We deal with the viscous profiles for a class of mixed hyperbolic-parabolic
systems. We focus, in particular, on the case of the compressible Navier Stokes
equation in one space variable written in Eulerian coordinates. We describe the
link between these profiles and a singular ordinary differential equation in
the form $$ dV / dt = F(V) / z (V) . $$ Here $V \in R^d$ and the function F
takes values into $R^d$ and is smooth. The real valued function z is as well
regular: the equation is singular in the sense that z (V) can attain the value
0.Comment: 6 pages, minor change

Bianchini, Stefano

Spinolo, Laura V.

English

arXiv

We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems in one space dimension. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form$\frac{dV}{dt} =  \frac{1}{\zeta (V)}  F(V).$ Here $V \in \mathbb{R}^d$ and the function $F$ takes values into $\mathbb{R}^d$ and is smooth. The real valued function $\zeta $ is  as well regular: the equation is singular in the sense that $\zeta (V)$ can attain the value $0$

OpenstarTs

Rend. Istit. Mat. Univ. TriesteVolume 41 (2009), 35–41.A Connection betweenViscous Profilesand Singular ODEsStefano Bianchini and Laura V. SpinoloAbstract. We deal with the viscous profiles for a class of mixedhyperbolic-parabolic systems in one space dimension. We focus, in par-ticular, on the case of the compressible Navier Stokes equation in onespace variable written in Eulerian coordinates. We describe the linkbetween these profiles and a singular ordinary di!erential equation inthe formdVdt=1!(V )F (V ). (1)Here V ! Rd and the function F takes values into Rd and is smooth.The real valued function ! is as well regular: the equation is singularin the sense that !(V ) can attain the value 0.Keywords: Mixed Hyperbolic-Parabolic Systems, Singular ODE, Boundary Layers, Trav-eling Waves, Navier Stokes Equation.MS Classification 2000: 35M10, 35L65, 34A99We focus on mixed hyperbolic-parabolic systems in one space variable inthe formE(u)ut + A(u, ux)ux = B(u)uxx. (2)Here the function u takes values in RN and depends on the two scalar variables tand x. We focus on the case the matrix B is singular, namely its rank is strictlysmaller than N . In particular, a conservative system ut + f(u)x =!B(u)ux"xcan be written in the form (2). Indeed, one can set A(u, ux) = D f(u) "B(u)x, where D f denotes the Jacobian matrix of f . In the following we willconsider explicitly the case of the compressible Navier Stokes equation in onespace variable:#$%$&"t + ("v)x = 0("v)t +!"v2 + p"x=!#vx"x'"e + "v22(t+'v)12"v2 + "e + p*(x='k$x + #vvx(x.(3)36 STEFANO BIANCHINI AND LAURA V. SPINOLOHere the unknowns "(t, x), v(t, x) and $(t, x) are the density of the fluid, thevelocity of the particles in the fluid and the absolute temperature respectively.The function p = p(", $) > 0 is the pressure and satisfies p! > 0, while e is theinternal energy. In the following we will focus on the case of a polytropic gas,so that e satisfies e = R$/(% " 1), where R is the universal gas constant and% > 1 is a constant specific of the gas. Finally, by #(") > 0 and k(") > 0 wedenote the viscosity and the heat conduction coe!cients respectively.In the following, we assume that system (2) satisfies a set of hypothesesintroduced by Kawashima and Shizuta in [6] and satisfied, up to a change inthe dependent variables, by the equations of the hydrodynamics. First, weassume that the rank of the matrix B(u) is constant and we denote it by r.Also, B(u) admits the block decompositionB(u) =+0 00 b(u),. (4)The block b(u) belongs to Mr!r and there exists a constant cb > 0 such that,for every &' ! Rr, one has #b(u)&', &'$ % cb|&'|2, where #·, ·$ denotes the standardscalar product in Rr.Also, we assume that for every u the matrix A(u, 0) is symmetric. Wedenote byA(u, ux) =+A11(u) AT21(u)A21(u) A22(u, ux),E(u) =+E11(u) ET21(u)E21(u) E22(u),(5)the block decomposition of A and E corresponding to (4). Note that only theblock A22 can depend on ux. Finally, we assume that for every u, the matrixE(u) is symmetric and positive definite.In the following, we focus on two classes of solutions of (2): traveling wavesand steady solutions. A traveling wave is a one variable function U(y) satisfying-A(U, U ")"(E(U).U " = B(U)U "", where ( is a real constant. From a travelingwave U one obtains a solution of (2) by setting u(t, x) = U(x"(t). The steadysolutions of (2) are one variable functions U(x) satisfyingA(U, U ")U " = B(U)U "". (6)We also require that U is bounded on x ! [0, +&[ and admits a limit asx ' +&. Steady solutions in this class are sometimes called boundary layers.In the applications, it is often interesting to focus on the case the speed ( ofthe traveling wave is close to an eigenvalue of E#1A. Since in general 0 is notan eigenvalue of E#1A, it is often useful to