Local smooth positive-density polytropic Euler

For a positive smooth solution of the mass equation, $D_t\rho =-\rho \mathrm{div}u$, where $D_t={\mathrm{\partial }}_t+u\cdot \mathrm{\nabla }$. Since $r^{\mathrm{\prime }}(\rho )=\sqrt{\kappa \gamma }{\rho }^{a-1}=c(\rho )\mathrm{/}\rho$ and $c(\rho )=ar$, the chain rule gives $D_{t}r=-ar\mathrm{div}u$. Also ${\rho }^{-1}{p}^{\mathrm{\prime }}(\rho )\mathrm{\nabla }\rho =c(\rho {)}^2\mathrm{\nabla }\rho \mathrm{/}\rho =c(\rho )\mathrm{\nabla }r=ar\mathrm{\nabla }r$. Conversely, for $r>0$, the inverse $\rho (r)=(ar\mathrm{/}\sqrt{\kappa \gamma }{)}^{1\mathrm{/}a}$ has ${\rho }^{\mathrm{\prime }}(r)=\rho \mathrm{/}(ar)$; the first symmetric equation therefore gives $D_t\rho =-\rho \mathrm{div}u$, and the second gives $D_{t}u+{\rho }^{-1}\mathrm{\nabla }p(\rho )=0$ using ${\rho }^{-1}\mathrm{\nabla }p=c(\rho )\mathrm{\nabla }r=ar\mathrm{\nabla }r$. Thus the two smooth systems are equivalent on the positive-density region.

Put $q_0=(r({\rho }_0),u_0)$. For $N\ge 1$, let $S_N$ be the componentwise orthogonal Fourier projection to $\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le N$ and solve ${\mathrm{\partial }}_t{q}_N+S_N{\sum }_{j=1}^d{A}_j(q_N){\mathrm{\partial }}_j{q}_N=0$, $q_N(0)=S_N{q}_0$. Each coefficient of its finite-dimensional vector field is polynomial. On any bounded coefficient box, the identity $x^n-y^n=(x-y){\sum }_{i=0}^{n-1}{x}^{n-1-i}{y}^i$, telescoped over the finitely many factors of each monomial, bounds every monomial difference by a constant times the Euclidean distance; summing the finitely many monomials proves local Lipschitz continuity. Thus [F7] gives a unique local real solution, and the real Fourier cutoff preserves conjugate symmetry.

The analytic suppliers [F2]–[F4] apply under the standing Countable Choice assumption [F6, A1]. For integer $m\ge s$, put $Q={\mathrm{\Lambda }}^m{q}_N$ with $\mathrm{\Lambda }$ the multiplier $(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Pair the projected equation with $Q$. Since $S_N$ is self-adjoint and commutes with $\mathrm{\Lambda }$, the projection disappears in the inner product. For each $j$, expand ${\mathrm{\Lambda }}^m(A_j(q_N){\mathrm{\partial }}_j{q}_N)=A_j(q_N){\mathrm{\partial }}_{j}Q+[{\mathrm{\Lambda }}^m,A_j(q_N)]{\mathrm{\partial }}_j{q}_N$. Symmetry and periodic integration by parts give $\int Q^{\mathsf{T}}{A}_j(q_N){\mathrm{\partial }}_{j}Q=-\frac{1}{2}\int Q^{\mathsf{T}}({\mathrm{\partial }}_j{A}_j(q_N))Q$, whose absolute value is at most $C\mathrm{\parallel }\mathrm{\nabla }{q}_N{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^m}^2$. The embedding in [F3] bounds this by $C_m\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^s}\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^m}^2$. Because each $A_j$ is linear in $q_N$, [F4] bounds all commutator terms by the same expression. Thus $\frac{1}{2}\frac{d}{dt}\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^m}^2\le C_m\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^s}\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^m}^2$.

Define a canonical extraction $E$ for a bounded real sequence $(b_n{)}_{n\ge 1}$. Let $B\ge 1$ be the least integer with $\mathrm{\mid }{b}_n\mathrm{\mid }\le B$ for every $n$, and put $I_0=[-B,B]$. Having defined $I_{\mathrm{\ell }}$, bisect it into its two closed halves and define $I_{\mathrm{\ell }+1}$ to be the left half if that half contains $b_n$ for infinitely many indices $n$, and the right half otherwise. At least one half contains infinitely many such indices, so this rule is defined and unique. Set $e_0=0$ and let $e_{\mathrm{\ell }}$ be the least $n>e_{\mathrm{\ell }-1}$ with $b_n\in I_{\mathrm{\ell }}$. These minima exist because the corresponding index sets are infinite. Then $E(b)(\mathrm{\ell })=e_{\mathrm{\ell }}$ is strictly increasing and $(b_{e_{\mathrm{\ell }}})$ is Cauchy: for $p,q\ge \mathrm{\ell }$, both values lie in $I_{\mathrm{\ell }}$, whose length is $2B{2}^{-\mathrm{\ell }}$. It converges, since [F10] gives a convergent subsequence of this bounded Cauchy sequence and the Cauchy inequality forces the entire sequence to have the same limit. All intervals and indices are uniquely specified; this defines a function $E$ without any choice principle.

