As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to be shape densities (characteristic functions). The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: Any two shapes of equal volume can be approximately connected by an Euler spray---a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays. Each such Wasserstein geodesic is also the unique minimizer of a relaxed least-action principle for a two-fluid mixture theory corresponding to incompressible fluid mixed with vacuum.

It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. We discuss its recent extensions given in \cite{TiDu16} in some heterogeneously localized nonlocal function spaces. The new trace theorems are stronger and more general than the classical result. They can be established essentially for all functions having only square integrability away from the boundary or in any compact subset of interior domain. Yet, the heterogeneous localization offers the necessary regularity precisely at the boundary to have well-defined traces. A consequence is that we may study associated Dirichlet type boundary value problems, as well as the coupling of local and nonlocal equations through co-dimension-1 interfaces. %Their derivations not only involve various extensions of classical inequalities but also %require new techniques absent from traditional approaches.

We deal with a stationary problem of a reaction–diffusion system with a conservation law under the Neu-mann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coef-ficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling with κthe asymptotic profile exhibits a parabola in the nonvanishing region. We also prove the existence of an unstable monotone solution when the mass is small.

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In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.

In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler’s equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.

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This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. Several multiscale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a secondorder elliptic equations with random potential in two dimensions of space. We show that the random fluctuations of such solutions are correctly estimated by the heterogeneous multi-scale algorithm when appropriate fine-scale problems are solved on subsets that cover the whole computational domain. However, when the fine-scale problems are solved over patches that do not cover the entire domain, the random fluctuations may or may not be estimated accurately. In the case of random potentials with short-range interactions, the variance of the random fluctuations is amplified as the inverse of the fraction of the medium covered by the patches. In the case of random potentials with long-range interactions, however, such an amplification does not occur and random fluctuations are correctly captured independent of the (macroscopic) size of the patches. These results are consistent with those obtained in [9] for more general equations in the one-dimensional setting and provide indications on the loss in accuracy that results from using coarser, and hence computationally less intensive, algorithms.

The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is highly singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or at some isolated interior points, it was unknown whether the entropy remains its boundedness. The results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, if the vacuum occurs at the far field only and the density decays slowly enough at the far field. Precisely, we consider the Cauchy problem to the one-dimensional full compressible Navier-Stokes equations without heat conduction, and establish the local and global existence and uniqueness of entropy-bounded solutions, in the presence of vacuum at the far field only. It is also shown that, different from the case that with compactly supported initial density, the compressible Navier-Stokes equations, with slowly decaying initial density, can propagate the regularities in inhomogeneous Sobolev spaces.

We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^{d}$ : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha } > 0$ with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative $\langle \nabla f, \alpha \rangle $ does not detect any oscillation in directions orthogonal to $\alpha $ , however, for certain $\alpha $ the geodesic flow in direction $\alpha $ is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$ the inequality holds for $\alpha = (1, \sqrt{2})$ but is not true for $\alpha = (1,e)$ . Similar results should hold at a great level of generality on very general domains.

An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.

The inflow problem of full compressible NavierStokes equations is considered on the half-line (0,+\infty). First, we give the existence (or nonexistence) of the boundary layer solution to the inflow problem when the right end state (0,+\infty) belongs to the subsonic, transonic, and supersonic regions, respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave is not necessarily small. The proofs are given by the elementary energy method.

We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time H\"older continuity of the weak solution is also discussed as a byproduct.

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordstr\"om manifolds.

The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier–Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semiimplicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semiimplicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method.

In this article, we continue the work in Guan-Li and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magneto-hydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.

We establish local higher integrability estimates for upper gradients of vector-valued parabolic quasi-minimizers in metric measure spaces, satisfying a doubling property and supporting a weak Poincaré inequality.

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In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure pe(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.

A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. The minimal two species periodic assembly is the one with the least energy per lattice cell area. There is a parameter $b$ in $[0,1]$ and the type of the lattice associated with the minimal assembly varies depending on $b$. There are several threshold defined by a number $B=0.1867...$ If $b \in [0, B)$, the minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, the minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, the minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. Only when $b=1$, this rhombic lattice is the hexagonal lattice. None of the other values of $b$ yields the hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where the hexagonal lattice is ubiquitously observed.

In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.

In this paper, we prove the uniqueness of topological multivortex solutions for the self-dual Maxwell–Chern–Simons U(1)U(1) model if the Chern–Simons coupling parameter is sufficiently large and the charge of electron is sufficiently small or large. On the other hand, we also establish the sharp region of the flux for non-topological solutions and provide the classification of radial solutions of all types in the case of one vortex point.