distinguish between traveling wavesand boundary layers.VISCOUS PROFILES AND SINGULAR ODEs 37Travelling waves and steady solutions are powerful tools to study the viscousapproximation of hyperbolic problems: being the literature concerning thistopic extremely big, we just refer to the books by Dafermos [3] and by Serre[9] and to the references in there.In a previous work [2], the authors studied a problem related to (2) byimposing a new condition of block linear degeneracy on the structure of thematrices E, A and B: let the block A11(u) and E11(u) be as in (5). The blocklinear degeneracy precribes that for every real number ( the dimension of thekernel of [A11(u) " (E11(u)] is constant with respect to u. In other words,such a dimension can in general vary when ( varies, but it cannot vary whenu varies.If the condition of block linear degeneracy is violated, then one might face“pathological” behaviors, in the following sense. There are systems satisfyingall the Kawashima Shizuta hypotheses and violating the block linear degeneracywhich admit non continuously di"erentiable steady solutions (note that in thiscase it is not completely clear, a priori, what we mean by solution of (6), becausewe are dealing with non regular functions: see [2, Section 2] for the details).On the other side, imposing the block linear degeneracy is restrictive in view ofsome applications. In particular, as pointed out by Frédéric Rousset [7], sucha condition is satisfied by the compressible Navier Stokes equation writtenin Lagrangian coordinates, but is violated by the same equation written inEulerian coordinates. The problem is the following.As pointed out for example in [8], the Navier Stokes equation written inLagrangian coordinates can be reduced to the form (2). In particular, the rankof the matrix B(u) is constantly equal to two and the block A11 defined asin (5) is a real valued function and it is actually identically equal to 0, whichimplies that the condition of block linear degeneracy is satisfied. By directcomputations one can then verify that the viscous profiles satisfy an ordinarydi"erential equation which does not have the singularity exhibited by (1).Let us consider now (3), the Navier Stokes equation written in Euleriancoordinates. We want to write it in the form (2) requiring that the matrixA(u, 0) is symmetric. In the following, we assume that " is bounded away from0, say " % c! > 0 for a suitable constant c!. This implies that the systemdoes not reach the vacuum. We proceed as in Kawashima and Shizuta [6] andby multiplying (3) on the left by a suitable nonsingular matrix we eventuallyobtain that (3) can be written in the form (2) forE(", v, $) =1$/0p!/" 0 00 " 00 0 "e"/$12B(", v, $) =1$/00 0 00 # 00 0 k/$12 (7)38 STEFANO BIANCHINI AND LAURA V. SPINOLOandA(", v, $, "x, vx, $x) =1$/0p!v/" p! 0p! "v " #""x p"0 p" " #vx/$ "ve"/$ " k""x/$12 (8)In the previous expression, we denote by p! and p" the partial derivative of pwith respect to " and $ respectively, and, by recalling that e = R$/(% " 1), weget that e" = R/(% " 1). Also, we setA21 =1$+p!0,A22 =1$+"v " #""x p"p" " #vx/$ "ve"/$ " k""x/$,(9)andb =1$+# 00 k/$,a11 = p!/($").The condition of block linear degeneracy is violated here. To see this, let usfocus on the case ( = 0: the dimension of the kernel of A11 = a11v is 0 if v (= 0,but it is 1 when v = 0 (we recall that p! > 0).To see what in principle can go wrong, we focus on steady solutions, the sit-uation for traveling waves being analogous. We set w = "x and &z =!vx, $x"T.Then (6) becames3a11vw + AT21&z = 0A21w + A22&z = b&zxAssume v (= 0, then (6) can be written as#$$%$$&w = "AT21&za11v&zx = b#1)A22 "A21AT21a11v*&z(10)Note that the matrix b is invertible and hence the previous expression iswell defined.By recalling that w = "x and &z =!vx, $x"Tand by relying on (10), oneobtains that the steady solutions of the Navier Stokes written in Eulerian co-ordinates satisfy a singular ordinary di"erential equation in the formdUdx=1vF (U) (11)provided that U =!", v, $, &z"TandF (U) =/40AT21&z/a11v &zb#1)A22v " A21AT21/a11*&z152VISCOUS PROFILES AND SINGULAR ODEs 39We say that equation (11) is singular since in general v, the velocity of thefluid, can attain the value 0.Let us now go back to the example in [2] concerning a system with noncontinuously di"erentiable steady solutions. It turns out that, as a consequenceof the fact that the system violates the block linear degeneracy, the steadysolutions V satisfy a singular ODE in the form (1), where ! is a real-valued,smooth function that can attain the value 0. The loss of regularity experiencedby V is actually due to the fact that the equation is singular. More precisely,the problem is that !