Weak asymptotic solutions and their Radon measure limits for the compressible Euler equations

Abstract

In Colombeau (Z Angew Math Phys 66:2575–2599, 2015), Colombeau (Z Angew Math Phys 71:112, 2020) we constructed weak asymptotic solutions to some systems of fluid dynamics on the n-D torus ${\mathbb {T}}^n$. In Colombeau (Z Angew Math Phys 71:112, 2020), we passed to the limit by compactness to obtain Radon measures on ${\mathbb {T}}^n$ that satisfy the equations in a natural sense as sum of distributions. The construction was done on physical ground and, though we obtained full rigorous mathematical proofs, we had to introduce arbitrary ingredients such as an approximation of the initial condition or a function to obtain bounds on the variables of density and velocity, which are used in the demonstrations. In this paper, from a more precise construction and under the assumptions of finiteness of velocity and absence of void region in the flow, we prove that the Radon measure limits are independent on all arbitrary ingredients used in their construction: they depend only on the initial condition $\rho _0\in L^1({\mathbb {T}}^n), \textbf{u}_0\in L^\infty ({\mathbb {T}}^n)^n$. We describe them as particular vanishing viscosity limits.

Appendices

Appendix 1. Detailed proof of Proposition 1 weak asymptotic solutions.

In Sect. 4 we have given a sketch of proof of Proposition 1. In this Appendix we give the proof in its entirety. We give the proof in 1-D to simplify the notation since the n-D proof is identical. We start with a given function $\mu : \epsilon \mapsto \mu (\epsilon )$ and with approximate initial conditions $(\rho _{0,\epsilon },u_{0,\epsilon })$ satisfying the properties in Sect. 2. We recall system (23, 24, 25) setting $h(\epsilon )=\epsilon \mu (\epsilon )$

$$\begin{aligned}{} & {} \frac{\partial }{\partial t}\rho _\epsilon +\frac{\partial }{\partial x}(\rho _\epsilon u_\epsilon )=h(\epsilon )\Delta \rho _\epsilon , \end{aligned}$$

(58)

$$\begin{aligned}{} & {} \frac{\partial }{\partial t}(\rho _\epsilon u_\epsilon )+\frac{\partial }{\partial x}(\rho _\epsilon u_\epsilon ^2)+\frac{\partial }{\partial x}(K\rho _\epsilon )=h(\epsilon )\Delta (\rho _\epsilon u_\epsilon ), \end{aligned}$$

(59)

$$\begin{aligned}{} & {} \rho _\epsilon (x,0)=\rho _{0,\epsilon }(x),u_\epsilon (x,0)=u_{0,\epsilon }(x). \end{aligned}$$

(60)

$\bullet$ First step. Local solution to system (23, 24, 25), (58, 59, 60). We recall a classical proof obtained by the contractive fixed point method. System (58, 59, 60) is stated in its usual integral form from the heat kernel

$$\begin{aligned} \rho _\epsilon (x,t)= & {} \frac{1}{\sqrt{\pi }}\int \limits _ye^{-y^2}\rho _{0,\epsilon }(x-2y\sqrt{h(\epsilon )t})dy+\nonumber \\{} & {} \frac{1}{\sqrt{\pi h(\epsilon )}}\int \limits _{s=0}^t\int \limits _y\frac{ye^{-y^2}}{\sqrt{t-s}}\rho _\epsilon u_\epsilon (x- 2y\sqrt{h(\epsilon )(t-s)},s)dyds, \end{aligned}$$

