions to fractional parabolic equations in the whole space. Our results are in particular applicable to entire solutions, by which we mean solutions defined for all t ∈ R. We believe that the new ideas and methods introduced here can be adapted to study many other nonlocal parabolic problems with more general operators and nonlinear terms. [ABSTRACT FROM AUTHOR]

Reaction-diffusion equations in unbounded domain are used to study the propagation phenomena of biological species. When propagation can happen in different directions, an interesting question arises: In which direction is propagation the fastest? For the one-dimensional KPP equation in almost periodic media with advection: (⁎) { u t = (a (x) u x) x + b (x) u x + f (x , u) t > 0 , x ∈ R , u (0 , x) = u 0 (x) ∈ [ 0 , 1 ] is nonzero with compact support , let ω + and ω − be the spreading speeds of (⁎) in the positive and negative directions respectively. The above question becomes this: Which is larger, ω + or ω − ? In this paper, after establishing the existence of ω + and ω − , we give a complete answer to this question: sgn (ω − − ω +) = sgn (lim x → ∞ ⁡ 1 x ∫ 0 x b (s) a (s) d s). [ABSTRACT FROM AUTHOR]

We consider the CR Yamabe equation with critical Sobolev exponent on a closed contact manifold M of dimension 2 n + 1. The problem of finding solutions with minimum energy has been resolved for all dimensions except for dimension 5 (n = 2). In this paper we prove the existence of minimum energy solutions in the 5-dimensional case when M is spin. The proof is based on a positive mass theorem built up through a spinorial approach. [ABSTRACT FROM AUTHOR]

In this paper, we study existence, regularity, classification, and asymptotic behaviors of solutions of some Monge–Ampère equations with isolated and line singularities. We classify all solutions of det ⁡ ∇ 2 u = 1 in R n with one puncture point. This can be applied to characterize ellipsoids, in the same spirit of Serrin's overdetermined problem for the Laplace operator. In the case of having k non-removable singular points for k > 1 , modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold. We also establish existence of global solutions with general singular sets, regularity properties, and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line. The geometric motivation comes from singular semi-flat Calabi–Yau metrics. [ABSTRACT FROM AUTHOR]

We consider the limiting extremal process X of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale Z , the rescaled "density" of particles, which are at distance n + x from a position close to the tip of X , converges in probability to a multiple of the exponential e x as n → + ∞. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a 1-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in [9,10] and further developed in the present paper. The proofs of the limit theorems for X rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of "size" 0 < ε ≪ 1 , up to terms of the order [ (log ⁡ ε − 1) ] − 1 − γ , with some γ > 0. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in Pan et al. (2018) [24] , we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the H 2 -norm of the initial perturbation value of both the velocity and magnetic field. Then, we establish time-decay rates of strong solutions. Moreover, we find that H 2 -norm of the velocity decays faster than the perturbed magnetic field. Finally, by developing some new analysis techniques, we show that the strong solution converges in a rate of the field strength to a solution of the corresponding linearized problem as the strength of the impressive magnetic field goes to infinity. In addition, an extension of similar results to the corresponding inviscid case with damping is presented. [ABSTRACT FROM AUTHOR]

In this paper, we prove the boundary layer separation of the steady Prandtl equation for Oleinik's data ensuring the local existence of the solution under suitable adverse pressure gradient, and confirm Goldstein's hypothesis on the separation rate. We also study the local behavior of the solution near the separation point, which is related to Open problem 5 proposed by Oleinik and Samokhin (P.501, [25]). [ABSTRACT FROM AUTHOR]

In this paper we introduce a new symmetrization with respect to mixed volume or anisotropic curvature integral, which generalizes the one with respect to quermassintegral due to Talenti [26] and Tso [28]. We show a Pólya-Szegő type principle for such symmetrization — it diminishes the anisotropic Hessian integral for quasi-convex functions. We achieve this by a systematic study of invariants on non-symmetric matrices with real eigenvalues and the higher order anisotropic mean curvatures of level sets, which may be of independent interest. As applications, we establish a comparison principle for anisotropic Hessian equations and sharp anisotropic Sobolev inequalities. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces with . Local results for arbitrary initial data are also given. [Copyright &y& Elsevier]

