In this paper, we show that under the assumptions of low regularity on the "wave part" (for more details on the "wave part") of the initial data, see Section 3.5, the regularity of the solutions of the relativistic Euler equations can be preserved for a short time. The details of the assumptions for the data in [9] and [37] will be discussed in Section 1.1.
Overview of Previous Low-Regularity Results
We emphasize again that for the general quasi-linear wave equation of the form (1.5) it is impossible to prove a good position result with data inH2. For the non-relativistic compressible Euler flow with vorticity and entropy, under the H2+ assumptions of "wave part" and "transport part" of the data, Disconzi-Luo-Mazzone-Speck [9] and Wang [37] proved locally good -positivity result for 3D compressible Euler equations.
A Brief Overview of the Strategy of the Proof
Inspired by the analysis of the 3D non-relativistic compressible Euler case in [9], we derive elliptic and Schauder estimates for the transport div-curl systems to find the H1+ε and CO,αnorms of C, D when the wave section is rough. We first note that the v-directional derivative of the vorticity and entropy gradient is favorable due to the transport phenomena.
Geometric Formulation of the Relativistic Euler Equations
For the 3D nonrelativistic compressible Euler equations with any equation of state, Speck [28] derived a system consisting of geometric wave equations and transport-div-curvature equations. See also Luk-Speck [22] and [23] for the geometric formulation of the compressible Euler in the barotropic case and its application to the shock formation problem.
Comparision with Low-regularity Results for 3D Non-relativistic Compressible Euler Equa-
The "transport-div-curl-part" formulation allows one to control the vorticity and entropy at a derivative level over the standard estimates. We take a similar approach when deriving elliptic div-curl estimates in L2space, where we must control the vorticity derivatives and entropy gradient with ∂∂∂~Ψ, the modified fluid variables C and D.
Main Idea of the Proof of Theorem 1.1
- Overview of elliptic and energy estimates
- Bootstrap assumptions
- Transport-Schauder estimates for the transport-div-curl system
- Reductions of the Strichartz type estimates
- Structures for the causal geometry of the acoustic space-time
- Control of the conformal energy
- Control of the acoustic geometry
This part of the result is well known and standard (see Section 1.1 for the introduction of the previous results). We must control them using the transport and Hodge-type equations for geometric quantities.
Paper Outline
Finally, we define the conformal energy and state the limit theorem of the conformal energy in Theorem 10.2, which plays a crucial role in the derivation of the decay estimates stated in Theorem 6.9. First, we define the conformal energy in Definition 10.1 followed by the limit theorem in Theorem 10.2.
Notations
In this section, we give the standard first-order relativistic Euler equations and a geometric reformulation of them, derived in [10].
Definitions of the Fluid Variables and Related Quantities
- The basic fluid variables
- v-orthogonal vorticity
- Auxiliary fluid variables
- Equation of state and the speed of sound
In the rest of the article, we see that the speed of sound is a function of rok:c=c(h,s). We limit ourselves to the physically relevant regime where the speed of sound does not exceed the speed of light:.
Standard First-Order Equations
Modified Fluid Variables and the Geometric Wave-Transport Formulation
The geometric wave-transport formulation of the relativistic Euler equations
L[A](B) denotes any scalar-valued function that is linear in the components of B with coefficients that are a function of the components of A. The div-curl system (2.26) in the geometric formulation of the relativistic Euler equations is the div-curl system of space- time.
Littlewood-Paley Projections
We refer readers to [9, Lemma 5.4] for the detailed proofs, where the structure of the equations is the same as in this paper.
Statement of Main Theorem
Choice of Parameters
Assumptions on the Initial Data
Bootstrap Assumptions
Our proofs are based on the bootstrapping argument, where the bootstrapping assumptions are in Section 3.6. See Section 1.5 for the logic of the bootstrap argument. (4.1a) is proved by the following set of reductions, see subsection 1.5.4 for an overview of the logic: Strichart estimates ←−Decomposition estimates ←− Conformal energy estimates ←−Acoustic null geometry control.
Similarities and Differences Compared to the 3D Compressible Euler Equations
The main goal for us is to improve the bootstrap assumptions to the following Strichartz-type estimates:. To prove (4.1b), we prove a transport-Schauder type estimate in Section 5, which is independent of the proof of (4.1a).
Energy, L 2 Elliptic and Schauder Estimates in Section 5
Reduction of Strichartz Estimates to Decay Estimates in Section 6
Geometric Setup in Section 7
Then we check the acoustic geometry based on the null frame we just constructed. Note that given the control over the acoustic geometry, we have favorable estimates for the conformal energy as we discuss in Chapter 10.