(V ) (= 0 at t = 0, but ! reaches the singular value 0 at afinite time t0, which is exactly the time when the first derivative blows up.Summing up, we have the following: the condition of block linear degeneracyis satisfied by the compressible Navier Stokes equation written in Lagrangiancoordinates, but it is violated by the same equation written in Eulerian co-ordinates. As a consequence, the viscous profiles of the equation written inEulerian coordinates satisfy a singular ODE and hence might in principle ex-perience a loss of regularity. This suggests that we should look for more generalconditions than the block linear degeneracy, namely conditions weak enoughto apply to the Navier Stokes equation written in Eulerian coordinates and byother systems violating the block linear degeneracy. On the other side, theseconditions should be su!ciently strong to rule out any loss of regularity.A set of conditions satisfying the previous requirements is defined in [1,Section 2] by relying on the study of the singular equation (1) in a small enoughneighborhood of a point V̄ satisfying !(V̄ ) = 0 and G(V̄ ) = &0. The analysisin [1] is indeed local, and hence the results only apply to the study of theviscous profiles with small enough total variation.In [1] we rely on the notions of fast and slow dynamics of the singularODE (1): the formal definition is technical, but we can get an heuristic ideaby considering the toy model#$%$&dv1dt= "v1dv2dt= "v2).Here the singularity ) is just a parameter, ) ' 0+ and the solution can beexplicitly computed. Both v1 and v2 are exponentially decaying to 0, but thespeed of exponential decay of v2 gets faster and faster as ) ' 0+ and hence v2is regarded as a fast dynamic. Conversely, the speed of exponential decay of v1is bounded with respect to ) and hence v1 is regarded as a slow dynamic.These notions can be extended to the general nonlinear case (1) and arereminiscent of the notions of slow and fast time scale discussed by Fenichel [4](see also the lecture notes by Jones [5]).40 STEFANO BIANCHINI AND LAURA V. SPINOLOIn [1] we impose on (1) several hypotheses, the key one being that thesingular set{U : !(U) = 0} ) Rdis invariant for both the slow and the fast dynamics. Namely, if U(t) is asolution of (1) which is either a slow or a fast dynamic and !!U(0)"= 0 then!!U(t)"= 0 for every t ! R. As a matter of fact, this condition and the otherhypotheses introduced in [1] are satisfied by the equation (11) describing theviscous profiles of the Navier Stokes equation written in Eulerian coordinates.In [1] the analysis focuses on the solutions of (1) lying on suitable invariantmanifolds (a manifold M is invariant for (1) if any solution U(t) such thatU(0) ! M satisfies U(t) ! M for every t) satisfying the following property:if U(t) is a solution lying on the manifold and !!U(0)"(= 0 then !!U(t)"(= 0for every t. Also, the viscous profiles with su!ciently small total variation lieon these manifolds, hence by applying our results to (11) we rule out the lossof regularity and we obtain that the traveling waves and the boundary layerswith small total variation are continuously di"erentiable.For a di"erent approach to the analysis of the viscous profiles of theNavier Stokes equation in Eulerian coordinates see Wagner [10] and the ref-erences therein.References[1] S. Bianchini and L.V. Spinolo, Invariant manifolds for a singular ordinarydi!erential equation, submitted, arXiv:0801.4425.[2] S. Bianchini and L.V. Spinolo, The boundary Riemann solver coming from thereal vanishing viscosity approximation, Arch. Rational Mech. Anal. 191 (2009),1–96.[3] C.M. Dafermos, Hyperbolic conservation laws in continuum physics, seconded., Springer, Berlin (2005).[4] N. Fenichel, Geometric singular perturbation theory for ordinary di!erentialequations, J. Di!erential Equations 31 (1979), 53–98.[5] C.K.R.T. Jones, Geometric singular perturbation theory, Dynamical systems(Montecatini Terme, 1994), Lecture Notes in Math. vol. 1609, Springer, Berlin(1995), pp. 44–118.[6] S. Kawashima and Y. Shizuta, On the normal form of the symmetrichyperbolic-parabolic systems associated with the conservation laws, TôhokuMath. J. 40 (1988), 449–464.[7] F. Rousset, Navier Stokes equation and block linear degeneracy, personal com-munication.[8] F. Rousset, Characteristic boundary layers in real vanishing viscosity limits, J.Di!erential Equations 210 (2005), 25–64.VISCOUS PROFILES AND SINGULAR ODEs 41[9] D. Serre, Systems of conservation laws, I, II, Cambridge University Press,Cambridge (2000).[10] D.H. Wagner, The existence and behavior of viscous structure for plane deto-nation waves, SIAM J. Math. Anal. 20 (1989), 1035–1054.Authors’ addresses:Stefano BianchiniSISSAVia Bonomea 26534136 Trieste, ItalyE-mail: bianchin@sissa.itLaura V. SpinoloCentro De GiorgiScuola Normale SuperiorePiazza dei Cavalieri 3, 56126 Pisa, ItalyE-mail: laura.spinolo@sns.itReceived October 13, 2009Revised November 16, 2009