(61)

$$\begin{aligned} (\rho u)_\epsilon (x,t)= & {} \frac{1}{\sqrt{\pi }}\int \limits _ye^{-y^2}(\rho _{0,\epsilon }u_{0,\epsilon })(x-2y\sqrt{h(\epsilon )t})dy+\nonumber \\{} & {} \frac{1}{\sqrt{\pi h(\epsilon )}}\int \limits _{s=0}^t\int \limits _y\frac{ye^{-y^2}}{\sqrt{t-s}}(\rho _\epsilon u_\epsilon ^2)(x- 2y\sqrt{h(\epsilon )(t-s)},s)dyds +\nonumber \\{} & {} \frac{1}{\sqrt{\pi h(\epsilon )}}\int \limits _{s=0}^t\int \limits _y\frac{ye^{-y^2}}{\sqrt{t-s}}(K\rho _\epsilon )(x- 2y\sqrt{h(\epsilon )(t-s)},s)dyds. \end{aligned}$$

(62)

Let $T>0$ and $X={\mathcal {C}}([0,T],(L^\infty ({\mathbb {T}})))^2$. We state $M:X\mapsto X$ the map obtained with the second members of (61, 62) where we replace $\rho _\epsilon$ and $(\rho u)_\epsilon$ by $\rho$ and $(\rho u)$, respectively.

Let $\beta >0$ and

$$\begin{aligned} B=\{(\rho ,\rho u)\in X \hbox { such that }\Vert (\rho ,\rho u)-({\overline{\rho }}_0,\overline{(\rho u)}_0)\Vert _X\le \beta \} \end{aligned}$$

where $\rho$ and $\rho u$ are independent variables and where ${\overline{\rho }}_0$ and $\overline{(\rho u)}_0$ are the first terms in second member of (61) and (62), respectively.

From the first term in the second member of (61), ${\overline{\rho }}_0(x,t)\ge min_{x\in {\mathbb {T}}}\rho _{0,\epsilon }(x) \forall x \hbox { and } \forall t\le T$. We recall the strict positiveness of $\rho _{0,\epsilon }(x,t)$ (27). We choose $\beta <min_{x\in {\mathbb {T}}} \rho _{0,\epsilon }(x)$. Then

$$\begin{aligned} (\rho ,\rho u)\in B \Rightarrow \rho (x,t)\ge min_{x\in {\mathbb {T}}} \rho _{0,\epsilon }(x)-\beta >0 \ \forall x \hbox { and } \forall t\le T. \end{aligned}$$

Therefore, if $(\rho ,\rho u)\in B, \ \Vert u\Vert _\infty \le \frac{\Vert \overline{(\rho u)}_0\Vert _\infty +\beta }{min_{x\in {\mathbb {T}}} \rho _{0,\epsilon }(x)-\beta }.$

Therefore, the map M maps B into B and is a contraction in B for $t>0$ small enough depending on $\epsilon$. Therefore, the equation $(\rho ,\rho u)=M(\rho ,\rho u)$ has a unique solution in B if $t>0$ is small enough. $\square$

$\bullet$ Second step. A priori estimates. Here the assumptions (26, 27) could be dropped as done in Ref. [8] by stating the state law in form (11, 12, 16, 18) in Ref. [8] which permits to construct and prove weak asymptotic solutions even in presence of void regions and unbounded velocity [8, 10]. Then one has to select those weak asymptotic solutions which satisfy the assumptions (26, 27) for the proofs in the present paper.

Lemma 3

A sup bound in density. There exists $const>0$ such that

$$\begin{aligned} \Vert \rho _\epsilon (.,t)\Vert _\infty \le \gamma (\epsilon )^\alpha exp\frac{const}{h(\epsilon )} \end{aligned}$$

(63)

$\forall \epsilon >0$ small enough and $\forall t\in [0,\delta [, 0<\alpha<\frac{1}{3}, \delta <+\infty$.

Proof of Lemma 3

The value $\gamma (\epsilon )$ in (63) will stem from assumption (22) on the approximation of the initial condition. From formula (61) and from the assumption of boundedness of velocity (26), we obtain

$$\begin{aligned} \Vert \rho (.,t,\epsilon )\Vert _\infty \le \Vert \rho _{0,\epsilon }\Vert _\infty +\frac{const}{\sqrt{h(\epsilon )}}\int \limits _{s=0}^t \frac{\Vert \rho (.,s,\epsilon )\Vert _\infty }{\sqrt{t-s}}ds. \end{aligned}$$

(64)

We use the following standard lemma that we recall for convenience.

