We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.

Let G=(V,E) be a finite weighted graph, and Ω⊆V be a domain such that Ω^∘≠∅. In this paper, we study the following initial boundary problem for the non-homogenous wave equation ∂^2_t u(t,x) − Δ_Ω u(t,x) = f(t,x), (t,x)∈[0,∞)×Ω^∘ u(0,x)=g(x), x∈Ω^∘, ∂_t u(0,x)=h(x), x∈Ω^∘, u(t,x)=0, (t,x)∈[0,∞)×∂Ω, where Δ_Ω denotes the Dirichlet Laplacian on Ω^∘. Using Rothe's method, we prove that the above wave equation has a unique solution.

For the free surface problem of the highly subsonic heat-conducting inviscid flow in 2-D and 3-D, a priori estimates for geometric quantities of free surfaces, such as the second fundamental form and the injectivity radius of the normal exponential map, and the Sobolev norms of fluid variables are proved by investigating the coupling of the boundary geometry and the interior solutions. An interesting feature for the free surface problem studied in this paper is the loss of one more derivative than the problem of incompressible Euler equations for which a geometric approach was introduced by Christodoulou and Lindblad in [11]. Due to the loss of one more derivative and loss of symmetry of equations, the geometric approach in [11] needs to be substantially developed by exploring the interaction of large variation of temperature, heat-conduction, non-zero divergence of the fluid velocity and the evolution of free surfaces.

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We study an asymptotic estimate on the number of negative eigenvalues of the Schr\"odinger operators on unbounded fractal spaces which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpi\'nski gasket and unbounded generalized Sierpi\'nski carpets. Final, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

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.

The Cauchy problem for the barotropic compressible Navier--Stokes equations on the whole two-dimensional space with vacuum as far field density is considered. When the shear viscosity is a positive constant and the bulk one is a power function of density with the power bigger than four-thirds, the global existence and uniqueness of strong and classical solutions is established. It should be remarked that there are no restrictions on the size of the data.

This paper concerns the physical behaviors of any solutions to the one dimensional compressible Navier-Stokes equations for viscous and heat conductive gases with constant viscosities and heat conductivity for fast decaying density at far fields only. First, it is shown that the specific entropy becomes not uniformly bounded immediately after the initial time, as long as the initial density decays to vacuum at the far field at the rate not slower than $O\left(\frac1{|x|^{\ell_\rho}}\right)$ with $\ell_\rho>2$. Furthermore, for faster decaying initial density, i.e., $\ell_\rho\geq4$, a sharper result is discovered that the absolute temperature becomes uniformly positive at each positive time, no matter whether it is uniformly positive or not initially, and consequently the corresponding entropy behaves as $O(-\log(\varrho_0(x)))$ at each positive time, independent of the boundedness of the initial entropy. Such phenomena are in sharp contrast to the case with slowly decaying initial density of the rate no faster than $O(\frac1{x^2})$, for which our previous works \cite{LIXINADV,LIXINCPAM,LIXIN3DK} show that the uniform boundedness of the entropy can be propagated for all positive time and thus the temperature decays to zero at the far field. These give a complete answer to the problem concerning the propagation of uniform boundedness of the entropy for the heat conductive ideal gases and, in particular, show that the algebraic decay rate $2$ of the initial density at the far field is sharp for the uniform boundedness of the entropy. The tools to prove our main results are based on some scaling transforms, including the Kelvin transform, and a Hopf type lemma for a class of degenerate equations with possible unbounded coefficients.

