Relative Entropy and the Navier-Stokes-Smoluchowski System for Compressible Fluids

Transcript

1 Relative Entropy and the Navier-Stokes-Smoluchowski System for Compressible Fluids Joshua Ballew Joint work with Konstantina Trivisa Carnegie Mellon University CNA Seminar October 28, 2014

2 Outline Relative Entropy Navier-Stokes-Smoluchowski System Weak-Strong Uniqueness Weakly Dissipative Solutions Relative Entropy Inequality Weak-Strong Uniqueness Result Low Stratification Low Mach Number Limit Concluding Remarks

3 Entropy/Entropy Flux Pairs Consider the hyperbolic equation t U + div x G(U) = 0 (1) where U : R n R m. Examples include the inviscid Burgers equation (G(U) = 1 2 U2 ). Consider functions E(U, x, t) and Q(U, x, t) such that DQ = (DG)(DE) E is called an entropy and Q and entropy flux for (1). Together, they are called an entropy/entropy flux pair. For weak solutions to the hyperbolic problem (1), t E + div x Q 0.

4 Examples of Entropy/Entropy Flux Pairs I Consider t U + x G(U) = 0. (2) If E is any function, and Q(U) = U E (U)G (U) du, then E and Q are an entropy/entropy flux pair.

5 Examples of Entropy/Entropy Flux Pairs II Consider the one-dimensional Euler equations for compressible fluids: with p F (ϱ) = aϱ γ. We have the entropy/entropy flux pair t ϱ + x (ϱu) = 0 (3) t (ϱu) + x ( ϱ u 2 + p F (ϱ) ) = 0 (4) E(ϱ, u) := 1 2 ϱ u 2 + a γ 1 ϱγ and Q(ϱ, u) = 1 2 ϱ u 3 + aγ γ 1 ϱγ u. If ψ is a continuous function, we also have the entropy/entropy flux pair E ψ (ϱ, u) := ϱ 1 ( Q ψ (ϱ, u) := ϱ 1 where λ = 3 γ 2(γ 1). 1 1 ψ(u + ϱ γ 1 s)(1 s 2 ) λ ds u + γ 1 ϱ γ 1 s 2 ) ψ(u + ϱ γ 1 s)(1 s 2 ) λ ds

6 Relative Entropy Consider an entropy/entropy flux pair (E, Q). We define the relative entropy H(U U) as H(U U) := E(U) E(U) DE(U) (U U) (5) Note that this definition will only consider quadratic terms, but not linear terms in the entropy.

7 Example of a Relative Entropy Consider the Euler model (3)-(4). The relative mechanical entropy is where E F (ϱ, ϱ) := H(ϱ, u ϱ, u) = 1 2 ϱ u u 2 + E F (ϱ, ϱ) (6) a γ 1 (ϱγ ϱ γ ) aγ γ 1 ϱγ 1 (ϱ ϱ).

8 Fluid-Particle Interaction Fluid-particle interaction models are of interest to engineers and scientists studying biotechnolgy, medicine, waste-water recycling, mineral processing, and combustion theory. The macroscopic model considered in this talk, the Navier-Stokes-Smoluchowski system, is formally derived from a Fokker-Planck type kinetic equation coupled with fluid equations. This coupling is from the mutual frictional forces between the particles and the fluid, assumed to follow Stokes Law. The fluid is a viscous, Newtonian, compressible fluid.

9 Navier-Stokes-Smoluchowski System t ϱ + div x (ϱu) = 0 (7) t (ϱu) + div x (ϱu u) + x (aϱ γ + η) = div x S( x u) (βϱ + η) x Φ (8) t η + div x (ηu η x Φ) = x η (9) 1 2 ϱ u 2 + a γ 1 ϱγ + η ln η + (βϱ + η)φ dx(τ) τ + µ x u 2 + λ div x u 2 + x η + η x Φ 2 dx dt ϱ 0 u a γ 1 ϱγ 0 + η 0 ln η 0 + (βϱ 0 + η 0 ) x Φ dx (10)

10 Constitutive Relations and Boundary and Initial Conditions Newtonian Condition for a Viscous Fluid: S( x u) := µ( x u + T x u) + λ div x ui µ > 0, λ µ 0 Pressure Conditions: γ > 3 2, a > 0 Boundary and Initial Conditions: u = ( x η + η x Φ) n = 0 (11) ϱ 0 L γ () L 1 +() (12) m 0 L 6 5 (; R 3 ) L 1 (; R 3 ) (13) η 0 L 2 () L 1 +() (14)

11 Confinement Hypotheses Take Φ : R + where is C 2,ν. Bounded Domain Φ is bounded and Lipschitz on. β 0. The sub-level sets [Φ < k] are connected in for all k > 0. Unbounded Domain Φ W 1, loc (). β > 0. The sub-level sets [Φ < k] are connected in for all k > 0. e Φ/2 L 1 (). x Φ(x) c 1 x Φ(x) c 2 Φ(x) for x with sufficiently large magnitude.