Lemma 4

A Gronwall formula adapted to the heat equation. Let $a,b>0$ and let $w:[0,T[\longmapsto {\mathbb {R}}^+$ continuous such that

$$\begin{aligned} w(t)\le a+b\int \limits _{s=0}^t\frac{w(s)}{\sqrt{t-s}}ds \ \forall t\in [0,T[. \end{aligned}$$

(65)

Then, one has

$$\begin{aligned} w(t)\le a(1+2b\sqrt{t})exp(\pi b^2t) \ \forall t\in [0,T[, \end{aligned}$$

(66)

that we will use always under the simplified form $w(t)\le a \ exp(const \ b^2 )$ to shorten the formula. Indeed we always use this simplified form when $t\in [0,T],T<+\infty$, $\epsilon >0$ small enough and $b=\frac{1}{\sqrt{h(\epsilon )}}$.

Proof of Lemma 4

Using (65) twice one obtains

$$\begin{aligned} w(t)\le a+b\int \limits _{s=0}^t\frac{a}{\sqrt{t-s}} ds+b^2\int \limits _{s=0}^t\int \limits _{\sigma =0}^s\frac{w(s)}{\sqrt{(t-s)(s-\sigma )}}d\sigma ds. \end{aligned}$$

Computing the integrals gives

$$\begin{aligned} w(t)\le a+2ab\sqrt{t}+\pi b^2\int \limits _{s=0}^t w(s)ds \end{aligned}$$

and we conclude by applying a usual Gronwall lemma. $\square$

From Lemma 4, (64) implies (63) on $[0,\delta (\epsilon )]$ taking (22) into account. From Lemma 3 and boundedness of velocity (26)

$$\begin{aligned} \Vert \rho _\epsilon u_\epsilon (.,t)\Vert _\infty \le const \ \gamma (\epsilon )^\alpha exp\frac{const}{h(\epsilon )} \end{aligned}$$

(67)

for small enough $\epsilon$ and for $t\in [0,\delta (\epsilon )[,\delta (\epsilon )<+\infty$. $\square$

$\bullet$ Third step. Global solution to system (23, 24, 25). Existence of a global solution for fixed $\epsilon >0$ is exposed in section 4.

$\bullet$ Fourth step. Bounds (20) on the solutions of the nonlinear heat Eqs. (23, 24, 25) (58, 59, 60). Adding sup bounds from the integral formulas (61, 62) for solutions $\rho _\epsilon$ and $\rho _\epsilon u_\epsilon$, using the bound $\Vert \rho _\epsilon u_\epsilon ^2\Vert _\infty \le const \Vert \rho _\epsilon u_\epsilon \Vert _\infty$ from (26), we obtain the bound

$$\begin{aligned}{} & {} \Vert \rho _\epsilon (.,t)\Vert _\infty +\Vert \rho _\epsilon u_\epsilon (.,t)\Vert _\infty \le \nonumber \\{} & {} \quad \Vert \rho _{0,\epsilon }\Vert _\infty +\Vert \rho _{0,\epsilon }u_{0,\epsilon }\Vert _\infty +\frac{const}{\sqrt{h(\epsilon )}}\int \limits _{s=0}^t\frac{\Vert \rho _\epsilon (.,s)\Vert _\infty +\Vert \rho _\epsilon u_\epsilon (.,s)\Vert _\infty }{\sqrt{t-s}} ds. \end{aligned}$$

(68)

The Gronwall formula (66) gives, if $t\in [0,T], T>0$ given,

$$\begin{aligned} \Vert \rho _\epsilon (.,t)\Vert _\infty +\Vert \rho _\epsilon u_\epsilon (.,t)\Vert _\infty \le (\Vert \rho _{0,\epsilon }\Vert _\infty +\Vert \rho _{0,\epsilon }u_{0,\epsilon }\Vert _\infty ) exp\frac{const }{\epsilon \mu (\epsilon )}. \end{aligned}$$

