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Dissipative Solutions to Compressible Navier–Stokes Equations with General Inflow–Outflow Data: Existence, Stability and Weak Strong Uniqueness

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Abstract

So far existence of dissipative weak solutions for the compressible Navier–Stokes equations (i.e. weak solutions satisfying the relative energy inequality) is known only in the case of boundary conditions with non zero inflow/outflow (i.e., in particular, when the normal component of the velocity on the boundary of the flow domain is equal to zero). Most of physical applications (as flows in wind tunnels, pipes, reactors of jet engines) requires to consider non-zero inflow–outflow boundary condtions. We prove existence of dissipative weak solutions to the compressible Navier–Stokes equations in barotropic regime (adiabatic coefficient \(\gamma >3/2\), in three dimensions, \(\gamma >1\) in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded piecewise regular Lipschitz domain, without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain. It is well known that the relative energy inequality has many applications, e.g., to investigation of incompressible or inviscid limits, to the dimension reduction of flows, to the error estimates of numerical schemes. In this paper we deal with one of its basic applications, namely weak–strong uniqueness principle.

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Notes

  1.  Inspired by this paper and [4], some questions related to this problem were recently discussed in [1].

  2.   We say that \(f\in C_\mathrm{weak}([0,T], L^p(\Omega ))\) iff \(f:[0,T]\rightarrow L^p(\Omega )\) is defined everywhere on [0, T], \(f\in L^\infty (0,T; L^p(\Omega ))\) and the map \(t\mapsto {\int _{\Omega }} f\varphi (t,\cdot ) \ \,{\mathrm{d}} {x}\in C[0,T]\) for all \(\varphi \in L^{p'}(\Omega )\).

  3.   In contrast with [4], our approach does not require neither (2.13) nor (2.14).

  4.   The requested regularity of some of this derivatives can be slightly weaken by a more detailed (elementary) analysis as in [15].

  5.   The exact choice of \(\beta \) is irrelevant from the point of view of the final result provided it is sufficiently large.

  6.   For bounded Lipschitz domains such extension always exists, cf. Remark 2.1.

  7.   Tensor \({{\mathbb {Z}}}_\varepsilon \) corresponds to the weak limit of the term \(\varepsilon |\nabla _{x}(\mathbf{u}_N-\mathbf{u}_\infty )|^2\nabla _{x}(\mathbf{u}_N-\mathbf{u}_\infty )\) in the weak formulation of the momentum equation, cf. (3.8), (4.24).

  8.   The only difference with respect to [4, Lemma 4.1] in Lemma 3.1 is the fact, that the test function in equation (3.7) does not vanish at the outflow boundary and the energy inequality is more precise containing also all boundary terms. This is essential for the construction of dissipative solutions. We also do not need the belonging of \(\mathbf{u}_\infty \) to the class (2.13).

  9.   The passage to the limit in the energy inequality in [4] requires the property \(\mathrm{div}\mathbf{u}_\infty \ge 0\) in a (small) inner neighborhood of \(\partial \Omega \), cf. (2.13), when dealing with the term \(\int _0^\tau {\int _{\Omega }} p_\delta (\varrho )\mathrm{div}\mathbf{u}_\infty \ \,{\mathrm{d}} {x}\). We can get rid of this requirement by using Lemma 6.1 by the same type of reasoning used to handle the similar term in the relative energy inequality, cf. (6.10).

  10.   This is the key point in the existence theory of compressible Navier–Stokes equations, whatever is the geometrical setting and whatever are the boundary conditions, cf. Lions [17, 24].

  11.   Here and hereafter, the standard compactness arguments include the Sobolev imbeddings, Arzela-Ascoli theorem and Lions-Aubin lemma.

  12.   This Lemma holds without condition \(p(0)=0\), \(p'(\varrho )>0\); regularity assumption of (2.2) is enough.

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Authors and Affiliations

  1. Department of Mathematics, Dong-A University, Busan, 604-714, Republic of Korea

    Young-Sam Kwon

  2. IMATH, EA 2134, Université du Sud Toulon-Var, BP 20132, 83957, La Garde, France

    Antonin Novotny

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  1. Young-Sam Kwon
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Correspondence to Antonin Novotny.

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Communicated by G. P. Galdi.

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Y.-S. Kwon: The work of the first author was partially supported by NRF-2017R1D1A1B03030249 and NRF-2019H1D3A2A01101128. A. Novotny: The work of the second author was supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea, NRF-2019H1D3A2A01101128.

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Kwon, YS., Novotny, A. Dissipative Solutions to Compressible Navier–Stokes Equations with General Inflow–Outflow Data: Existence, Stability and Weak Strong Uniqueness. J. Math. Fluid Mech. 23, 23 (2021). https://doi.org/10.1007/s00021-020-00553-z

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