We study homogenization of G-equation with a ow straining term (or the strain G-equation) in two dimensional periodic cellular ow. The strain G-equation is a highly non-coercive and non-convex level set Hamilton-Jacobi equation. The main objective is to investigate how the ow induced straining (the nonconvex term) in uences front propagation as the ow intensity A increases. Three distinct regimes are identified. When A is below the critical level, homogenization holds and the turbulent ame speed sT (effective Hamiltonian) is well-defined for any periodic ow with small divergence and is enhanced by the cellular ow as $s_T \ge O(A/logA)$. In the second regime where A is slightly above the critical value, homogenization breaks down, and $s_T$ is not well defined along any direction. Solutions become a mixture of fast moving part and a stagnant part. When $A$ is sufficiently large, the whole ame front ceases to propagate forward due to the flow induced straining. In particular, along directions $p = (1; 0)$ and $(0;1)$, $s_T$ is well-defined again with a value of zero (trapping). A partial homogenization result is also proved. If we consider a similar but relatively simpler Hamiltonian, the trapping occurs along all directions. The analysis is based on the two-player dierential game representation of solutions, selection of game strategies and trapping regions, and construction of connecting trajectories.

We review the existence results of traveling wave solutions to the reaction-diffusion equations with periodic diffusion (convection) coefficients and combustion (bistable) nonlinearities. We prove that whenever traveling waves exist, the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions. In the case of bistable nonlinearity and one space dimension, we give an example of nonexistence of traveling waves which causes quenching (localization) of wavefront propagation. Quenching (localization) only occurs when the variations of the media from their constant mean values are large enough. Our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example. Finally, we comment on the role of the effective wave speeds in

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

We study convolution groups generated by completely monotone sequences and completely monotone functions. Using a convolution group, we define a fractional calculus for a certain class of distributions. When acting on causal functions, this definition agrees with the traditional Riemann-Liouville definition for t>0 but includes some singularities at t=0 so that the group property holds. Using this group, we are able to extend the definition of Caputo derivatives of order in (0,1) to a certain class of locally integrable functions without using the first derivative. The group property allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gronwall inequality (or comparison principle) with the most general conditions. This then opens the door of a priori energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions. Besides, we also obtain some interesting results about completely monotone sequences.

In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models. To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than |t|−5/6) and the presence of a large, codimension-1, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.

We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

This note considers the hyper-contractivity and the uniqueness of the weak solutions to the two dimensional Keller–Segel equations. We prove a refined hyper-contractive property and consequently obtain the uniqueness of global weak solutions provided that the initial data satisfy , and . We also extend the result to higher dimensions.

We consider the following mean field equation: $$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M, where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M, and  $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological degree counting formula for noncritical values of $\rho$. We also give several applications of this formula, including existence of curvature +1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.

We study the effective fronts of first order front propagations in two dimensions (n=2) in the periodic setting. Using PDE-based approaches, we show that for every α∈(0,1), the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in C^{1,α}(T^2,(0,∞)). This result can also be formulated in the language of stable norms corresponding to periodic metrics in T^2. Similar results were known long time ago when n≥3 for front speeds in C^∞(T^n,(0,∞)). Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is C^1, which cannot be a polygon, for given C^{1,1}(T^2,(0,∞)) front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

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For periodic initial data with density allowed to vanish initially, we establish the global existence of strong and weak solutions to the two-dimensional barotropic compressible Navier–Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is λ =ρ^β with β>4/3. These results generalize and improve the previous ones due to Vaigant–Kazhikhov [Sib. Math. J. 36 (1995) 1283–1316] who required β>3. Moreover, we also prove that the densities for both the strong and weak solutions remain bounded from above independently of time. As a consequence, it is shown that both the strong and weak solutions converge to the equilibrium state as time tends to infinity.

We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that, for n ≥ 3, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established by I. Babenko and F. Balacheff [Sur la forme de la boule unit\'e de la norme stable unidimensionnelle, Manuscripta Math. 119 (2006) 347--358] and M. Jotz [Hedlund metrics and the stable norm, Diff. Geometry Appl. 27 (2009) 543--550] in the form of stable norms as an extension of G. A. Hedlund's classical result [Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932) 719--739]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in$H$^{1}close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18].

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and NavierStokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation

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.

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.