(69)

From (19), the left member of (69) is $\le \gamma (\epsilon )$ for $\epsilon >0$ small enough as soon as $\Vert \rho _{0,\epsilon }\Vert _\infty \le \gamma (\epsilon )^\alpha$ and $\Vert \rho _{0,\epsilon } u_{0,\epsilon }\Vert _\infty \le \gamma (\epsilon )^\alpha$ for some $0<\alpha <1$. We have obtained (20) at zero order provided (22).

Now, we consider bounds for first-order derivatives. We differentiate (61,62) in the variable x and we add the formulas so obtained. Considering (59), from

$$\begin{aligned} \frac{\partial }{\partial x}(\rho u^2)=2u\frac{\partial }{\partial x}(\rho u)-u^2\frac{\partial }{\partial x}\rho , \end{aligned}$$

and the assumed boundedness of velocity (26), one obtains

$$\begin{aligned}{} & {} \Vert \frac{\partial }{\partial x}\rho _\epsilon (.,t)\Vert _\infty +\Vert \frac{\partial }{\partial x}(\rho _\epsilon u_\epsilon )(.,t)\Vert _\infty \le \Vert \frac{\partial }{\partial x}\rho _{0,\epsilon }\Vert _\infty +\Vert \frac{\partial }{\partial x}(\rho _{0,\epsilon } u_{0,\epsilon })\Vert _\infty +\nonumber \\{} & {} \quad \frac{const}{\sqrt{\epsilon \mu (\epsilon )}}\int \limits _{s=0}^t\frac{\Vert \frac{\partial }{\partial x}\rho _\epsilon (.,s)\Vert _\infty +\Vert \frac{\partial }{\partial x}(\rho _\epsilon u_\epsilon )(.,s)\Vert _\infty }{\sqrt{t-s}} ds. \end{aligned}$$

(70)

The Gronwall formula (66) gives

$$\begin{aligned} \Vert \frac{\partial }{\partial x}\rho _\epsilon (.,t)\Vert _\infty +\Vert \frac{\partial }{\partial x}(\rho _\epsilon u_\epsilon )(.,t)\Vert _\infty \le (\Vert \frac{\partial }{\partial x}\rho _{0,\epsilon }\Vert _\infty +\Vert \frac{\partial }{\partial x}(\rho _{0,\epsilon } u_{0,\epsilon })\Vert _\infty ) exp\frac{const }{\epsilon \mu (\epsilon )}. \end{aligned}$$

(71)

The conclusion is similar to that for 0 order derivatives: from (19) the left member of (71) is less than $\gamma (\epsilon )$ for $\epsilon$ small enough as soon as $\Vert \frac{\partial }{\partial x}\rho _{0,\epsilon }\Vert _\infty$ and $\Vert \frac{\partial }{\partial x}(\rho _{0,\epsilon } u_{0,\epsilon })\Vert _\infty$ are less than $\gamma (\epsilon )^\alpha ,$ for some $0<\alpha <1$. We obtain (20) at first order.

We establish bounds for second order derivatives. We differentiate (61,62) once again. We use the formula

$$\begin{aligned} \frac{\partial ^2}{\partial {x^2}}(\rho u^2)=-u^2\frac{\partial ^2}{\partial {x^2}}\rho +2u\frac{\partial ^2}{\partial {x^2}}(\rho u)+\frac{2}{\rho }((\frac{\partial }{\partial {x}}(\rho u))^2-2u\frac{\partial }{\partial {x}}\rho \frac{\partial }{\partial {x}}(\rho u)+u^2(\frac{\partial }{\partial {x}}\rho )^2), \end{aligned}$$