At $m=s$, let $Y_N=\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^s}$. If $Y_N(0)>0$, step 1.3 gives $Y_N^{\mathrm{\prime }}\le C_s{Y}_N^2$ while $Y_N>0$; integrating $(1\mathrm{/}{Y}_N{)}^{\mathrm{\prime }}\ge -C_s$ yields $Y_N(t)\le 2\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^s}$ for $0\le t\le T_0=(2C_s\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^s}{)}^{-1}$. If $q_0=0$, the unique finite-dimensional solution is zero and take $T_0=1$. For each $m>s$, substituting this $H^s$ bound into step 1.3 and multiplying the squared-norm inequality by the corresponding decreasing exponential gives $\mathrm{\parallel }{q}_N(t){\mathrm{\parallel }}_{H^m}\le \mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^m}{e}^{2C_m\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^s}{T}_0}$. Parseval [F2] bounds the Euclidean coefficient norm by $\mathrm{\parallel }{q}_N{\mathrm{\parallel }}_{H^s}$; since the finite-dimensional state space is ${\mathbb{R}}^M$ with $M\ge 1$, completeness [F9] supplies the endpoint limit used by the continuation argument [F8]. Thus every finite-dimensional polynomial trajectory continues throughout $[0,T_0]$.

Let $q,v$ be two smooth solutions of the symmetric system with the same initial data and positive first component. Their difference $w=q-v$ satisfies $w_t+{\sum }_j{A}_j(q){\mathrm{\partial }}_{j}w=-{\sum }_j(A_j(q)-A_j(v)){\mathrm{\partial }}_{j}v$. Pair with $w$, integrate over the torus, and use symmetry of $A_j(q)$ to integrate the principal term into $-\frac{1}{2}\int w^{\mathsf{T}}({\mathrm{\partial }}_j{A}_j(q))w$. Linearity of $A_j$ bounds the right side by $C(\mathrm{\parallel }\mathrm{\nabla }q{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\parallel }w{\mathrm{\parallel }}_2^2$, so $z(t)=\mathrm{\parallel }w(t){\mathrm{\parallel }}_2^2$ satisfies $z^{\mathrm{\prime }}\le C(t)z$, with continuous nonnegative $C$. Multiplying by $e^{-{\int }_0^{t}C(s)ds}$ gives a nonincreasing nonnegative function initially zero, hence $w=0$. The inverse change of variables then makes the positive-density Euler solution unique.

For every integer $m\ge s+1$, the product estimate [F3] and the bounds in step 2.1 bound ${\mathrm{\partial }}_t{q}_N=-S_N{\sum }_j{A}_j(q_N){\mathrm{\partial }}_j{q}_N$ uniformly in $C([0,T_0],H^{m-1})$. Hence each fixed Fourier coefficient is uniformly bounded and equicontinuous in time.

Enumerate tuples $(k,A,B,j,ϵ)$ with $k\in {\mathbb{Z}}^d$, integers $A\ge 0$, $B\ge 1$, $\gcd (A,B)=1$, $A\mathrm{/}B\le T_0$, $0\le j\le d$, and $ϵ\in \{0,1\}$ by increasing $\mathrm{\mid }k{\mathrm{\mid }}_1+A+B+j+ϵ$, then lexicographically within each finite level. At tuple $i$, let $b_N^i$ be the real ($ϵ=0$) or imaginary ($ϵ=1$) part of the $j$th component of ${\widehat{q}}_N(k,A\mathrm{/}B)$. Starting with $h_0(n)=n$, define recursively $h_i=h_{i-1}\circ E(b^i\circ h_{i-1})$ and put $N_{\mathrm{\ell }}=h_{\mathrm{\ell }}(\mathrm{\ell })$. The successor in this recursion is a specified function of $h_{i-1}$, so this is ordinary recursion, not a succession of unspecified dependent selections. Each $h_i$ is strictly increasing and its range is contained in every preceding range. Moreover $N_{\mathrm{\ell }+1}=h_{\mathrm{\ell }}(E(b^{\mathrm{\ell }+1}\circ h_{\mathrm{\ell }})(\mathrm{\ell }+1))\ge h_{\mathrm{\ell }}(\mathrm{\ell }+1)>N_{\mathrm{\ell }}$. For each fixed $i$, the tail $(N_{\mathrm{\ell }}{)}_{\mathrm{\ell }\ge i}$ is an increasing subsequence of the range of $h_i$, on which $b^i$ converges. Thus this one subsequence converges at every enumerated coefficient-time coordinate. Countable Choice remains the standing analytic Fourier assumption; the diagonal construction itself does not require dependent choice.