12 Weak Formulation I Carrillo et al. (2010) established the existence of renormalized weak solutions in the following sense: Assume that Φ, satisfy the confinement hypotheses. Then {ϱ, u, η} represent a renormalized weak solution to (7)-(10) if and only if ϱ 0 in L (0, T ; L γ ()) represents a renormalized solution of (7) on (0, ), i.e., for any test function φ D([0, T ) ), T > 0 and any b, B such that b L ([0, )) C([0, )), B(ϱ) := B(1) + the renormalized continuity equation T 0 holds. ϱ 1 b(z) z 2 dz, B(ϱ) t φ+b(ϱ)u x φ b(ϱ)φ div x u dx dt = B(ϱ 0 )φ(0, ) dx (15)

13 Weak Formulation II The balance of momentum holds in the sense of distributions, i.e., for any w D([0, T ); D(; R 3 )), T 0 = 0 ϱu t w + ϱu u : x w + (p F (ϱ) + η) div x w dx dt µ x u x w + λ div x u div x w (βϱ + η) x Φ w dx dt m 0 w(0, ) dx (16) All quantities are required to be integrable, so in particular, u L 2 (0, T ; W 1,2 (; R 3 )), thus the velocity field can be required to vanish on in the sense of traces. η 0 is a weak solution of (9), i.e., T η t φ+ηu x φ η x Φ x φ x η x φ dx dt = η 0 φ(0, ) dx 0 (17) Again, terms in this equation must be integrable on (0, T ), so in particular η L 2 (0, T ; L 3 ()) L 1 (0, T ; W 1, 3 2 ()).

14 Weak Existence The existence result of Carrillo et al. is established by implementing a time-discretization approximation supplemented with an artificial pressure approximation. Their paper also handles the case of unbounded domains and proves the convergence to a steady-state solution as t.

15 Weakly Dissipative Solutions I Next, we define a new version of solution. Definition (Weakly Dissipative Solutions) {ϱ, u, η} are called a weakly dissipative solution to the NSS system if and only if {ϱ, u, η} form a renormalized weak solution with the energy inequality 1 2 ϱ u 2 + a γ 1 ϱγ + η ln η + ηφ dx(τ) τ + S( x u) : x u + 2 x η + η x Φ 2 dx dt ϱ 0 u a γ 1 ϱγ 0 + η 0 ln η 0 + η 0 Φ dx τ β ϱu x Φ dx dt (18) for all τ. 0

16 Weakly Dissipative Solutions II Definition (Weakly Dissipative Solutions) for all suitably smooth solutions {r, U, s} of the NSS system, the following relative entropy inequality holds for all τ. 1 2 ϱ u U 2 + E F (ϱ, r) + E P (η, s) dx(τ) τ + [S( x u) S( x U)] : x (u U) dx dt ϱ 0 u 0 U E F (ϱ 0, r 0 ) + E P (η 0, s 0 ) dx + τ 0 R(ϱ, u, η, r, U, s) dt (19)

17 Remainder Term The remainder term in (19) has the form R(ϱ, u, η, r, U, s) := div x (S( x U)) (U u) dx ϱ( t U + u x U) (u U) dx t P F (r)(ϱ r) + x P F (r) (ϱu ru) dx [ϱ(p F (ϱ) P F (r)) E F (ϱ, r)] div x U dx t P P (s)(η s) + x P P (s) (ηu su) dx [η(p P (η) P P (s)) E P (η, s)] div x U dx x (P P (η) P P (s)) ( x η + η x Φ) dx [ (βϱ + η) x Φ + η ] xs (u U) dx s

18 Here we define H F (ϱ) := a γ 1 ϱγ P F (ϱ) := H F (ϱ) = aγ γ 1 ϱγ 1 E F (ϱ, r) := H F (ϱ) H F (r)(ϱ r) H F (r) H P (η) := η ln η P P (η) := H P(η) = ln η + 1 E P (η, s) := H P (η) H P(s)(η s) H P (s)

