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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]

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]

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]

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]