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 RPUiyj PRROPTRO ꏊ PSURPinzقUK~[eBOPj u r J icEHj u 1kS̒̎_^ uv| 1kS̒^鎿_lĈƎ_̒ԋ𒲂ׂĎnɂĐlvZsƁC $u$ $L^{\infty}$-m $t^{-1/2}$ ̃I[_[Ō邪C_̑x $V(t)$ $t^{-3/2}$ ̃I[_[Ō邱Ƃώ@łD{̎藝͉̊e_]^̂ŁČۂɐ邱ƂłDؖ̃|CǵC 1kNavier-Stokes̊{ɑ΂e_] [T.-P. Liu and Y. Zeng, 1997] ܂p邱ƂłDuł͎藝Ɗ֘AЉCؖ̊TvD

 RPUiyj POOOPQOO ꏊ PSURPinzقUK~[eBOPj u ] K icEHj u Leray-Hopf̎̍\ɊւV݂ɂ uv| lZɊÂIȍ@ɂāC3LË̔񈳏kNavier-Stokesɑ΂Leray-Hopf̎\D̕@ʂāC񈳏kNavier-Stokes̎ԔWɂččlC@Leray-Hopf̗̎ɂǂ̂悤Ɋ^邩qׂD

 RPTij PRROPTRO ꏊ PSURPinzقUK~[eBOPj u c F iE_j u ̕Ɖ̕ uv| MΗグāCp[^[̕ωɂp^[C萫C̕ωȂǂɊւ镪lDǂ𖾂ĂȂȂǂqׂD

 RPTij POOOPQOO ꏊ PSURPinzقUK~[eBOPj u r J icEHj u C̒̍̉^̕qC̗͊wɂƂÂ uv| {uł́CC̒̍̉^𕪎qC̗͊ẘϓ_璲ׂЉDȂ킿CĈ𑽐̕qWcƂ݂ȂC̓vIȋLqpċĈƍ̂̑ݍp𒲂ׂDu҂̌ʂЉCw͂␔lvZɂČݒmĂ邱ƂЉD

 PPWij PUST ꏊ PSVRRinzقVK~[eBORj u 㓡c ikEdqj u Vortex dynamics of measure-valued solutions for the filtered Euler system uv| 񎟌t鐫ƂĔSɌɂ闬̂̃GXgtB[U킪, S̕łEuler̎U퐫𗝉邤ŏdvƍl. , 񎟌EulerɂĂ͈Ӊ𐫂lł͉̃GXgtB[U킪NȂƂؖĂ, U퐫̍\ɂ͂萳̎アlKv. {łEuler̐fłfiltered-Eulerl, ɓ_QQwȂǂ̑xl̉Qxɒڂ. {uł͓񎟌filtered-Euleȓxlɂ, p[^̋Ɍł̓񎟌Euler̎Ƃ̊֌W, QwɂQp^[(~VKw Robert KrasnyƂ̋), ܂_QՓ˂ɂGXgtB[U(sw MVƂ̋)̌ʂɂĂb\ł.

 PQQPij PUST ꏊ PSVRRinzقVK~[eBORj u J V icE_j u Classical Solvability of Radial Viscous Fingering Problems in a Hele-Shaw Cell uv| We discuss two-phase radial viscous fingering problem in a Hele-Shaw cell, which is a nonlinear problem with a free boundary for elliptic equations. Unlike the Stefan problem for heat equations Hele-Shaw problem is of hydrodynamic type. In this talk the classical solvability of two-phase Hele-Shaw problem with radial geometry is established by applying the same method as for the Stefan problem and justifying the vanishing the coefficients of the derivative with respect to time in parabolic equations. Included historical survey, present and future problems.