Abstract: In this paper we define time dependent parabolic Reifenberg domains and study estimates for weak solutions of uniformly parabolic equations in divergence form on these domains. The basic assumption is that the principal coefficients are of parabolic BMO space with small parabolic BMO seminorms. It is shown that estimates hold for time dependent parabolic δ-Reifenberg domains. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function . An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan''s lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary. [Copyright &y& Elsevier]

Abstract: As is well known, h-vectors of simplicial convex polytopes are characterized. Those h-vectors satisfy Dehn–Sommerville equations and some inequalities conjectured by P. McMullen and first proved by R. Stanley using toric geometry. The boundary of a simplicial convex polytope determines a Gorenstein* simplicial poset but there are many Gorenstein* simplicial posets which do not arise this way. However, it is known that h-vectors of Gorenstein* simplicial posets still satisfy Dehn–Sommerville equations and that every component in the h-vectors is non-negative. In this paper we prove that h-vectors of Gorenstein* simplicial posets must satisfy one more subtle condition conjectured by R. Stanley and complete the characterization of h-vectors of Gorenstein* simplicial posets. Our proof is purely algebraic but the idea of the proof stems from topology. [Copyright &y& Elsevier]

In this paper we establish a global existence of weak solutions to the two-dimensional Prandtl''s system for unsteady boundary layers in the class considered by Oleinik (J. Appl. Math. Mech. 30 (1966) 951) provided that the pressure is favourable. This generalizes the local well-posedness results due to Oleinik (1966; Mathematical Models in Boundary Layer Theory, Chapman & Hall, London, 1999). For the proof, we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations. [Copyright &y& Elsevier]

In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal parabolic problems with more general operators and more general nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we study singular solutions of linear problems with fractional Laplacian. First, we establish Bôcher type theorems on a punctured ball via distributional approach. Then, we develop a few interesting maximum principles on a punctured ball. Our distributional approach only requires the basic L loc 1 -integrability. Furthermore, several basic lemmas are introduced to unify the treatments of Laplacian and fractional Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we consider the V -soliton equation which is a degenerate fully nonlinear equation introduced by La Nave and Tian in their work on Kähler-Ricci flow on symplectic quotients. One can apply the interpretation to study finite time singularities of the Kähler-Ricci flow. As in the case of Kähler-Einstein metrics, we can also reduce the V -soliton equation to a scalar equation on Kähler potentials, which is of Monge-Ampère type. We formulate some preliminary estimates for such a scalar equation on a compact Kähler manifold M. [ABSTRACT FROM AUTHOR]

In this paper, we prove a fundamental inequality for the algebraic structure of Δ v Δ ∞ v : for every v ∈ C ∞ , | | D 2 v D v | 2 − Δ v Δ ∞ v − 1 2 | D 2 v | 2 − (Δ v) 2 | D v | 2 | ≤ n − 2 2 | D 2 v | 2 | D v | 2 − | D 2 v D v | 2 , where Δ is the Laplacian and Δ ∞ is the ∞-Laplacian. Based on this, we prove the following results: (i) For any p -harmonic functions u with p ∈ (1 , 2) ∪ (2 , ∞) , we have | D u | p − γ 2 D u ∈ W loc 1 , 2 with γ < min ⁡ { p + n n − 1 , 3 + p − 1 n − 1 }. As a by-product, when p ∈ (1 , 2) ∪ (2 , 3 + 2 n − 2) , we give a new proof of the known W loc 2 , q -regularity of p -harmonic functions for some q > 2. (ii) When n ≥ 2 and p ∈ (1 , 2) ∪ (2 , 3 + 2 n − 2) , the viscosity solutions to the parabolic normalized p -Laplace equation have the W loc 2 , q -regularity in the spatial variable and the W loc 1 , q -regularity in the time variable for some q > 2. Especially, when n = 2 an open question in [18] is completely answered. (iii) When n ≥ 1 and p ∈ (1 , 2) ∪ (2 , 3) , the weak or viscosity solutions to the parabolic p -Laplace equation have the W loc 2 , 2 -regularity in the spatial variable and the W loc 1 , 2 -regularity in the time variable. The range of p here (including p = 2 from the classical result) is sharp for the W loc 2 , 2 -regularity. [ABSTRACT FROM AUTHOR]