Energy along Acoustic Null Hypersurfaces and Structures of the Acoustic Geomtry in Sec-
Control of the Acoustic Geometry in Section 9
Control of the Conformal Energy in Section 10
By the estimates derived via the multiplier method described above, we prove decay estimates for the conformal energy in Section 10.6, thereby concluding the limit theorem for the conformal energy. Note that we obtain the same results as in [9, Sections 4–5], where ρ in [9] plays the same role as in this paper.
Energy and L 2 Elliptic Estimates
The basic energy inequality for wave equations and transport equations
We note that for our implementation of the geometric energy method for wave equations, the time-like vector field T (defined in Definition 2.11) plays the same role as B (Note that B=∂t+va∂ain [9] is not the same as B=vα not v0∂α in this paper.) in [9, section 4.1]. All the arguments for geometric energy method for wave equations proceed in the same way as in [9, section 4.1].
Elliptic div-curl estiamtes in L 2 space
Regarding elliptic estimates, there is a big difference in Proposition 5.8 compared to the Hodge system for the non-relativistic 3D compressible Euler equations. Given Proposition 5.9 and Proposition 5.11 with xlandφl, l∈N, defined as above, let X be the solution of the equations (5.29).
Proof of Proposition 5.1
Schauder Estimates
Now let's look at the first term on the right side of the last line of (5.94).
We then give a series of analytic reductions from the Strichartz estimates of Theorem 6.1 to the decay estimates of Theorem 6.9. We first reduce the proof of Theorem 6.1 to the proof of Strichartz estimates on small time intervals.
Partitioning of the Bootstrap Time Interval
Dicussion of the reduction to Theorem 6.2 from Theorem 6.1
The proof that Theorem 6.1 follows from Theorem 6.2 is exactly the same as the proof of [9, Theorem 7.2]. The reason is that the proof relies only on: 1) Duhamel's principle; 2) our estimates of the high-order energy (5.1), which is the same as the corresponding estimates in [9], and 3) the Littlewood-Paley estimates for the inhomogeneous terms in a frequency-projected version of the wave equations, when the wave equations in this paper have a schematic form identical to those in [9].
Rescaled Quantities and Rescaled Relativistic Euler Equations
Consequences of the bootstrap assumptions
Further Reduction of the Strichartz Estimates
The proof of Theorem 6.8 using Theorem 6.9 can be done via an approach involving the Bernstein inequalities by LP projection, partitioning of unity6ofϕ and Sobolev embeddingW2,1,→L2.
Construction of the Acoustical Function
Point z and integral curve γ z (t)
The interior and exterior solution u
To propagate Lω along the cone point axisγz(t), we define for anyp∈γz(t)(as defined in subsection 7.1.1) andω∈S2 Lω|p by solving the parallel transport equationDTLω=0 with initial conditions Lω |z . The curvet →ϒu,ω(t) is a non-affine parameterized zero geodesic such that the vector field Lαu,ω:= dtdϒαu,ω is zero and normalized by L0u,ω =1. We note that Cu's are the outgoing truncated zero cones, i.e. Cu:= S. For any ω∈S2andu∈[−w∗,0) we define the angular coordinate functions{ωA}A=1,2 as constant along zero geodesicsϒu ,ω(t)and to coincide with the angular coordinates{ωA}A=1,2onΣ0which are provided by the above construction; note that atΣ0\{z}, by construction, the.
Geometric Quantities
7In the rest of the question paper, we sum automatically if there are two A's in the expression. We use the notation|ξ|g/to denote the norm of the St,u tangent tensor fieldξ=Π/ξ with respect to g/, that is,. The proof of estimates for geometric quantities is obtained by transport equation and div-curl estimates for the acoustic quantities, decomposition of Ricci curvature components, trace and Sobolev inequalities.
Energy Estimates along Acoustic Null Hypersurfaces
In this section, we derive energy estimates along acoustic null hypersurfaces, which are necessary for obtaining the mixed-norm estimates in Proposition 9.1. We omit the proof of these estimates since they are the same as in [9, Section 10]. That is, our estimate is sufficient to obtain the mixture-norm estimates for fluid variables, which serve the same purpose as the estimates in [9, 36].
Connection Coefficients
Levi-Civita connections, angular operators and curvatures
Furthermore, we have the following relationships between some connection coefficients:. 8.17) The calculation of the above identities is based on the following fact: Let X then be a vector field.