 PPPUij PUST ꏊ PSVRRinzقVK~[eBORj u gA iHEj u 􉽊wIx_gϋȗ uv| ԂɈˑēȖʂϋȗłƂ́A̋Ȗʂ̑xϋȗɊe_eœƂłDϋȗ͋Ȗʂ̔M̂悤Ȃ̂ŁAԂɊւċȖʐςzƂȂĂD̕ϕ\pēٓ_ȖʂɂĂłAʖ@ȂǗlXȒ莮邪A̒ł􉽊wIx_pBrakkeɂ莮́iɏȖʂ̏ꍇɂ͒mĂjʓIȐ_ȂǁA[_DʂɊ炩Ƃ͌ȂȖʓIȕWiႦΈꎟ̏ꍇ̓lbg[N̏Wj^ƂAf[^ɂϋȗ̎ԑ𑶍݂͒NsłA2015NɎƂłDuł͂̉̍\@ɂĊTvqׁAɂ̍\@琬藧̐ɂĐD

 PPQij PUST ꏊ PSVRRinzقVK~[eBORj u 쓇 G iEHj u Mathematical analysis for a model system of viscoelastic fluids uv| ĜƌĂ΂闬̂̃fƂāAkSễfnl@B͒ʏ̗̕nɂāA̓e\^\e\iconfiguration tensorjŋLqAɂ̌e\όe\ɈˑĎԔW郂fłAwIɂ͊ɘaIoȌ^nɕނ̂łB̌nɑ΂āA쓇EYong (2004, 2009) ̐w͂̈ʘ_Kp\؂BȂ킿AwIGgs[݂邩A萫𖞂𒲂ׂBɁAԂPnɑ΂Ă͏ڍׂȌʂ^B܂AԂRnɑ΂萫ɂĂyB̐wf̍\zɊւǍۑɂĂyB

 POQSij PUST ꏊ PSVRSinzقVK~[eBOSj u c iEHj u Kinetic theory for a simple modeling of phase transition uv| A simple kinetic model, which is presumably minimum, for the phase transition of the van der Waals fluid is presented. In the model, intermolecular collisions for a dense gas has not been treated faithfully. Instead, the expected interactions as the non-ideal gas effect are confined in a self-consistent force term. Collision term plays just a role of thermal bath. Accordingly, it conserves neither momentum nor energy, even globally. It is demonstrated that (i) by a natural separation of the mean-field self-consistent potential, the potential for the non-ideal gas effect is determined from the equation of state for the van der Waals fluid, with the aid of the balance equation of momentum, (ii) a functional which monotonically decreases in time is identified by the H theorem and is found to have a close relation to the Helmholtz free energy in thermodynamics, and (iii) the Cahn--Hilliard-type equation is obtained in the continuum limit of the present kinetic model. Numerical simulations have also been carried out for both the Cahn--Hilliard-type equation and the kinetic equation to demonstrate the occurrence of phase transition from unstable uniform equilibrium states. The present talk is based on the references below. [1] S. Takata and T. Noguchi, J. Stat. Phys., Vol. 172, 880 (2018) [2] S. Takata, T. Matsumoto, A. Hirahara, and M. Hattori, arxiv:1807.04630

 VQOij PUST ꏊ PSVRRinzقVK~[eBORj u BY icEHj u Initial value problem to a shallow water model with a floating solid body uv| In this talk we are concerned with the well-posedness of the initial value problem to a shallow water model for two-dimensional water waves with a floating solid body. We consider three cases: the body is fixed, the motion of the body is prescribed, and the body moves freely according to Newton's laws. The difficulty of the analysis comes from the fact that we have to treat the contact points, where the water, the air, and the solid body meet. This model yields a new type of free boundary problems for a quasilinear hyperbolic system. We will report that the initial value problem to this model is in fact well-posed. This result is based on the joint research with David Lannes at University of Bordeaux.

 VPPij PVPTPWPT ꏊ PSURP/ainzقUK~[eBOP/Paj u Professor Mark Groves (University of Saarlandes) u Small-amplitude static periodic patterns at a fluid-ferrofluid interface uv| We establish the existence of static doubly periodic patterns (in particular rolls, squares and hexagons) on the free surface of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. A novel formulation of the ferrohydrostatic equations in terms of Dirichlet-Neumann operators for nonlinear elliptic boundary-value problems is presented. We demonstrate the analyticity of these operators in suitable function spaces and solve the ferrohydrostatic problem using an analytic version of Crandall-Rabinowitz local bifurcation theory. Criteria are derived for the bifurcations to be sub-, super- or transcritical with respect to a dimensionless physical parameter.

 VPPij PUPVOO ꏊ PSURP/ainzقUK~[eBOP/Paj u Professor Snorre Christiansen (University of Oslo) u A justification of the definition of curvature in Regge Calculus uv| Regge calculus was introduced in 1961 as a coordinate free and discrete analogue of Einstein's theory of gravitation. Yet, in spite of its beautiful geometric features, the bulk of numerical computations in general relativity is, as of today, carried out by other methods, probably because of a lack of understanding of its stability and convergence properties. Surprisingly, Regge defined a curvature of simplicial manifolds, equipped with a piecewise constant metric, with a partial continuity requirement between simplices. I will provide a justification of this definition by a smoothing procedure: the curvature of the smoothened metrics converges to the curvature defined combinatorially by Regge, in the sense of measures, as the smoothing parameter goes to zero.

 UQXij PUST ꏊ PSVRRinzقVK~[eBORj u O iLEȊwj u On transverse stability of line solitary waves of the Benney-Luke equation uv| ԂQBenney-Luke̐Ǘg̉f萫ɂčuBBenney-LukeKPlARʔg̒gߎfłB{uł͕\ʒ͂̎アꍇɁAǗgɂȂ邱ƂЉAǗg̕ϒLqmodulation equation̓osB

 UQQij PUST ꏊ PSVRRinzقVK~[eBORj u Professor Yue-Jun Peng (University of Clermont Auvergne) u Stability of non-constant equilibrium solutions for Euler-Maxwell systems uv| Euler-Maxwell systems are fluid models arising in plasma physics. In both isentropic and non-isentropic cases, such systems admit non-constant steady-state solutions with zero velocity. For the Cauchy problem or the periodic problem with initial data near the steady-states, we show global existence and the convergence of smooth solutions toward these states as the time goes to infinity. In the proof of the above result, we mainly use three techniques to yield energy estimates. These techniques are the choice of symmetrizer of the systems, the existence of anti-symmetric matrices and an induction argument on the order of space-time derivatives of solutions.

 UQOij PUST ꏊ PSURP/ainzقUK~[eBOP/Paj u Professor Shih-Hsien Yu (National University of Singapore) u Wave motions around a 2-D viscous Burgers' shock profile uv| In this talk we introduce the Laplace wave trains to form a basis for the 2-D wave scattering around the 2-D inviscid Burgers' shock wave for the viscous Burgers' profile for a 2-D Burgers' equation. With all those wave trains around the inviscid shock wave, one can construct the wave trains for the problem linearized around the viscous shock profile. After the complete structure of the wave scattering in terms of the Laplace wave train, one can invert the wave train information into a pointwise space-time structure of the Green's function for the problem linearized around the Burgers' shock profile. With the pointwise structure of the Green's function, the nonlinear wave scattering follows.

 TROij PUST ꏊ PSURP/ainzقUK~[eBOP/Paj u Professor David Lannes iUniversity of Bordeauxj u The shoreline problem for the nonlinear shallow water and Green-Naghdi equations uv| The nonlinear shallow water equations and the Green-Naghdi equations are the most commonly used models to describe coastal flows. A natural question is therefore to investigate their behavior at the shoreline, i.e. when the water depth vanishes. For the nonlinear shallow water equations, this problem is closely related to the vacuum problem for compressible Euler equations, recently solved by Jang-Masmoudi and Coutand-Shkoller. For the Green-Naghdi equation, the analysis is of a different nature due to the presence of linear and nonlinear dispersive terms. We will show in this talk how to address this problem. This is a joint work with G. Métivier.

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