together with finiteness of velocity and absence of void region (26, 27). Then, from the integral formulas and the Gronwall formula (66), we obtain for adequate values const

$$\begin{aligned}{} & {} \Vert \frac{\partial ^2}{\partial {x^2}}\rho _\epsilon (.,t)\Vert _\infty +\Vert \frac{\partial ^2}{\partial {x^2}}(\rho _\epsilon u_\epsilon )(.,t)\Vert _\infty \le \{ \Vert \frac{\partial ^2}{\partial {x^2}}\rho _{0,\epsilon }\Vert _\infty +\Vert \frac{\partial ^2}{\partial {x^2}}(\rho _{0,\epsilon } u_{0,\epsilon })\Vert _\infty +\nonumber \\{} & {} \quad const\int \limits _{s=0}^t(\Vert \frac{\partial }{\partial {x}}\rho (.,s)\Vert _\infty +\Vert \frac{\partial }{\partial {x}}(\rho (.,s) u(.,s))\Vert _\infty )^2ds\}\times exp\frac{const }{\epsilon \mu (\epsilon )}. \end{aligned}$$

(72)

From (19), to have bounds (20), it suffices that $\Vert \frac{\partial ^2}{\partial {x^2}}\rho _{0,\epsilon }\Vert _\infty , \Vert \frac{\partial ^2}{\partial {x^2}}(\rho _{0,\epsilon }u_{0,\epsilon })\Vert _\infty \le \gamma (\epsilon )^\alpha ,$ for some $0<\alpha <1$ and that $\Vert \frac{\partial }{\partial {x}}\rho _\epsilon \Vert _\infty$, $\Vert \frac{\partial }{\partial x}(\rho _\epsilon u_\epsilon )\Vert _\infty \le \gamma (\epsilon )^\alpha ,$ for some $0<\alpha <\frac{1}{2}$. From (71) this is implied by $\Vert \frac{\partial }{\partial {x}}\rho _{0,\epsilon }\Vert _\infty$, $\Vert \frac{\partial }{\partial {x}}(\rho _{0,\epsilon } u_{0,\epsilon })\Vert _\infty \le \gamma (\epsilon )^\alpha ,$ for some $0<\alpha <\frac{1}{2}$. At order 3 one finds $\alpha \le \frac{1}{3}$. Finally, it suffices that $\Vert D\rho _{0,\epsilon }\Vert _\infty$ and $\Vert D(\rho _{0,\epsilon }u_{0,\epsilon })\Vert _\infty \le \gamma (\epsilon )^\alpha ,$ for some $0<\alpha <\frac{1}{3}$ when the order of D is less than or equal to 3.

$\bullet$ Fifth step. The family $(\rho _\epsilon , u_\epsilon )$ solution of (23, 24, 25) which satisfies (20) satisfies the ODEs (15,16) with remainder terms $(r_\epsilon ,R_\epsilon )$ satisfying (21). From Taylor’s formula in x, from (6,20, by comparison between Eqs. (9,10 and Eqs. (23, 24)

$$\begin{aligned}{} & {} \frac{1}{\epsilon }\{\rho _\epsilon u_\epsilon ^+(x-\epsilon ,t)- \rho _\epsilon u_\epsilon ^+(x,t)-\rho _\epsilon u_\epsilon ^-(x,t)+\rho _\epsilon u_\epsilon ^-(x+\epsilon ,t)\}= \nonumber \\{} & {} \quad -\frac{\partial }{\partial {x}}(\rho _\epsilon (u_\epsilon ^+-u_\epsilon ^-))(x,t)+O(const \ \epsilon \gamma (\epsilon ))=-\frac{\partial }{\partial {x}}(\rho _\epsilon u_\epsilon )(x,t)+O(const \ \epsilon \gamma (\epsilon )). \end{aligned}$$

(73)

Similarly, the 1-D first term in second member of (10) gives

$$\begin{aligned} -\frac{\partial }{\partial {x}}(\rho _\epsilon u^2_\epsilon )(x,t)+O(const \ \epsilon \gamma ^3(\epsilon )) \end{aligned}$$

(74)

where $\gamma ^3$ stems from $\frac{\partial ^2}{\partial x^2}(\rho _\epsilon u_\epsilon u^\pm _\epsilon )$ from (26,20). From (73,74), equations (23–24) imply (15, 16).