19 Approximation Scheme A three-level approximation scheme is employed Artificial pressure parameterized by small δ Vanishing viscosity parameterized by small ε Faedo-Galerkin approximation where test functions for the momentum equation are taken from n-dimensional function spaces X n of smooth functions on

20 Approximate System t ϱ n + div x (ϱ n u n ) = ε x ϱ n (20) t η n + div x (η n u n η n x Φ) = x η n (21) t (ϱ n u n ) w dx = ϱ n u n u n : x w + (aϱ γ n + η n + δϱ α n ) div x w dx S( x u n ) : x w + ε x ϱ n x u n w dx (βϱ n + η n ) x Φ w dx with the additional conditions x ϱ n n = 0 on (0, T ) u n = ( x η n + η n x Φ) n = 0 on (0, T ) (22)

21 Existence of Approximate Solutions Existence of u n is obtained from the Faedo-Galerkin approximation and an iteration argument. ϱ n, η n obtained from u n using fixed point arguments in the spirit of Ladyzhenskaya.

22 Approximate Energy Inequality Using u n as a test function in (22) and some straight-forward manipulations: 1 2 ϱ n u n 2 + a γ 1 ϱγ n + δ α 1 ϱα n + η n ln η n + η n Φ dx(τ) τ + S( x u n ) : x u n + 2 x ηn + η n x Φ 2 dx dt 0 + ε β τ 0 τ x ϱ n 2 (aγϱ γ 2 n + δaϱ α 2 n ) dx dt 1 2 ϱ 0,δ u 0,δ 2 + a γ 1 ϱγ 0,δ + δ α 1 ϱα 0,δ + η 0,δ ln η 0,δ + η 0,δ Φ dx ϱ n u n x Φ dx dt (23) 0

23 Uniform Bounds From the energy inequality, we find that {u} n,ε,δ b L 2 (0, T ; W 1,2 0 (; R 3 )) { ϱu} n,ε,δ b L (0, T ; L 2 (; R 3 )) {ϱ} n,ε,δ b L (0, T ; L γ ()) {η ln η} n,ε,δ b L (0, T ; L 1 ()) { x η}n,ε,δ b L 2 (0, T ; L 2 (; R 3 )) {η} n,ε,δ b L 2 (0, T ; W 1, 3 2 ())

24 Faedo-Galerkin Limit I From the approximate energy balance, the term T εδ x ϱ n 2 ϱ α 2 n dx dt 0 is bounded independently of n. Thus by Poincaré s inequality, {ϱ} n b L 2 (0, T ; W 1,2 ()). From this, x ϱ n u n b L 1 (0, T ; L 3/2 ()). To get higher time integrability, multiply (20) by G (ϱ n ) where G(ϱ n ) := ϱ n ln ϱ n. Then T x ϱ n 2 ε 0 ϱ n dx dt is bounded independently of n. Using Hölder s and interpolation, { x ϱ n u n } n b L q (0, T ; L p ()) for some p ( 1, 3 2) and q (1, 2). Thus, ϱε, u ε obey t ϱ ε + div x (ϱ ε u ε ) = ε x ϱ ε.

25 Faedo-Galerkin Limit II Strong convergence of x ϱ n x ϱ ε follows from letting G(z) = z 2. Similar techniques show convergence of η n η ε and x η n x η ε to allow t η ε + div x (η ε u ε η ε x Φ) = x η ε. Terms in the momentum equation converge as we want using the bounds and the above convergences, except for the convective term. Convergence of the convective term ϱ n u n u n in L q ((0, T ) ; R 3 ) follows from convergence of ϱ n u n and Arzela-Ascoli. The following lemma is of use throughout the analysis for convergence of the η terms: Lemma (Simon) Let X B Y be Banach spaces with X B compactly. Then, for 1 p <, {v : v L p (0, T ; X ), v t L 1 (0, T ; Y )} is compactly embedded in L p (0, T ; B). Thus, {η} n,ε η δ in L 2 (0, T ; L 3 ()).

