On the heat flow of equation of surfaces of constant mean curvatures

We consider the initial and boundary value problem of heat flow of equation of surfaces of constant mean curvatures. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solutions to the heat flow that converges to zero in H ¹ exponentially as time goes to infinity.     

Expression of curvature tensors and some applications

We get an explicit expression of curvature operators in terms of at most eight terms of sectional curvatures. Some applications of this result are also given, particularly we improve a result of Chen-Tian related to the first Chern class of admissible surfaces in pinched manifolds. We also characterize in a simple way all functionsk(x, y) which can be sectional curvatures of some curvature operatorR.Supported by CNPq, Brazil and NNSFC.     

Trapping quasiminimizing submanifolds in spaces of negative curvature

LetM be a Hadamard manifold with all sectional curvatures bounded above by some negative constant. A well-known lemma essentially due to M. Morse states that every quasigeodesic segment inM lies within an a priori bounded distance from the geodesic arc connecting its endpoints. In this paper we establish an analogue of this fact for quasiminimizing surfaces in all dimensions and codimensions; the only additional requirement is that the sectional curvatures ofM be bounded from below as well. We apply this estimate to obtain new solutions to the asymptotic Plateau problem in various settings. The second author was supported by the Swiss National Science Foundation and enjoyed the hospitality of the University of Bonn. The collaboration between the authors was facilitated by the program GADGET II.     

Limiting angle of Brownian motion on certain manifolds

Summary. Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire S ^m−1 . This improves a result of Hsu and March. Received: 7 December 1994/In revised form: 2 September 1995     

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c ₁/{r ² ln r} and below by −c ₂ r ², where c ₁ and c ₂ are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M. Received: 19 September 1997     

Concentration on minimal submanifolds for a Yamabe-type problem

We construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDE's on the submanifold with a singular term of attractive or repulsive type is established.     

Global solutions of the compressible navier-stokes equations with larger discontinuous initial data

We prove the global existence of weak solutions to the Navier-Stokes equations for compressible, heat-conducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy. and that the magnitudes of singularities in the solution decay to zero.     

On the Asymptotic Dirichlet Problem for the Minimal Hypersurface Equation in a Hadamard Manifold

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold M of dimension n ≥ 2 for a large class of operators containing, in particular, the p-Laplacian and the minimal graph operator. We extend several existence results obtained for the p-Laplacian to our class of operators. As an application of our main result, we prove the solvability of the asymptotic Dirichlet problem for the minimal graph equation for any continuous boundary data on a (possibly non rotationally symmetric) manifold whose sectional curvatures are allowed to decay to 0 quadratically.     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.     

Asymptotical flatness and cone structure at infinity

We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends, and we classify (except for the case dim M=4, where it remains open if one of the theoretically possible cones can actually arise) for simply connected ends all possible cones at infinity. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat m-manifold with nonnegative sectional curvature is isometric to , whereS is an asymptotically flat surface. Received: 5 January 2000 / Published online: 19 October 2001     

Dirichlet Problem at Infinity for $\mathcal A$-Harmonic Functions

We study the Dirichlet problem at infinity for -harmonic functions on a Cartan–Hadamard manifold M and give a sufficient condition for a point at infinity x ₀∈M(∞) to be -regular. This condition is local in the sense that it only involves sectional curvatures of M in a set U∩M, where U is an arbitrary neighborhood of x ₀ in the cone topology. The results apply to the Laplacian and p-Laplacian, 1<p<∞, as special cases.      

Self-similar solutions for active scalar equations in Fourier–Besov–Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of positive-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier–Besov–Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large \(L^{2}\) -norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burgers’ equation, 2D vorticity equation, 2D generalized SQG, 3D magneto-geostrophic equations, among others.     

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.     

Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in L^p and the velocity u converges to the steady state velocity in W^1,p for any 1p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. Based on the existence results of global classical solutions, we prove that when t tends to infinity, the solution approaches a combination of C¹ travelling wave solutions, provided that L¹ ∩ L^∞ norm of the initial data as well as its derivative are bounded. Application is given for the time‐like extremal surface in Minkowski space. Copyright © 2006 John Wiley & Sons, Ltd.     

On the global existence and small dispersion limit for a class of complex Ginzburg–Landau equations

In this paper we consider a class of complex Ginzburg–Landau equations. We obtain sufficient conditions for the existence and uniqueness of global solutions for the initial‐value problem in d‐dimensional torus ??^d, and that solutions are initially approximated by solutions of the corresponding small dispersion limit equation for a period of time that goes to infinity as dispersive coefficient goes to zero. Copyright © 2008 John Wiley & Sons, Ltd.     

Real Solutions of the First Painlevé Equation with Large Initial Data

We consider three special cases of the initial value problem of the first Painlevé (PI) equation. Our approach is based on the method of uniform asymptotics introduced by Bassom et al. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained.     

Global dynamics of the Boussinesq‐Burgers system with large initial data

In this paper, we study the global existence and asymptotic behavior of the Boussinesq‐Burgers system subject to the Dirichlet boundary conditions. Based on the L^p(p > 2) estimates of the solution, which are different from the standard L²‐based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd.     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.