## Abstract

We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and transport-div-curl equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular, for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type.

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## Notes

Barotropic equations of state are such that the pressure is a function of the proper energy density \(\uprho \) alone.

In observing many of the cancellations, the precise numerical coefficients in the equations are important; roughly, these cancellations lead to the presence of the null-form structures described below. However, for most applications, the overall coefficient of the null forms is not important; what matters is that the cancellations lead to null forms.

Here we further explain how standard first-order formulations of the relativistic Euler equations limit the available energy estimates. In deriving energy estimates for the relativistic Euler equations in their standard first-order form, one is effectively controlling the wave and transport parts of the system at the same time, and, up to a scalar function multiple, there is only one energy estimate available for transport equations. To see this limitation in a more concrete fashion, one can rewrite the relativistic Euler equations in first-order symmetric hyperbolic form as \(A^{\alpha }({\mathbf {V}}) \partial _{\alpha } {\mathbf {V}} = 0\), where \({\mathbf {V}}\) is the array of solution variables and the \(A^{\alpha }\) are symmetric matrices with \(A^0\) positive definite; see, for example, [27] for a symmetric hyperbolic formulation of the general relativistic Euler equations in the barotropic case. The standard energy estimate for symmetric hyperbolic systems is obtained by taking the Euclidean dot product of both sides of the equation with \({\mathbf {V}}\) and then integrating by parts over an appropriate spacetime domain foliated by spacelike hypersurfaces. The key point is that for systems without additional structure, no other energy estimate is known, aside from rescaling the standard one by a scalar function.

By “auxiliary,” we mean that they are determined by \(h\), \(s\), and

*u*.Relative to arbitrary coordinates, for scalar functions

*f*, we have$$\begin{aligned} \square _g f= \frac{1}{\sqrt{|\hbox {det} g|}} \partial _{\alpha }\left( \sqrt{|\hbox {det} g|} (g^{-1})^{\alpha \beta } \partial _{\beta } f \right) . \end{aligned}$$For solutions with vanishing vorticity and constant entropy, one can introduce a potential function \(\Phi \) and reformulate the relativistic Euler equations as a quasilinear wave equation in \(\Phi \).

One of the key results of [4] is conditional: For small data, the only possible singularities that can form are shocks driven by the intersection of the acoustic characteristics. Here “small” means a small perturbation of the data of a non-vacuum constant fluid state, where the size of the perturbation is measured relative to a high-order Sobolev norm. Another result of [4] is that there is an open subset of small data, perhaps strictly contained in the aforementioned set of data, such that the acoustic characteristics do in fact intersect in finite time. The results of [4] leave open the possibility that there might exist some non-trivial small global solutions.

In [5], Christodoulou solved the “restricted” shock development problem, in which he ignored the jump in entropy and vorticity across the shock hypersurface.

In one spatial dimension, the vorticity must vanish, but the entropy can be dynamic.

In all known shock formation results, at the location of shock singularities, the geometric partial derivative vectorfield \(\frac{\partial }{\partial U}\) has vanishing Euclidean length (i.e., \(\updelta _{ab} \left( \frac{\partial }{\partial U}\right) ^a \left( \frac{\partial }{\partial U}\right) ^b = 0\), where \(\left\{ \left( \frac{\partial }{\partial U}\right) ^a \right\} _{a=1,2,3}\) denotes the rectangular spatial components of \(\frac{\partial }{\partial U}\) and \(\updelta _{ab}\) is the Kronecker delta). That is, at the shock singularities, \(\frac{\partial }{\partial U}\) degenerates with respect to the rectangular coordinates. Due to this degeneracy, the solution’s \(\frac{\partial }{\partial U}\) derivatives can remain bounded all the way up to the shock, even though \(\frac{\partial }{\partial U}\) is transversal to the characteristics.

Roughly, these covariant wave operators are equivalent to divergence-form wave operators. In this way, one could say that a better theory is available for divergence-form wave operators than for non-divergence-form wave operators. This reminds one of the situation in elliptic PDE theory, where better results are known for elliptic PDEs in divergence form compared to ones in non-divergence form.

In one spatial dimension, one can rely exclusively on the method of characteristics and thus avoid energy estimates.

As is explained in [22], in the known framework for proving shock formation, one crucially relies on the fact that the derivatives of the solution blow up at a linear rate, that is like \(\frac{C}{T_{(\mathrm{Lifespan})} - t}\), where

*C*is a constant and \(T_{(\mathrm{Lifespan})} > 0\) is the (future) classical lifespan of the solution; if one perturbs the equation by adding terms that are expected to alter this blowup rate, then one should expect that the known approach for proving shock formation will not work (at least in its current form).Actually, it is not known whether or not the derivative-loss-avoiding procedure can be implemented for general systems of wave equations featuring more than one distinct wave operator. From this perspective, we find it fortunate that the equations of Theorem 1.2 feature only one wave operator.

In the absence of special structures, solutions to transport equations are not more regular than their source terms.

The evolution equation is in fact the famous

*Raychaudhuri equation*, which plays an important role in general relativity.See Sect. 9.6.1 for additional details regarding the multiplier method in the context of wave equations.