Proof of (8.14)-(8.17)
Conformal metric, initial conditions on Σ 0 and on the cone-tip axis for the acoustical
In the following definition we define the mass aspect function µ and its modified version ˇµ, as well as the modified torsion ˜ζ. Initial conditions on the cone point axis associated with the acoustic function] On each zero cone Cuinitiating from a point on the time axis0≤t=u≤T∗;(λ)there holds. The reason is that Prop.8.12 yields a foliation and estimates on the hypersurface Σ0 with respect to the rescaled coordinates, and this hypersurface corresponds to the hyperfacesΣtk for each k with respect to the original spacetime (see Note 6.4 and Subsection 7.1. 2 for the description of Σ0in rescaled space-time).
PDEs Verified by Geometric Quantities
Proof of Lemma 8.14
Curvature Decompositions
We only need to compare terms of the highest order, since terms of the form ΓΓΓ·ΓΓΓ are absorbed by Q(~Ψ)[∂∂∂~Ψ,∂∂∂~Ψ]. Applying the same method to all Dπ and comparing with (8.100), we get the desired equation.
Main version of the PDEs Verified by the Acoustical Quantities
The following identities apply where the terms on the right are shown schematically:. Transport equation forg/Along the integral curves of L, parameterized by t, we have, with e/the standard. round metric on the Euclidean unit sphereS2, the following identity:. Since we have LβLγDβDγLα−LβLγDβDγLα=RiemαLLL, the decomposition of the wave operator can be written as follows:.
Restatement of Bootstrap Assumptions and Estimates for Quantities Constructed out of the
The fixed number p
In this section, we prove the estimates for the geometric quantities stated in Proposition 9.1.
Bootstrap assumptions for geometric quantities
Finally, assume that the following estimates hold in the inner range of M(Int) (defined in Section 7.1):.
Discussion of the proof of Proposition 9.1
After the substitution, we are faced with the control of the source terms in the geometric equations in mixed space-time norms. After controlling the source terms in the geometric PDEs for the acoustic geometry from step 1, we use the transport lemma and the div-curl estimates to obtain various estimates of the mixed spatiotemporal norms for the erp-weighted acoustic quantities in Proposition 9.1. Although these differences necessitate changes to some of the proofs given earlier in the article (such as the proof of energy estimates on constant-time hypersurfaces and acoustic null hypersurfaces), these changes do not affect the proofs of the estimate for acoustic geometries; for this reason, we refer to [9, Section 10] for details behind the proof of Statement 9.1.
Preparations for the Proofs of Main Estimates for the Eikonal Function Quatities
Proof of Proposition 9.3
We use the same approach as in the proof of (9.33b), for the case of the Euclidean sphere we have: 9.65) The proof of (9.65) can be reduced by a unit partition to the case where ξ has compact support in a local mapU ⊂St,u. We obtain the results by comparing the energy estimates along constant-time hypersurfaces (5.1) and along zero acoustic hypersurfaces. Under the bootstrap assumptions, the following estimates hold whenever q>2 is sufficiently close to 2, where p is the same as in the previous statement.
Discussion of the proof of Prop.9.7
Discussion of the proof of Lemma 9.8, Prop.9.9 and Corollary 9.10
The proof of Theorem 9.11 relies on the Fundamental Theorem of Calculus and the fact that L(ωA) =0. Assume that Fis is the St,u projection of a space-time tensor field F˜ or a contraction of a space-time tensor field F˜ against L, L or N. If Q>2,1≤c<∞, andδ0>0 is sufficiently small, then the following estimates apply, where~F˜ denotes the array of (scalar) Cartesian component function ofF:˜. 9.114) is a perturbation result of the standard Schauder-type estimate for Hodge system onS2 where the error terms are governed by (9.113) and the bootstrap assumptions.
Proofs of the Main Estimates for the Eikonal Function Quatities
Proof of Proposition 9.1
Taking into account the initial condition for the two separate casesu<0 andu≥0, this is the case. By absorbing the second term on RHS of (9.256) to the left, we conclude the desired result. Based on the Hardy-Littlewood maximum function estimate (9.28) and bootstrap assumptions (6.13), we arrive at the desired estimate.
Setup of the Conformal Energy
Reduction of Theorem 6.9
Bootstrap assumptions for the conformal energy
The proof of Theorem 6.9 using Theorem 10.2 is done via product evaluations and the Berstein inequality of the Littlewood-Paley theory.
Discussion of the proof of Theorem 10.2
Integrated Energy Estimates
We use the divergence theorem over the range Mττ12,R with modified current(X)switch=f Where f is defined in (10.15).
Control of Lower Order Terms
Proof of Lemma 10.14
Comparison Results
Multiplier Approach
Bounding the weighted energy
Decay Estimates for the Conformal Energy
Proof of the Boundness Theorem of the Conformal Energy