A Connection between Viscous Profiles and Singular ODEs

Abstract. We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form dV/dt = F (V )/ζ(V ). Here V ∈ Rd and the function F takes values into Rd and is smooth. The real

valued function ζ is as well regular: the equation is singular in the sense that ζ(V ) can attain the value 0

Bianchini, S.

Spinolo, L. V.

SISSA Digital Library

A connection between viscous profiles and singular ODEs

We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems in one space dimension. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form dV/dt = 1/ζ(V) F(V). (1) Here V ∈ Rd and the function F takes values into Rd and is smooth. The real valued function ζ is as well regular: the equation is singular in the sense that ζ(V) can attain the value 0

Sissa Digital Library

A CONNECTION BETWEEN VISCOUS PROFILES ANDSINGULAR ODESSTEFANO BIANCHINI AND LAURA V. SPINOLOAbstract. We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressibleNavier Stokes equation in one space variable written in Eulerian coordinates.We describe the link between these profiles and a singular ordinary differentialequation in the form(1)dVdt=1ζ(V )F (V ).Here V ∈ Rd and the function F takes values into Rd and is smooth. The realvalued function ζ is as well regular: the equation is singular in the sense thatζ(V ) can attain the value 0.We focus on mixed hyperbolic-parabolic systems in one space variable in theform(2) E(u)ut +A(u, ux)ux = B(u)uxx.Here the function u takes values in RN and depends on the two scalar variables tand x. We focus on the case the matrix B is singular, namely its rank is strictlysmaller than N . In particular, a conservative systemut + f(u)x =(B(u)ux)xcan be written in the form (2). Indeed, one can setA(u, ux) = Df(u)−B(u)x,where Df denotes the jacobian matrix of f . In the following we will considerexplicitly the case of the compressible Navier Stokes equation in one space variable:(3)ρt + (ρv)x = 0(ρv)t +(ρv2 + p)x=(νvx)x(ρe+ ρv22)t+(v[12ρv2 + ρe+ p])x=(kθx + νvvx)x.Here the unknowns ρ(t, x), v(t, x) and θ(t, x) are the density of the fluid, thevelocity of the particles in the fluid and the absolute temperature respectively. Thefunction p = p(ρ, θ) > 0 is the pressure and satisfies pρ > 0, while e is the internalenergy. In the following we will focus on the case of a polytropic gas, so that esatisfies(4) e =Rθγ − 1,2000 Mathematics Subject Classification. 35M10, 35L65, 34A99.Key words and phrases. mixed hyperbolic-parabolic systems, singular ODE, boundary layers,travelling waves, Navier Stokes equation.12 STEFANO BIANCHINI AND LAURA V. SPINOLOwhere R is the universal gas constant and γ > 1 is a constant specific of the gas.Finally, by ν(ρ) > 0 and k(ρ) > 0 we denote the viscosity and the heat conductioncoefficients respectively.In the following, we assume that system (2) satisfies a set of hypotheses intro-duced by Kawashima and Shizuta in [8]. These conditions were defined relying onexamples with a physical meaning, in particular, up to a change in the dependentvariables, they are satisfied by the equations of the hydrodynamics. For complete-ness, we recall these conditions here. First, we assume that the rank of the matrixB(u) is constant and we denote it by r. Also, B(u) admits the block decomposition(5) B(u) =(0 00 b(u)).The block b(u) belongs to Mr×r and there exists a constant cb > 0 such that forevery ~ξ ∈ Rr(6) 〈b(u)~ξ, ~ξ〉 ≥ cb|~ξ|2.In the previous expression, 〈·, ·〉 denotes the standard scalar product in Rr.Also, we assume that for every u the matrix A(u, 0) is symmetric. We denoteby(7) A(u, ux) =(A11(u) AT21(u)A21(u) A22(u, ux))E(u) =(E11(u) ET21(u)E21(u) E22(u))the block decomposition of A and E corresponding to (5). Note that only the blockA22 can depend on ux. Finally, we assume that for every u, the matrix E(u) issymmetric and positive definite.In the following, we focus on two classes of