26 Vanishing Viscosity Approximation I t ϱ ε + div x (ϱ ε u ε ) = ε x ϱ ε (24) t η ε + div x (η ε u ε η ε x Φ) = x η ε (25) t (ϱ ε u ε ) w dx = ϱ ε u ε u ε : x w + (aϱ γ ε + η ε + δϱ α ε ) div x w dx S( x u ε ) : x w + ε x ϱ ε x u ε w dx (βϱ ε + η ε ) x Φ w dx x ϱ ε n = 0 u ε = ( x η ε + η ε x Φ) n = 0 (26)

27 Vanishing Viscosity Approximation II 1 2 ϱ ε u ε 2 + a γ 1 ϱγ ε + δ α 1 ϱα ε + η ε ln η ε + η ε Φ dx(τ) τ + S( x u ε ) : x u ε + 2 x ηε + η ε x Φ 2 dx dt 0 τ + ε x ϱ ε 2 (aγϱ γ 2 ε + δaϱ α 2 ε ) dx dt ϱ 0,δ u 0,δ 2 + a γ 1 ϱγ 0,δ + δ α 1 ϱα 0,δ + η 0,δ ln η 0,δ + η 0,δ Φ dx τ β ϱ ε u ε x Φ dx dt (27) 0

28 Vanishing Viscosity Limit I We begin by using the uniform bounds and obtaining weak limits ϱ δ, u δ, η δ. We show that since ε x ϱ ε 0 in L 2 (0, T ; L 2 (; R 3 )), and using Arzela-Ascoli, we have that ϱ δ, u δ solve the continuity equation weakly. Similar analysis shows that η δ, u δ solve the Smoluchowski equation weakly.

29 Vanishing Viscosity Limit II Convergence of the momentum equation is fairly straight-forward except for the pressure-related terms. Using the Bogovskii operator (analogous to an inverse divergence operator) and an appropriate test function, we find that aϱ γ ε + η ε + δϱ α ε has a weak limit. To show this weak limit is aϱ γ δ + η δ + δϱ α δ, we have to show the strong convergence of the fluid density (strong convergence of the particle density follows from the lemma of Simon). This is obtained by using the test function ψ(t)ζ(x)ϕ 1 (x) where ψ Cc (0, T ), ζ Cc (), ϕ 1 (x) := x 1 x (1 ϱ ε ), and analysis involving the double Reisz transform and the Div-Curl Lemma.

30 Artificial Pressure Approximation I T 0 T = 0 T 0 ϱ δ B(ϱ δ )( t φ + u δ x φ) dx dt + ϱ 0,δ B(ϱ 0,δ )φ(0, ) dx b(ϱ δ ) div x u δ φ dx dt (28) η δ t φ + (η δ u δ η δ x Φ x η δ ) x φ dx dt = η 0,δ φ(0, ) dx (29) t (ϱ δ u δ )w dx = ϱ δ u δ u δ : x w + (aϱ γ δ + η δ + δϱ α δ ) div x w dx S( x u δ ) : x w dx (βϱ δ + η δ ) x Φ w dx (30)

31 Artificial Pressure Approximation II 1 2 ϱ δ u δ 2 + a γ 1 ϱγ δ + δ α 1 ϱα δ + η δ ln η δ + η δ Φ dx(τ) τ + S( x u δ ) : x u δ + 2 x ηδ + η δ x Φ 2 dx dt ϱ 0,δ u 0,δ 2 + a γ 1 ϱγ 0,δ + δ α 1 ϱα 0,δ + η 0,δ ln η 0,δ + η 0,δ Φ dx τ β ϱ δ u δ x Φ dx dt (31) 0

32 Artificial Pressure Limit Again, from uniform bounds, we are able to obtain the existence of weak limits ϱ, u, η. Much of the difficulty in taking the artificial pressure limit is controlling the oscillation defect measure for the fluid density ϱ.

33 Oscillation Defect Measure and Strong Convergence Definition Let Q and q 1. Then osc q [ϱ δ ϱ](q) := sup k 1 ( lim sup δ 0 + Q ) T k (ϱ δ ) T k (ϱ) q dx. Here, {T k } is a family of appropriately concave cutoff functions. Using these cutoff functions, we can control the oscillation defect measure and obtain strong convergence of the fluid density.

34 Approximate Relative Entropy Inequality We formulate an approximate relative entropy inequality for each fixed n, ε, δ. We define smooth functions U m C 1 ([0, T ]; X m ) zero on the boundary and positive r m, s m on [0, T ]. We take u n U m as a test function on the Faedo-Galerkin approximate momentum equation and perform some calculations to obtain an approximate relative entropy inequality. We take the limits to obtain the relative entropy inequality.

