Schedule-at-a-Glance

17 PRIMA 2013 AbstractsSpecial Session 8Hyperbolic Conservation Laws and RelatedApplicationsSome counterexamples in the theory of conservationlawsAlberto BressanPennsylvania State University, USAbressan@math.psu.eduThe first part of the talk is concerned with sticky particlemodels in two space dimensions. Examples of Cauchyproblems can be constructed with two and with zero solutions,respectively. Similar ideas apply to the equationsof pressureless gases in two space dimensions, with L ∞initial data.The second set of examples concern the p-system of isentropicgas dynamics. For initial data with large totalvariation, one can arrange repeated wave-front interactionsin such a way that the total strength of all wavefronts becomes arbitrarily large.Nonlinear mixed type equations arisen inMach reflectionShuxing ChenFudan University, Chinasxchen@fudan.edu.cnIn the study of Mach reflection we find a kind of nonlinearmixed type equations. The solvability of the generalizedTricomi problem of the mixed type equation leads thestability of corresponding Mach configuration.Self-similar vortex spiral solutions of the 2dincompressible Euler EquationsVolker EllingUniversity of Michigan, Ann Arbor, USAvelling@umich.eduVortex spirals are ubiquitous in fluid flow, for example asturbulent eddies or as trailing vortices at aircraft wings.However, there are few proofs of existence for any of thecommon fluid models. We consider solutions of the incompressibleEuler equations that have vorticity stratifyinginto algebraic spirals. The solutions are selfsimilar:velocity v(t, x) = t m−1 v(t −m x), for similarity exponent1< m < ∞. Selfsimilar flows are special solutions of the2full initial-value problem, but obtained by solving moretractable boundary value problems. The key to the existenceproof is an coordinate change which is implicit,depending on the a priori unknown solution. We willalso discuss the importance of the program for showingnon-uniqueness in the initial-value problem for the 2d incompressibleEuler solutions.Some results about compressible Oldroyd-BDaoyuan FangZhejiang University, Chinadyf@zju.edu.cnIn this talk, I will show some results about the compressibleOldroyd-B model, which is the joint work withRuizhao Zi. For such medel, we proved that the systemadmits a unique local strong solution with initial densityvanishes from below, and give a blow-up criterion; Inthe framework of critical spaces, we establish the globalsolutions if the initial data and coupling constant are sufficientlysmall. We also proved that as the Mach numbertends to zero, the global solution converges to the solutionto the corresponding incompressible model in somefunction spaces,and obtained a kind of the converge rates.Shock reflection and von Neumann conjecturesMikhail FeldmanUniversity of Wisconsin-Madison, USAfeldman@math.wisc.eduWe discuss shock reflection problem for compressible gasdynamics, and von Neumann conjectures on transitionbetween regular and Mach reflections. Then we describerecent results on existence of regular reflection solutionsfor potential flow equation up to the detachment angle,and discuss some techniques. The approach is to reducethe shock reflection problem to a free boundary problemfor a nonlinear equation of mixed elliptic-hyperbolic type.Open problems will also be discussed. The talk will bebased on the joint work with Gui-Qiang Chen.Normal forms and a Burgers-Hilbert equationJohn K. HunterUniversity of California, Davis, USAjkhunter@ucdavis.eduThe Burgers-Hilbert equation arises as a model equationfor the motion of a vortex patch or vorticity discontinuityin a two-dimensional, inviscid, incompressible fluid flow,and describes the effect of nonlinear steepening on an interfaceor wave that oscillates at a constant backgroundfrequency. For small amplitudes, these oscillations delaywave breaking. We will explain how non-standardnormal form methods can be used to prove an enhancedlife-span of small smooth solutions of the Burgers-Hilbertequation in comparison with the inviscid Burgers equation.These normal form methods can be applied to otherquasilinear wave equations, for which the Burgers-Hilbertequation provides a useful test case. This is joint workwith M. Ifrim, D. Tataru, and D. Wong.Incompressible limit of the non-isentropicideal magnetohydrodynamic equationsSong JiangInstitute of Applied Physics and Computational Mathematics,Chinajiang@iapcm.ac.cnWe first give a short review of results on the low Machnumber limit for the compressible magnetohydrodynamicequations. Then, we study the incompressible limit of thecompressible non-isentropic ideal magnetohydrodynamicequations with general initial data in the whole space IR d(d = 2, 3), and establish the existence of classic solutionson a time interval independent of the Mach number. Finally,by deriving uniform a priori estimates, we obtainthe convergence of the solution to that of the incompressiblemagnetohydrodynamic equations as the Mach numbertends to zero.(joint-work with Q.C. Ju and F.C. Li)The Cauchy problem to the Kazhikhov-Vaigant model in compressible flowQuansen JiuCapital Normal University, Chinajiuqs@mail.cnu.edu.cnIn this talk, we will present some recent progresses ofthe global well-posedness of the Cauchy problem to the