The remainders $r_\epsilon$ and $R_\epsilon$ in (15,16) stem from the two above O terms and also similar O terms from the discretization of $\Delta \rho _\epsilon$ and $\Delta (\rho _\epsilon u_\epsilon )$. From (18) $\epsilon ^a\gamma ^3(\epsilon )\rightarrow 0 \ \forall a>0$; therefore, the remainder terms satisfy (21).

Therefore, the solution of (23, 24) satisfies (15, 16) with $\Vert r_\epsilon \Vert _\infty =O(const \ \epsilon \gamma ^3(\epsilon ))$ and $\Vert \textbf{R}_\epsilon \Vert _\infty =O(const \ \epsilon \gamma ^3(\epsilon ))$ (21). These terms $r_\epsilon$ and $\textbf{R}_\epsilon$ reflect the difference between the auxiliary system (23, 24) and the ODEs (15, 16). We obtain a $L^\infty$ bound on the solutions of (15, 16) through the auxiliary system (23, 24) thanks to the integral formulas (61, 62) and the adapted Gronwall formula (66). Conversely, formulas (73, 74) permit as well with same $(\rho _\epsilon ,\textbf{u}_\epsilon )$ to pass from system (15, 16) with the bounds (20, 21) to system (23-24) with additional remainder terms $(r_\epsilon ), (\textbf{R}_\epsilon )$ still satisfying (21). With these additional remainder terms, system (23, 24) becomes the system (32, 33). The solutions $(\rho _\epsilon ,\textbf{u}_\epsilon )$ of system (15, 16) and system (32, 33) are the same, provided the remainder terms $r_\epsilon ,\textbf{R}_{j,\epsilon }$, which satisfy (21), change with the system.

$\bullet$ Sixth step. Weak asymptotic solutions. We refer to Sect. 4 for the proof of existence of the weak asymptotic solution.

Appendix 2. Physical considerations

Physicists consider particles of fluid, which are small imaginary bags of close enough molecules in which the physical variables are assumed practically constant. In Eqs. (9, 10) the value $\epsilon$ represents the side of cubic particles of fluid $\Pi _{i=1}^n[x_i-\frac{\epsilon }{2},x_i+\frac{\epsilon }{2}]$. If the mean free path, i.e., the average distance between molecules, is about $10^{-9}$ meters for instance then $\epsilon$ would be about $10^{-8}$ or $10^{-7}$ meters. As usual, we let $\epsilon \rightarrow 0$ to model the fact that $\epsilon$ is very small. Then a physical quantity $u_\epsilon$, $\epsilon$ corresponding to the size of a particle of fluid, is mathematically replaced by the weak limit $u:=lim \ u_\epsilon$ when $\epsilon \rightarrow 0$. Similarly, a physical variable $f(u_\epsilon ), f$ a nonlinear function, is replaced by the weak limit of the sequence $(f(u_\epsilon ))$. This limit needs not be the composed function $f(u)=f\circ u$ even when it makes sense. These particles of fluid exchange molecules with the neighbor particles of fluid. The first n terms in the second member of (9, 10) model this exchange when the particles of fluid are $\Pi _{i=1}^n[x_i-\frac{\epsilon }{2},x_i+\frac{\epsilon }{2}]$. To represent the velocity at the level of particles of fluid in (15,16) we introduce the average individual velocity of the molecules in each sense due to their irregular microscopic movements linked to the temperature, not taking into account the overall macroscopic movement of the fluid. In each direction, these two velocities in each sense have same absolute value $\mu (\epsilon ),$ which is modeled by replacing $u_i^\pm$ by ${\tilde{u}}_i^\pm =u_i^\pm +\mu (\epsilon )$ (11). By letting $\mu (\epsilon )\rightarrow +\infty$ very slowly when $\epsilon \rightarrow 0$ so that $\epsilon \mu (\epsilon )\rightarrow 0$ very slowly, we assume that we consider a fluid with a very strong molecular agitation (12). Indeed, the equations of fluid dynamics make sense physically when a local thermodynamic equilibrium is established at all time. This modeling of molecular agitation introduces mathematically a small vanishing viscous term of coefficient $\epsilon \mu (\epsilon )$ which plays an important role in the mathematical proofs. For short we have dropped the regularization of $u^\pm$ when u=0, see formula (12) in Ref. [10]. The small terms $r_\epsilon$ and $\textbf{R}_\epsilon$ in (15, 16) serve to pass in both senses between the two formulations (15,16) and (32, 33) so as to use the heat kernel for a study of the solutions of the ODEs (15, 16). Their presence is justified by the fact that the physical variables are not exactly constant inside the particles of fluid, which justifies the presence of these small remainder terms of higher order. Bounds (20) are justified by the fact that the physical variables are assumed practically constant inside the particles of fluid.