The equicontinuity from step 3.1 upgrades convergence at rational times to uniform-in-time convergence of each fixed Fourier coefficient. Given $\epsilon >0$, choose a finite rational time grid so that every time is close enough to a grid point to make each of the two equicontinuity errors below $\epsilon \mathrm{/}3$; choose the subsequence indices large enough that the finitely many grid values differ by less than $\epsilon \mathrm{/}3$. The triangle inequality makes each coordinate uniformly Cauchy, and scalar completeness supplies its uniform limit.

For every integer $m\ge s$, the $H^{m+1}$ bound from step 2.1 and the coefficient weight give $\mathrm{\parallel }(I-S_K)q_N(t){\mathrm{\parallel }}_{H^m}\le K^{-1}\mathrm{\parallel }{q}_N(t){\mathrm{\parallel }}_{H^{m+1}}$ for $K\ge 1$. This tail tends to zero uniformly in $N,t$. Combining it with finite-mode convergence from step 5.1 makes the diagonal subsequence Cauchy in $C([0,T_0],H^m)$ for every integer $m\ge s$. Completeness of the weighted coefficient space [F1] yields a limit $q\in {\bigcap }_{m\ge s}C([0,T_0],H^m)$ with all these uniform bounds; coefficientwise limits preserve reality and the initial data.

Since $q_N\to q$ in $C_t{H}^m$ for every $m\ge s$, the algebra estimate [F3] gives $A_j(q_N){\mathrm{\partial }}_j{q}_N\to A_j(q){\mathrm{\partial }}_{j}q$ in $C_t{H}^{m-1}$. The Fourier cutoffs $S_N$ converge strongly to the identity in that space. They converge uniformly on the compact time image of the continuous curve $t\mapsto {\sum }_j{A}_j(q){\mathrm{\partial }}_{j}q$: choose a finite $\epsilon$-net, use strong convergence at its finitely many points, and use $\mathrm{\parallel }{S}_N\mathrm{\parallel }\le 1$ for the net errors. Thus the Galerkin right-hand sides converge uniformly in $C_t{H}^{m-1}$.

Integrate each Galerkin equation from $0$ to $t$. A continuous curve into the complete weighted coefficient space is Riemann integrable: uniform continuity makes its Riemann sums Cauchy, and completeness supplies the limit; the triangle inequality gives $\mathrm{\parallel }\int G_N-\int G{\mathrm{\parallel }}_{H^{m-1}}\le T_0{\sup }_t\mathrm{\parallel }{G}_N(t)-G(t){\mathrm{\parallel }}_{H^{m-1}}$. The right-hand-side convergence in step 7.1 and $S_N{q}_0\to q_0$ therefore let the integrals pass to the limit, giving $q(t)=q_0-{\int }_0^t{\sum }_j{A}_j(q(s)){\mathrm{\partial }}_{j}q(s)ds$. The integrand is continuous in every $H^{m-1}$; its integral has derivative equal to the integrand because the difference quotient error is bounded by the maximum oscillation of that continuous integrand on the shrinking time interval. Therefore $q_t+{\sum }_j{A}_j(q){\mathrm{\partial }}_{j}q=0$ at every Sobolev order, and the embeddings in [F3] imply spatial smoothness. Differentiating this polynomial first-order identity inductively gives all time derivatives: each new derivative is a finite sum of products of already continuous spatial derivatives.

The first component satisfies $r_t=-u\cdot \mathrm{\nabla }r-ar\mathrm{div}u$. By [F3], the uniform $H^s$ bound in step 2.1 bounds the right side in $L^{\mathrm{\infty }}$ by $C\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^s}^2$. Hence $\mathrm{\parallel }r(t)-r(0){\mathrm{\parallel }}_{\mathrm{\infty }}\le Ct\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^s}^2$. Choose $0<T\le T_0$ so that this is less than $\frac{1}{2}\min r(0)$; then $r(t,x)\ge \frac{1}{2}\min r(0)>0$. The smooth inverse $\rho =(ar\mathrm{/}\sqrt{\kappa \gamma }{)}^{1\mathrm{/}a}$ and step 1.1 give a smooth positive-density solution of the stated Euler equations. On the compact spacetime cylinder, $r$ stays in a compact subinterval of $(0,\mathrm{\infty })$; all derivatives of this inverse power are therefore bounded on the range of $r$, and the repeated chain rule gives $\rho \in C_t{H}^m$ for every $m$. The choice of $T$ uses only the stated parameters, $\mathrm{\parallel }{q}_0{\mathrm{\parallel }}_{H^s}$, and $\min r(0)$.

The compactness, passage, positivity, and uniqueness arguments above give existence, smoothness, positivity, uniqueness, and all-order Sobolev bounds on the same interval. No argument extends the solution through vacuum, a shock, or a finite-time loss of the stated bounds.