American Institute of Mathematical Sciences

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.

In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations \begin{document}$ \sigma_k(\partial \bar{\partial} u) = \sum\limits _{l = 0}^{k-1} \alpha_l(z) \sigma_l (\partial \bar{\partial} u) $\end{document} with \begin{document}$ 2 \leq k \leq n $\end{document}, and establish the global \begin{document}$ C^1 $\end{document} estimates and reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global \begin{document}$ C^2 $\end{document} estimates and the existence theorems when \begin{document}$ k = n $\end{document}.

In this paper, we mainly show a close relationship between topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we prove that above result holds for rate distortion dimension of subshifts with respect to a lower rank subgroup and measure entropy. Furthermore, we present some examples.

In this paper we study the connectivity of Fatou components for maps in a large family of singular perturbations. We prove that, for some parameters inside the family, the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and we determine precisely these connectivities. In particular, these results extend the ones obtained in [5,6].

Let \begin{document}$ f $\end{document} be a \begin{document}$ C^{1+\alpha} $\end{document} diffeomorphism on a compact manifold \begin{document}$ M $\end{document} and let \begin{document}$ \mu $\end{document} be an ergodic measure. We use a special family of fake center-stable manifolds to bound the entropy of \begin{document}$ \mu $\end{document} in terms of positive Lyapunov exponents and the so called 'dimensional entropy', a notion related to the topological entropy of submanifolds.

We show that there are no general stability results for the logarithmic Sobolev inequality in terms of the Wasserstein distances and \begin{document}$ L^{p}(d\gamma) $\end{document} distance for \begin{document}$ p>1 $\end{document}. To this end, we construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not, which leads to instability for the logarithmic Sobolev inequality. As an application, we prove instability results for Talagrand's transportation inequality and the Beckner–Hirschman inequality.

We construct non-vanishing steady solutions to the Euler equations (for some metric) with analytic Bernoulli function in each three-manifold where they can exist: graph manifolds. Using the theory of integrable systems, any admissible Morse-Bott function can be realized as the Bernoulli function of some non-vanishing steady Euler flow. This can be interpreted as an inverse problem to Arnold's structure theorem and yields as a corollary the topological classification of such solutions. Finally, we prove that the topological obstruction holds without the non-vanishing assumption: steady Euler flows with a Morse-Bott Bernoulli function only exist on graph three-manifolds.

The main result of this paper is a construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, we construct Bufetov functionals on rectangles on \begin{document}$ (2g+1) $\end{document}-dimensional Heisenberg manifolds. We prove that deviation of the ergodic integral of higher rank actions is described by the asymptotic of Bufetov functionals for a sufficiently smooth function. As a corollary, the distribution of normalized ergodic integrals which have variance 1, converges along certain subsequences to a non-degenerate compactly supported measure on the real line.

We discuss how the higher-order term \begin{document}$ |u|^q $\end{document} \begin{document}$ (q>1+2/(n-1)) $\end{document} has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations

\begin{document}$ \partial_t^2 u-\Delta u = |v|^p, \qquad \partial_t^2 v-\Delta v = |\partial_t u|^{(n+1)/(n-1)} +|u|^q $\end{document}

in \begin{document}$ n\,(\geq 2) $\end{document} space dimensions. We show the existence of a certain "critical curve" in the \begin{document}$ pq $\end{document}-plane such that for any \begin{document}$ (p,q) $\end{document} \begin{document}$ (p,q>1) $\end{document} lying below the curve, nonexistence of global solutions occurs, whereas for any \begin{document}$ (p,q) $\end{document} \begin{document}$ (p>1+3/(n-1),\,q>1+2/(n-1)) $\end{document} lying exactly on it, this system admits a unique global solution for small data. When \begin{document}$ n = 3 $\end{document}, the discussion for the above system with \begin{document}$ (p,q) = (3,3) $\end{document}, which lies on the critical curve, has relevance to the study on systems satisfying the weak null condition, and we obtain a new result of global existence for such systems. Moreover, in the particular case of \begin{document}$ n = 2 $\end{document} and \begin{document}$ p = 4 $\end{document} it is observed that no matter how large \begin{document}$ q $\end{document} is, the higher-order term \begin{document}$ |u|^q $\end{document} never becomes negligible and it essentially affects the lifespan of small solutions.