Appendix 3. A particular example of independence on the ingredients with explicit bounds: Holder continuous initial conditions

In Theorem 1 A one has used a unique approximation of the initial condition made possible by the existence of a privileged approximation for initial conditions of class ${\mathcal {C}}^3$: $\forall \epsilon \ \rho _{0,\epsilon }=\rho _0,\textbf{u}_{0,\epsilon }=\textbf{u}_0$. Is it possible to obtain a result independent to some extent on the way the initial condition is approximated (which could apply when one considers general initial conditions $(\rho _0,\textbf{u}_0)\in L^1({\mathbb {T}}^n)\times L^\infty ({\mathbb {T}}^n)^n$ that do not have a privileged approximation by functions of class ${\mathcal {C}}^3$)? We have described how this is possible: Theorem 1 B. Since the needed slowness of convergence to 0 of the sequence $(\epsilon \mu (\epsilon ))_\epsilon$ depends on the accuracy of the approximation of the initial condition $(\rho _{0},\rho _0\textbf{u}_0)$ by the sequence $(\rho _{0,\epsilon },\rho _{0,\epsilon }\textbf{u}_{0,\epsilon })_\epsilon$ that serves as initial conditions in the system of ODEs (15,16), in Theorem 1 B one is forced to let $\epsilon \mu (\epsilon )$ tend to 0 arbitrarily slowly. In this Appendix, to describe a result having a more familiar formulation, we expose a particular case valid for Holder continuous initial conditions which will be formulated in Proposition 3 below.

From (19) $\forall A>0 \mu (\epsilon )>\frac{A}{\epsilon ln\gamma (\epsilon )}$ for $\epsilon >0$ small enough, therefore we first have to fix bounds for the function $\gamma$. The bound (18) suggests to set

$$\begin{aligned} \exists \ 0<b'<b<1 \hbox { such that } (ln\frac{1}{\epsilon })^{b'}\le \gamma (\epsilon )\le (ln\frac{1}{\epsilon })^b. \end{aligned}$$

(75)

Then to have (19) we set

$$\begin{aligned} \exists \ const >0 \hbox { such that } \mu (\epsilon )\ge \frac{const}{\epsilon ln(ln(ln\frac{1}{\epsilon }))} \hbox { together with } \epsilon \mu (\epsilon )\rightarrow 0. \end{aligned}$$

(76)

