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组 织 者:张晓、吴小宁

会议地点:北京 中国科学院






















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  目:The geometric aspect of shock formations

  要:We will present the geometric description of shock formations for 3D quasilinear wave equations. In particular, we show that how smooth solutions break down in finite time. This is a joint work with Shuang Miao (EPFL).



  目:Dynamical black holes with prescribed masses in spherical symmetry

  要:In this talk, I will review our recent work on a construction of spherically symmetric global solution to the Einstein–scalar field system with large bounded variation norms and large Bondi masses. We show that similar ideas, together with Christodoulou's short pulse method, allow us to prove the following result: Given M_i greater or equal to M_f, there exists a spherically symmetric (black hole) solution to the Einstein scalar field system such that up to an error, the initial Bondi mass is M_i and the final Bondi mass is M_f. Moreover, if one assumes a continuity property of the final Bondi mass (which in principle follows from known techniques in the literature), then for any M_i>M_f, this result holds without the error loss. This is the joint work with Jonathan Luk and Sung-jin Oh.



  目:Center stable manifold for nonlinear wave equation with potential

  要:In this talk, we consider the defocusing energy critical wave equation with a trapping potential. When the potential decays fast enough, it is easy to show that all finite energy solutions exist globally, hence our main interest is to describe the long time dynamics. In the radial case, our previous works gave a complete answer and we were able to classify all the long time dynamics. Here we partly extend previous result to the nonradial case, and show that the set of initial data for which solutions scatter to an unstable excited state forms a finite co-dimensional path connected C^1 manifold in the energy space. This gives us a better-understanding of the non-generic behavior of solutions, with the generic behavior left as an open problem. This talk is based on joint works with Hao Jia, Wilhelm Schlag and Guixiang Xu.



  目:Instability of spherical naked singularities of a scalar field under gravitational perturbations

  要:It was proven by Christodoulou that in spherical symmetric solutions of the Einstein equations coupled with a massless scalar field, the naked singularities are instable, in the sense that the space of the initial data leading to the formation of naked singularities is of codimension at least one. According to the proof of this theorem, we consider the following characteristic initial data problem of Einstein-scalar field equations: Let e be a singularity whose causal past is spherically symmetric and Ce be the boundary of the causal past, and C0 be the outgoing boundary of the causal future of some spherical section of Ce . The initial data is given on C0 and Ce and in particular the data on C0 is arbitrary with no symmetries assumed. We prove that if the initial shear tensor on C0 satisfies some additional condition, then the future development has a sequence of closed trapped surfaces approaching e, so that we may say e is not naked. We can find a space of initial shear tensors on C0 such that the subspace of initial shear tensors not satisfying the additional condition is of codimension at least 1. So in some certain sense, we may say a spherical naked singularity is not stable under gravitational perturbations. This work is joint with Jue Liu.



  目:Dirac equation in nonextreme Kerr-Newman-AdS spacetime

  要:In non-extreme Kerr-Newman-AdS spacetime, we prove that there is no nontrivial Dirac particle which is L^p for some p outside and away from the event horizon. In particular, the normalizable massive Dirac particle with mass greater than |Q|+ κ/2 must either disappear into the black hole or escape to infinity. Furthermore, we prove that any Dirac particle with eigenvalue |λ| <κ/2 must be L^2 outside and away from the event horizon.






  目:Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

  要:In this talk, we will present some new results of well posedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. It is a joint work with Kunio Hidano.



  目:Wave-Klein-Gordon Model for Einstein-massive scalar field system

  要:The global non-linear stability of Minkowski space has attracted lots of attention of the mathematicians. After the pioneer work of S.Klainerman and I. Rodnianski, People have made lots of effort and established similar results in various context. In this talk we are especially interested in the case where the gravitational field is coupled with a massive scalar field. We will formulate a simple but not trivial system to show a series of analytical tools including the hierarchy of energy bounds, the sup-norm estimates of Klein-Gordon equations in curved background metric, the secondary bootstrap argument etc.


8、报告人:魏昌华 (浙江理工大学)

  目:Classical solutions to the relativistic Euler equations for a linearly degenerate equation of state

  要:In this talk, I will introduce our recent results on the classical solutions of the Cauchy problem to the relativistic Euler equations for a linearly degenerate equation of state. I will show the relationship between the conditions “linearly degenerate” and “null condition” of the wave equation under suitable assumptions. Based on this, we mainly show the global existence of the 3D radial solution to the relativistic Euler equations. I will also introduce the blowup mechanism for 22 system when the initial data is large. These works are inspired by a conjecture of Majda on symmetric hyperbolic systems with totally linearly degenerate characteristics.