We study a class of gauged nonlinear Schrödinger equations in the plane

\begin{document}$ \left\{ \begin{array}{l} -\Delta u+V(|x|) u+\lambda\bigg(\int_{|x|}^\infty \frac{h_u(s)}{s}u^2(s)ds+\frac{h_u^2(|x|)}{|x|^2} \bigg)u\\\qquad \, = K(|x|)f(u)+\mu g(|x|)|u|^{q-2}u, \\ u(x) = u(|x|) \; {\rm{in}}\; \mathbb{R}^2, \\\\ \end{array} \right. $\end{document}

where \begin{document}$ h_u(s) = \int_0^s\frac{r}{2}u^2(r)dr $\end{document}, \begin{document}$ \lambda,\mu>0 $\end{document} are constants, \begin{document}$ V(|x|) $\end{document} and \begin{document}$ K(|x|) $\end{document} are continuous functions vanishing at infinity. Assume that \begin{document}$ f $\end{document} is of critical exponential growth and \begin{document}$ g(x) = g(|x|) $\end{document} satisfies some technical assumptions with \begin{document}$ 1\leq q<2 $\end{document}, we obtain the existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, with the help of the genus theory, we prove the existence of infinitely many solutions if \begin{document}$ f $\end{document} in addition is odd.

Based on the spectral analysis and the inverse scattering method, by introducing some spectral function transformations and variable transformations, the initial value problem for the modified complex short pulse (mCSP) equation is transformed into a \begin{document}$ 2\times2 $\end{document} matrix Riemann-Hilbert problem. It is proved that the solution of the initial value problem for the mCSP equation has a parametric expression related to the solution of the matrix Riemann-Hilbert problem. Various Deift-Zhou contour deformations and the motivation behind them are given. Through several appropriate transformations and strict error estimates, the original matrix Riemann-Hilbert problem can be reduced to the model Riemann-Hilbert problem, whose solution can be solved explicitly in terms of the parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the initial value problem for the mCSP equation is obtained by using the nonlinear steepest decent method.

We prove the comparison principle for viscosity sub and super solutions of degenerate nonlocal operators with general nonlocal gradient nonlinearities. The proofs apply to purely Hamilton-Jacobi equations of order \begin{document}$ 0<s<1 $\end{document}.

In this paper, we consider the Cauchy problem for the three dimensional axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion \begin{document}$ \Delta n^m $\end{document}. Taking advantage of the structure of axisymmetric flow without swirl, we show the global existence of weak solutions for the chemotaxis-Navier-Stokes equations with \begin{document}$ m=\frac{5}{3} $\end{document}.

This paper focuses on two-dimensional incompressible non-resistive MHD equations with only horizontal dissipation in \begin{document}$ \mathbb{T}\times\mathbb{R} $\end{document}. Invoking three Poincaré-type inequalities about the horizontal derivative, we study the global well-posedness of the system near a background magnetic via the structure of the perturbation MHD system and the symmetry condition imposed on the initial data. By a precise time-weighted energy estimate, we also establish the global well-posedness of the system with only horizontal magnetic damping. Here we overcome the difficulties brought by the absence of magnetic diffusion and the appearance of the boundary. We note that the stability of MHD equations with one-directional dissipation in \begin{document}$ \mathbb{R}^2 $\end{document} or a bounded domain appears to be unknown.

We focus on a class of derivative nonlinear Schrödinger equation with reversible nonlinear term depending on spatial variable \begin{document}$ x $\end{document}:

\begin{document}$ \begin{equation*} \mathrm{i} u_t+u_{xx}-\bar{u}u_{x}^2 + F(x, u, \bar{u}, u_{x}, \bar{u}_{x}) = 0, \quad x\in \mathbb{T}: = \mathbb{R}/2\pi\mathbb{Z}, \end{equation*} $\end{document}

where the nonlinear term \begin{document}$ F $\end{document} is an analytic function of order at least five in \begin{document}$ u, \bar{u}, u_{x}, \bar{u}_{x} $\end{document} and satisfies

\begin{document}$ \begin{equation*} F(x, u, \bar{u}, u_{x}, \bar{u}_{x}) = \overline{F(x, \bar{u}, u, \bar{u}_{x}, u_{x})}. \end{equation*} $\end{document}

Moreover, we also assume that \begin{document}$ F $\end{document} satisfies the homogeneous condition (6) to overcome the degeneracy. We prove the existence of small amplitude, smooth quasi-periodic solutions for the above equation via establishing an abstract infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with unbounded perturbation.

By using distribution dependent Zvonkin's transforms and Malliavin calculus, the Bismut type formula is derived for the intrinisc/Lions derivatives of distribution dependent SDEs with singular drifts, which generalizes the corresponding results derived for classical SDEs and regular distribution dependent SDEs.

In this paper, we consider the 3D Jordan–Moore–Gibson–Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model removing the \begin{document}$ L^1 $\end{document}-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

In this paper we study multiple ergodic averages for "good" variable polynomials. In particular, under an additional assumption, we show that these averages converge to the expected limit, making progress related to an open problem posted by Frantzikinakis ([13,Problem 10]). These general convergence results imply several variable extensions of classical recurrence, combinatorial and number theoretical results which are presented as well.