Equations (2.11), (2.20), and (2.13a) collectively imply that when \(\sum _{a=1}^3|u^a|\) is large, \(g(\partial _t,\partial _t) = g_{00} = - 1 + (c^{-2} - 1) u_a u^a\) can be positive, i.e., \(\partial _t\) can be spacelike with respect to the acoustical metric

*g*; it is well known that this can lead to indefinite energies if the standard partial time derivative vectorfield \(\partial _t\) is used as a multiplier in the wave equation energy estimates.The use of

*u*as a multiplier is likely familiar to researchers who have previously studied the relativistic Euler equations, but it might be unknown to the broader PDE community. We also remark that in searching the literature, we were unable to find results that, given our new formulation of the relativistic Euler equations, could be directly applied to establish points (i) and (ii) above. Moreover, we were not able to locate a local well-posedness result for elliptic–hyperbolic systems that can be directly applied to our new formulation of the equations. In particular, we could not locate a result that would directly imply continuous dependence of solutions on the initial data up to top order, i.e., a result that applies in the case when the vorticity and entropy enjoy the aforementioned extra regularity.That is, the signature of the \(4 \times 4\) matrix \(g_{\alpha \beta }\), viewed as a quadratic form, is \((-,+,+,+)\).

It is straightforward to check that \((g^{-1})^{\alpha \kappa } g_{\kappa \beta } = \updelta _{\beta }^{\alpha }\), where \(\updelta _{\beta }^{\alpha }\) is the Kronecker delta. That is, \(g^{-1}\) is indeed the inverse of

*g*.On might argue that it is more accurate to think of \(u^0\) as being “redundant” in the sense that it is algebraically determined in terms of \(\lbrace u^a \rbrace _{a=1,2,3}\) via the condition \(u^0 > 0\) and the normalization condition (2.20). In fact, in most of Sect. 9, we adopt this point of view. However, prior to Sect. 9, we do not adopt this point of view.

We stress that on LHS (3.3), the components \(u^{\alpha }\) are treated as scalar functions under the action of the covariant wave operator \(\square _g\).

The wave equation (3.5) is auxiliary in the sense that we do not use it in our proof of Theorem 9.12. However, in applications (for example, in the study of shock formation), one has to compute \(\square _g\) applied to the scalar component functions \(g_{\alpha \beta }\), and, by virtue of the chain rule, the quantity \(\square _g s\) arises in such computations. It is for this reason that we have included Eq. (3.5) in this paper.

Our labeling of the terms \({\mathscr {Q}}_2\), \({\mathscr {Q}}_3\), etc. is tied to the order in which terms appear in our proof of (8.41).

Such a \(\updelta > 0\) exists due to the compactness of \(w({\mathbb {T}}^3)\) and \({\mathscr {K}}\), where the compactness of \(w({\mathbb {T}}^3)\) follows from the assumption that the \(v_a\) are continuous.

The datum \(u^0|_{\Sigma _0}\) is determined from the other data by virtue of the constraint (2.20).

In fact, using additional arguments not presented here, one can show that for any fixed real number \(r > 5/2\), the time of existence can be controlled by a function of \({\mathfrak {K}}\), \(\Vert \mathring{h} \Vert _{H^r(\Sigma _0)}\), \(\Vert \mathring{s} \Vert _{H^r(\Sigma _0)}\), and \(\Vert \mathring{u}^i \Vert _{H^r(\Sigma _0)}\). Of course, if the initial data enjoy additional Sobolev regularity, then the additional regularity persists in the solution during its classical lifespan.

In particular, there is a \(\left( H^3(\Sigma _0) \right) ^5\)-neighborhood of \((\mathring{h},\mathring{s},\mathring{u}^i)\) such that all data in the neighborhood launch solutions that exist on the same slab \([0,T] \times {\mathbb {T}}^3\) and, assuming also that the data belong to \(\left( H^N(\Sigma _0) \right) ^5\), enjoy the regularity properties stated in the theorem.

More precisely, in [31], the spacetime metrics are scalar function multiples of the Minkowski metric on \({\mathbb {R}}^{1+3}\).

In deriving a priori estimates, in addition to the multiplier method, we will use only the simplest version of the vectorfield commutator method. Specifically, we will commute the equations only with the coordinate spatial derivative operators \(\partial _{\vec {I}}\).

By a “future-directed” vectorfield

*X*, we mean that \(X^0 > 0\).*X*is defined to be timelike with respect to*g*if \(g_{\alpha \beta } X^{\alpha } X^{\beta } < 0\).By a “causal hypersurface,” we mean a hypersurface whose future-directed unit normal is either timelike with respect to

*g*or null with respect to*g*at each point.A hypersurface is spacelike with respect to

*g*if, at each point, its unit normal is timelike with respect to*g*.For example, \(\nabla _{\alpha } X^{\beta } = \partial _{\alpha } X^{\beta } + \Gamma _{\alpha \ \gamma }^{\ \beta } X^{\gamma }\), where \(\Gamma _{\alpha \ \gamma }^{\ \beta }\) is defined by (9.53).

To see this, it is helpful to note the following identity, which holds relative to the standard coordinates: \(\frac{\mathrm{d} \mu _{{\underline{g}}}}{\sqrt{|(g^{-1})^{00}|}} = c^{-3} \, \mathrm{d}x^1 \mathrm{d}x^2 \mathrm{d}x^3\). This identity follows from (2.14a) and the linear algebraic identity \(\hbox {det} {\underline{g}} = (g^{-1})^{00} \hbox {det} g\).

By “strong solution,” we mean in particular that at each fixed \(t \in [0,T]\), the equations of Theorem 3.1 are satisfied for almost every \(x \in {\mathbb {T}}^3\).

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Communicated by Mihalis Dafermos.

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MMD gratefully acknowledges support from NSF Grant #1812826, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Discovery Grant administered by Vanderbilt University. JS gratefully acknowledges support from NSF Grant # 1162211, from NSF CAREER Grant #1454419, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology.

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Disconzi, M.M., Speck, J. The Relativistic Euler Equations: Remarkable Null Structures and Regularity Properties.
*Ann. Henri Poincaré* **20**, 2173–2270 (2019). https://doi.org/10.1007/s00023-019-00801-7

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DOI: https://doi.org/10.1007/s00023-019-00801-7