  目:Global smooth solutions to relativistic membrane equations with large data  要:This paper is concerned with the Cauchy problem for the relativistic membrane equation (RME) embedded in $\mathbb R^{1+(1+n)}$ with $n=2, 3$. We show that the RME with a class of large data (in energy norm) admits a uniquely global and smooth solution. The data is indeed given by the short plus type, which is introduced by Christodoulou in his work on formation of black holes. In particular, we construct two multiplier vector fields adapted to the membrane geometry and present an effective way for proving the global existence of quasilinear wave equations with double null structure. This work generalize the results of Miao, Pei and Yu on the semilinear wave equation to the quasilinear case. This is joint work with Changhua Wei.



  目:Uniqueness of the Mean Field Equation and Rigidity of Hawking Mass

  要:In this paper, we prove that the even solution of the mean field equation $\Delta u=\lambda (1-e^u) $ on $S^2$ must be axially symmetric when $4<\lambda \leq 8$. In particular, zero is the only even solution for $\lambda=6$. This implies the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere with even symmetry.






  目:Toroidal marginally outer trapped surfaces in the closed Friedmann-Lemaitre -Robertson-Walker universe

  要:We explicitly construct toroidal MOTS in the closed FLRW universe. This construction is used to assess the quality of certain isoperimetric inequalities recently proved in axial symmetry. We also show that these constructed toroidal MOTS are unstable. This talk is based on a joint work with Patryk Mach.



  目:A class of solutions to the constraint equations

  要:In this talk, I will study a class of solutions to the constraint equations, which is close to the Euclidean space in the asymptotic sense, as well as the space structure of this solution set.



  目:On The Negativity of Ricci Curvatures of Complete Conformal Metrics

  要:A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this talk, I will disscuss whether these metrics have negative Ricci curvatures. We will provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green's function and the positive mass theorem play essential roles in certain cases. On the other hand, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space.



  目:Well-posedness and regularity of viscosity solution to non-monotone weakly coupled system of evolutionary Hamilton-Jacobi equations

  要:In this talk, we will present a well-posedness result of the viscosity solution to Cauchy problem of certain non-monotone weakly coupled systems of first order evolutionary Hamilton-Jacobi equations. Moreover, for a typical model problem, we obtain the locally Lipschitz continuity of the viscosity solution and a series of corollaries from this property. This talk is based on a joint work with Lin Wang and Jun Yan.



  目:Peeling property of Bondi-Sachs metrics for nonzero cosmological constant

  要:We show that the peeling property still holds for Bondi-Sachs metrics with nonzero cosmological constant under the new boundary condition which satisfies the Sommerfeld's radiation condition. This should indicate the new boundary condition is natural. Moreover, we construct some nontrivial vacuum Bondi-Sachs metrics without the Bondi news, which gives a new feature of gravitational waves for nonzero cosmological constant. This is a joint work with Xiao Zhang.









  目:Relationship between BS quantities and source of gravitational radiation in asymptotically de Sitter spacetime

  要:Gravitational radiation plays an important role in astrophysics. Based on the fact that our universe is expanding, the gravitational radiation when a positive cosmological constant is presented has been studied along with two different ways recently, one is the Bondi-Sachs (BS) framework in which the result is shown by BS quantities in the asymptotic null structure, the other is the perturbation approach in which the result is presented by the quadrupoles of source. Therefore, it is worth to interpret the quantities in asymptotic null structure in terms of the information of the source. In this talk, we will discuss this problem and show the explicit expressions of BS quantities in terms of the quadrupoles of source in asymptotically de Sitter spacetime.



  目:Radiation Field of General Linear Wave Equations

  要:In this talk, we consider the Null-Timelike Problem and Radiation field of the linear wave equations with that the metric is asymptotically flat in Bondi-Sachs coordinate. This is joint work with Professor Qing Han.



  目:Some results on nonlinear elastic waves

  要:In this talk, I will give some results and their brief proof for compressible nonlinear elastic waves. In the 3-D case, some space-time L^2 estimates of Keel-Smith-Sogge type are established for perturbed linear elastic waves. In the 2-D case, based on the variational structural of nonlinear elastic waves, some null conditions are introduced using Zhou's suggestion. Under the null condition and the radial symmetry assumption on the initial data, global existence of small smooth solutions are proved for the Cauchy problem and exterior problem.



  目:Memory effect and soft theorem

  要:Soft theorem is an important theoretical result in quantum field theory. In 2014, Strominger and his colleague related this theorem with the famous gravitational effect based on the Gauge/gravity duality. We present a new type of electromagnetic memory. It is a `magnetic' type, or B mode, radiation memory effect. Rather than a residual velocity, we find a position displacement of a charged particle induced by the B mode radiation with memory. Our result show that Strominger's conjecture should be right up to second order.