Let be given initial conditions $(\rho _0,\textbf{u}_0)\in {\mathcal {C}}^{0,H}({\mathbb {T}}^n)\times {\mathcal {C}}^{0,H}({\mathbb {T}}^n)^n$ which are Holder continuous functions on ${\mathbb {T}}^n$ of variable exponent $H\in ]0,1]$. Since Holder continuous functions are not derivable in general, we will choose to approximate them in the form

$$\begin{aligned} w_{0,\epsilon }=w_0*\psi _{\theta (\epsilon )}, w=\rho ,\textbf{u}, \end{aligned}$$

(77)

where $\psi \in {\mathcal {C}}_c^\infty ({\mathbb {R}}^n)$ is a standard mollifier $(\int \psi =1,\psi _\epsilon (x)=\frac{1}{\epsilon ^n}\psi (\frac{x}{\epsilon })$) and $\theta (\epsilon )\rightarrow 0$ when $\epsilon \rightarrow 0$. To have that the approximations $w_{0,\epsilon }$ satisfy (17,22) and that the second member of formulas (51, 52, 53) tend to 0 we fix bounds for the function $\theta$

$$\begin{aligned} \exists \ 0<\alpha <\frac{1}{3}, \exists \ c\in ]0,\frac{\alpha b'}{3}[ \hbox { such that } \frac{const}{(ln\frac{1}{\epsilon })^{\frac{\alpha b'}{3}}}\le \theta (\epsilon )\le \frac{const}{(ln\frac{1}{\epsilon })^c}. \end{aligned}$$

(78)

The real numbers $H,b,b',\alpha ,c$ and the mollifier $\psi$ from formulas (75, 77, 78) above and the assumption on initial conditions chosen as Holder continuous functions are ingredients used only in Proposition 3. The requirements of Sect. 2 are satisfied with these values $H,b,b',\alpha ,c,\psi$; therefore, Propositions 1 and 2 apply. Then we will obtain

Proposition 3

Independence on all ingredients used in the construction in the case of Holder continuous initial conditions presented in Appendix 3. Under the above requirements (75-78) on a choice of ingredients, the set of accumulation points of any asymptotic solution is independent on all arbitrariness in the choice of ingredients, i.e., the ones listed in Theorem 1 A plus all the additional ones above: $H,b,b',\alpha ,c,\psi$. In particular, this set does not depend on the choice of an approximation of the initial condition in the general form (77, 78).

Proof of Proposition 3.

Properties (18) and (19) are satisfied from (75, 76). If D is a partial differential operator of order $\le 3$, $\Vert Dw_{0,\epsilon }\Vert _\infty \le \frac{const}{\theta (\epsilon )^3}$. For (22) we have to check that $\frac{const}{\theta (\epsilon )^3}<\gamma (\epsilon )^\alpha$. This is ensured by $\theta (\epsilon )>\frac{const}{(ln\frac{1}{\epsilon )}^{\frac{b'\alpha }{3}}}$ (78). Therefore, the properties in Sect. 2 are satisfied. Propositions 1 and 2 apply.

It remains to check that the second members of (51, 52, 53) tend to 0 when $\epsilon \rightarrow 0$. One has two sequences $w_{i,\epsilon }=(\rho _{i,\epsilon },\rho _{i,\epsilon }u_{i,\epsilon }), i=1,2$ that correspond to values $H_i,b_i,b'_i,\psi _i,\theta _i,c_i, i=1,2$. From the Holder property, $\Vert w_{i,0,\epsilon }-w_0\Vert _1\le const \ \theta (\epsilon )^{H_i}\le \frac{const}{(ln\frac{1}{\epsilon })^{c_iH_i}}.$ Therefore,

$$\begin{aligned} \Vert w_{i,0,\epsilon }-w_0\Vert _1 exp \frac{const}{\epsilon \mu _j(\epsilon )}\le \frac{const (lnln\frac{1}{\epsilon })^{const}}{(ln\frac{1}{\epsilon })^{c_iH_i}} \end{aligned}$$

(79)

which tends to 0 when $\epsilon \rightarrow 0$. We obtain that the second members of (51, 52, 53) tend to 0 when $\epsilon \rightarrow 0$.

$\square$