solutions of (2): travelling waves andsteady solutions. A travelling wave is a one variable function U(y) satisfying(8)[A(U, U ′)− σE(U)]U ′ = B(U)U ′′.In the previous expression, σ is a real number and is the speed of the wave. From(8) one obtains a solution of (2) by setting u(t, x) = U(x − σt). Steady solutionsare solutions of (2) that do not depend on time: namely, they are one variablefunctions U(x) satisfying(9) A(U, U ′)U ′ = B(U)U ′′.We also require that U is bounded on x ∈ [0, +∞[ and admits a limit as x→ +∞.Steady solutions in this class are sometimes called boundary layers. In the appli-cations, it is often interesting to focus on the case the speed σ in (8) is close to aneigenvalue of E−1A. Since in general 0 is not an eigenvalue of E−1A, it is useful todistinguish between (8) and (9).Travelling waves and steady solutions are powerful tools to study the parabolicapproximation of hyperbolic problems: since the literature concerning this topic isextremely big, we just refer to the books by Dafermos [6] and by Serre [11] andto the rich bibliography contained therein. Concerning the analysis of the viscousprofiles, we refer to Benzoni-Gavage, Rousset, Serre and Zumbrun [2], to Zumbrun[13] and to the references therein. Loosely speaking, the problem of the parabolicapproximation of an hyperbolic system is the following: consider the family ofsystems(10) E(uε)uεt +A(uε, εuεx)uεx = εB(uε)uεxx,VISCOUS PROFILES AND SINGULAR ODES 3which reduces to (2) via the change of variable u(t, x) = uε(εt, εx). Letting ε→ 0+,equation (10) formally reduces to(11) E(u)ut +A(u, 0)ux = 0.As we pointed out before, the compressible Navier Stokes equation in one spacevariable can be written in a form like (10): in this case, system (11), which formallyis obtained by setting ε = 0, is the Euler equation. The proof of the convergence ofuε to a solution of (11) is an open problem in the most general case, but convergenceresults are available in more specific situations (see e.g. Bianchini and Bressan [3],Ancona and Bianchini [1] and Gisclon [7]). Travelling waves have been exploitedto study the parabolic approximation (10) and its relations with the limit (11),especially to describe such phenomena as the formation of singularities. The studyof steady solutions has provided useful information to the analysis of (10) especiallyin the case of initial boundary value problems. In the following we will be mainlyconcerned with viscous profiles having sufficiently small total variation.In a previous work [4], the authors studied the approximation (10) for an initialboundary value problem with special data. Concerning the structure of the matricesE, A and B, we introduced a new condition of block linear degeneracy, which isthe following. Let the block A11(u) and E11(u) be as in (7). The block lineardegeneracy precribes that for every real number σ the dimension of the kernel of[A11(u) − σE11(u)] is constant with respect to u. In other words, the conditionof block linear degeneracy says that such a dimension can in general vary when σvaries, but it cannot vary when u varies.If the condition of block linear degeneracy is violated, then one might face“pathological” behaviors, in the following sense. In [4, Section 2] we discuss anexample of a system that satisfies all the Kawashima Shizuta hypotheses, but doesnot satisfy the block linear degeneracy. We exhibit a steady solution (9) of thissystem which is not continuously differentiable. Note that in this case it is notcompletely clear, a priori, what we mean by solution of (9), because we are dealingwith non regular functions. The details are given in [4].On the other side, imposing the block linear degeneracy is restrictive in viewof some applications. In particular, as pointed out by Fre´de´ric Rousset in [9],this condition is satisfied by the compressible Navier Stokes equation written inLagrangian coordinates, but is violated