We derive new, localized geometric integral identities for solutions to the $3D$ compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The identities are coercive in the first derivatives of the specific vorticity and the second derivatives of the entropy, and the error terms exhibit remarkable regularity and null structures. Our framework allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution's higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric and ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. Our main new contribution is our analysis of the positive co-dimension, spacelike boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation, we observe several crucial cancellations, which in total show that the boundary terms have a good sign.
We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the "wave part" of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify…
We derive a new formulation of the $3D$ compressible Euler equations with dynamic entropy exhibiting remarkable null structures and regularity properties. Our results hold for an arbitrary equation of state (which yields the pressure in terms of the density and the entropy) in non-vacuum regions where the speed of sound is positive. Our work is an extension of our prior joint work with J. Luk, in which we derived a similar new formulation in the special case of a barotropic fluid, that is, when the equation of state depends only on the density. The new formulation comprises covariant wave equa…
In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in the density or stress equation is everywhere in the …
In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, …
For one dimensional or multidimensional compressible Euler system of polytropic gases, it is well known that the smooth solution will generally develop singularities in finite time. However, for three dimensional Chaplygin gases, due to the crucial role of "null condition" in the potential equation which is derived by the irrotational and isentropic flow, P.Godin in [9] has proved the global existence of a smooth 3-D spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a constant state. It is noted that there are some…
We study the Cauchy problem for the $3D$ compressible Euler equations under an arbitrary equation of state with positive speed of sound, aside from that of a Chaplygin gas. For open sets of smooth initial data with non-trivial vorticity and entropy, our main results yield a constructive proof of the formation, structure, and stability of the singular boundary, which is the set of points where the solution forms a shock singularity, i.e., where some first-order Cartesian coordinate partial derivatives of the velocity and density blow up. We prove that in the solution regime under study, the sin…
Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime
Georgia Institute of Technology（佐治亚理工学院）；Vanderbilt University（范德堡大学）
We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space H^{\frac{n-1}{2}}(R^n) for n=2, 3. The proof proceeds by using the Fierz identities to rewrite the equation in a form where the null structure of the system is readily apparent. This null structure is then exploited via bilinear estimates in spaces based on the null frame spaces of Tataru. We hope that the spaces and estimates used here can be applied to other nonlinear Dirac equations in the scale invariant setting. Our work complements recent results of Bejenaru-Herr who pr…
We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \fw) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The classical local well-posedness result for the compressible Euler equations in $3$-D holds for the initial data $v, \varrho \in H^{s+\f12},\, s>2$. Due to the works of Smith-Tataru and Wang for the irrotational isentropic case, the local well-posedness can be achieved if the data satisfy $v, \varrho \in H^{s}$, with $s>2$. In the incompressible case the solution is prov…
We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in $Hm$, for $m$ large enough. To do so, we introduce a specific symmetrisation allowing areas of null density. Next, we make estimates in $Hm$, using for some terms the estimates done by M. Grassin, who proved the same theorem in the easier case of a perfect polytropic gas. We treat the remaining terms separately, due to their non-linearity.
In this first of four articles, we study a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). In CFT, these are null-state equations and conformal Ward identities. They govern partition functions for the continuum limit of a statistical cluster or loop model, such as percolation, or more generally the Potts models and O$(n)$ models, at the statistical mechanical critical point. (SLE partition functions also satisfy these equations.) For such a lattice model in a pol…
In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations, even for 3D case of compressible gas flow or 3D case of non-stationary flow of incompressible fluid. But there is an essential deficiency of non-stationary solutions indeed. We explore the ansatz of derivation of non-stationary solution for the Navier-Stokes equations in the case of incompressible flow, which was suggested earlier. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard…