by the same equation written in Euleriancoordinates.The problem is the following. Consider the Navier Stokes equation written inLagrangian coordinates:(12)τt − vx = 0vt + px =(ντvx)x(e+v22)t+(pv)x=(kτθx +ντvvx)x.Here the unknowns are τ(t, x), v(t, x) and θ(t, x): τ is the specific volume, andit is the inverse of the density, τ = 1/ρ. As in (3), v and θ denote the velocityand the absolute temperature of the fluid. As pointed out for example in Rousset[10], the compressible Navier Stokes equation written in Lagrangian coordinatescan be written in a form like (2), with E, A and B satisfying all the KawashimaShizuta [8] conditions. In particular, the rank of the viscosity matrix is constantlyequal to two. Let A11 be the same block as in (7). In the case of the compressible4 STEFANO BIANCHINI AND LAURA V. SPINOLONavier Stokes written in Lagrangian coordinates A11 is a real valued function andit is actually identically equal to 0. This implies, in particular, that the conditionof block linear degeneracy is satisfied by the compressible Navier Stokes writtenin Lagrangian coordinates. Indeed, let E11(u) as in (7): in this case, E11(u) is astrictly positive, real valued function. It follows that, for every real number σ, thedimension of the kernel of −σE11(u) does not depend on u.By direct computations one can then verify that the viscous profiles of the NavierStokes equation written in Lagrangian coordinates satisfy an ordinary differentialequation which does not have the singularity exhibited by (1).Let us consider now (3), the Navier Stokes equation written in Eulerian coordi-nates. We want to write it in the form(13) E(u)ut +A(u, ux)ux = B(u)uxx,requiring that the matrix A(u, 0) is symmetric. In the following, we assume thatρ is bounded away from 0, say ρ ≥ cρ > 0 for a suitable constant cρ. This impliesthat the system does not reach the vacuum.We proceed as in Kawashima and Shizuta [8] and by multiplying (3) on the leftby a suitable nonsingular matrix we eventually obtain that (3) can be written inthe form (13) for(14) E(ρ, v, θ) =1θ pρ/ρ 0 00 ρ 00 0 ρeθ/θ B(ρ, v, θ) = 1θ 0 0 00 ν 00 0 k/θand(15) A(ρ, v, θ, ρx, vx, θx) =1θ pρv/ρ pρ 0pρ ρv − ν′ρx pθ0 pθ − νvx/θ ρveθ/θ − k′ρx/θIn the previous expression, we denote by pρ and pθ the partial derivative of p withrespect to ρ and θ respectively, while by exploiting (4) we get that eθ = R/(γ− 1).Also, to simplify the notations we write(16) A21 =1θ(pρ0)A22 =1θ(ρv − ν′ρx pθpθ − νvx/θ ρveθ/θ − k′ρx/θ)andb =1θ(ν 00 k/θ)a11 = pρ/(θρ).The condition of block linear degeneracy is violated here. To see this, let us focuson the case σ = 0: the dimension of the kernel of A11 = a11v is 0 if v 6= 0, but it is1 when v = 0 (we recall that pρ > 0).To see what in principle can go wrong, we focus on steady solutions, the situationfor travelling waves being analogous. We set(17) w = ρx ~z =(vx, θx)T.Then (9) becames(a11v AT21(u)A21(u) A22(u, ux))(w~z)=(0 00 b(u))(wx~zx),VISCOUS PROFILES AND SINGULAR ODES 5i.e. {a11vw +AT21~z = 0A21w +A22~z = b~zxAssume v 6= 0, then (9) can be written as(18)w = −AT21~za11v~zx = b−1[A22 −A21AT21a11v]~zNote that the matrix b is invertible and hence the previous expression is well defined.By combining (17) and (18), one obtains that the steady solutions of the NavierStokes written in Eulerian coordinates satisfy a singular ordinary differential equa-tion in the form(19)dUdx=1vF (U)provided that U =(ρ, v, θ, ~z)TandF (U) =AT21~z/a11v ~zb−1[A22v −A21AT21/a11]~zWe say that equation (19) is singular since in general v, the velocity of the fluid,can attain the value 0.Let us now go back to the example in [4] we mentioned before: the exampledeals with a system with non continuously differentiable steady solutions. It turnsout that, as a consequence of the fact that the system violates the block lineardegeneracy, the steady solutions V satisfy a singular ODE in the form(20)dVdt=1ζ(V )G(V ),where ζ is a real-valued, smooth function that can attain the value 0. The loss ofregularity experienced by V is actually due to the fact that (20) is singular.Summing up, we have the following: the condition of block linear degeneracy issatisfied by the compressible Navier Stokes equation written in Lagrangian coordi-nates, but it is violated by the same equation written in Eulerian coordinates. Asa consequence, the viscous profiles of the equation written in Eulerian coordinatessatisfy a singular ODE and hence might in principle experience a loss of regularity.This suggests that we should look for more general conditions than the block lineardegeneracy. Namely, we want to find a condition on the singular equation (20)which is sufficiently weak to be satisfied by the Navier Stokes equation written inEulerian coordinates and by other systems violating the block linear degeneracy.On the other side, this condition should be sufficiently strong to rule out any lossof regularity.The definition of this condition and the corresponding analysis of the singularequation (20) is done in [5]. The conditions defined there apply to a class of systemsthat do not satisfy the block linear degeneracy and, in particular, to the NavierStokes in Eulerian coordinates.6 STEFANO BIANCHINI AND LAURA V. SPINOLORemark 1. For a different approach to the analysis of the viscous profiles of theNavier Stokes equation in Eulerian coordinates we refer for example to Wagner [12]and to the references therein.References[1] F. Ancona and S. Bianchini. Vanishing viscosity solutions of hyperbolic systems of conserva-tion laws with boundary. In “WASCOM 2005”—13th Conference on Waves and Stability inContinuous Media, pages 13–21. World Sci. Publ., Hackensack, NJ, 2006.[2] S. Benzoni-Gavage, F. Rousset, D. Serre, and K. Zumbrun. Generic types and transitions inhyperbolic initial-boundary-value problems. Proc. Roy. Soc. Edinburgh Sect. A, 132(5):1073–1104, 2002.[3] S. Bianchini and A. Bressan. Vanishing viscosity solutions of non linear hyperbolic systems.Ann. of Math., 161:223–342, 2005.[4] S. Bianchini and L.V. Spinolo. The boundary Riemann solver coming from the real vanishingviscosity approximation. Arch. Rational Mech. Anal., 191(1):1–96, 2009.[5] S. Bianchini and L.V. Spinolo. Invariant manifolds for a singular ordinary differential equa-tion. Submitted. Available in preprint version on www.arxiv.org.[6] C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag,Berlin, Second edition, 2005.[7] M. Gisclon. Etude des conditions aux limites pour un syste`me hyperbolique, vial’approximation parabolique. J. Maths. Pures & Appl., 75:485–508, 1996.[8] S. Kawashima and Y. Shizuta. On the normal form of the symmetric hyperbolic-parabolicsystems associated with the conservation laws. Toˆhoku Math. J., 40:449–464, 1988.[9] F. Rousset. Navier Stokes Equation and Block Linear Degeneracy, Personal Communication.[10] F. Rousset. Characteristic boundary layers in real vanishing viscosity limits. J. DifferentialEquations, 210:25–64, 2005.[11] D. Serre. Systems of Conservation Laws, I, II. Cambridge University Press, Cambridge, 2000.[12] D. H. Wagner. The Existence and Behavior of Viscous Structure for Plane Detonation Waves.SIAM J. Math. Anal., 20(5):1035–1054, 1989.[13] K. Zumbrun. Stability of large-amplitude shock waves of compressible Navier-Stokes equa-tions. In Handbook of mathematical fluid dynamics. Vol. III, pages 311–533. North-Holland,Amsterdam, 2004. With an appendix by Helge Kristian Jenssen and Gregory Lyng.S.B.: SISSA, via Beirut 2-4 34014 Trieste, ItalyE-mail address: bianchin@sissa.itL.V.S.: Centro De Giorgi, Collegio Puteano, Scuola Normale Superiore, Piazza deiCavalieri 3, 56126 Pisa, ItalyE-mail address: laura.spinolo@sns.it

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A connection between viscous profiles and singular ODEs

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