35 Relative Entropy Inequality Regularity of r, U, s are imposed to ensure that all integrals in the formula for the relative entropy are defined. r C weak ([0, T ]; L γ ()) U C weak ([0, T ]; L 2γ/γ 1 (; R 3 )) x U L 2 (0, T ; L 2 (; R 3 3 )), U = 0 s C weak ([0, T ]; L 1 ()) L 1 (0, T ; L 6γ/γ 3 ()) t U L 1 (0, T ; L 2γ/γ 1 (; R 3 )) L 2 (0, T ; L 6γ/5γ 6 (; R 3 )) 2 xu L 1 (0, T ; L 2γ/γ+1 (; R )) L 2 (0, T ; L 6γ/5γ 6 (; R )) t P F (r) L 1 (0, T ; L γ/γ 1 ()) x P F (r) L 1 (0, T ; L 2γ/γ 1 (; R 3 )) L 2 (0, T ; L 6γ/5γ 6 (; R 3 )) t P P (s) L 1 (0, T ; L ()) L (0, T ; L 3/2 ()) x P P (s) L (0, T ; L 3 (; R 3 )) x s L (0, T ; L 2 (; R 3 )) L 2 (0, T ; L 6γ/5γ+3 (; R 3 )). (32)

36 Uniqueness of Weakly Dissipative Solutions Theorem (Weak-Strong Uniqueness) Assume {ϱ, u, η} is a weakly dissipative solution of the NSS system. Assume that {r, U, s} is a smooth solution of the NSS system with appropriate regularity with the same initial data. Then {ϱ, u, η} and {r, U, s} are identical. Note that the following hypotheses are imposed on the smooth solutions where x r L 2 (0, T ; L q (; R 3 )) 2 xu L 2 (0, T ; L q (; R )) α := x s + s x Φ L 2 (0, T ; L q (; R 3 )) (33) { } 3 q > max 3, γ 1 The proof involves analysis bounding the remainder terms in terms of the relative entropy and using Gronwall s inequality on the result.

37 Remarks The result can be generalized to unbounded spatial domains by creating a sequence of bounded domains and passing the limits through using the confinement hypotheses. This result does not show the existence of appropriately smooth {r, U, s}, which is the focus of other work.

38 Low Stratification t ϱ ε + div x (ϱ ε u ε ) = 0 (34) ε 2 [ t (ϱ ε u ε ) + div x (ϱ ε u ε u ε )] + x (aϱ γ ε + D ) ζ η ε = ε 2 (µ x u ε + λ x div x u ε ) ε(βϱ ε + η ε ) x Φ (35) t η ε + div x (η ε u ε ) ε div x (ζη ε x Φ) D x η ε = 0 (36) d ε 2 dt 2 ϱ ε u ε 2 + a γ 1 ϱγ ε + Dη ε ln η ε + ε(βϱ ε + η ε )Φ dx ζ + D 2 xη ε 2 + 2εD x η ε x Φ + ε 2 ζη ε x Φ 2 dx ζη ε + ε 2 S( x u ε ) : x u ε dx 0 (37)

39 Formal Evaluation of the Low Stratification Low Mach Number Limit Assume the following expansions: ϱ ε = ϱ + η ε = η + u ε = u + i=1 i=1 i=1 ε i ϱ (i) ε ε i η (i) ε ε i u (i) ε By considering the energy inequality, x η = 0, so η = 1 η 0(x) dx. By ( equating terms ) of order 1 in the momentum equation, x aϱ γ + D ζ η = 0, implying ϱ = 1 ϱ 0(x) dx. Thus, u satisfies the incompressibility condition div x u = 0.

40 Low Stratification Limit η = 1 ϱ = 1 η 0 (x)dx (38) ϱ 0 (x)dx (39) div x u = 0 (40) ϱ[ t u + div x (u u)] + x Π = µ x u (βr + θ) x Φ (41) where r, θ satisfy x ( ar γ + D ζ θ ) = (βϱ + η) x Φ

41 Low Stratification System Weak Formulation I {ϱ ε, u ε, η ε } form a weak solution to the scaled low stratification equations if: ϱ ε 0 and u ε form a renormalized solution of the scaled continuity equation, i.e., T 0 = B(ϱ ε ) t ϕ + B(ϱ ε )u ε x ϕ b(ϱ ε ) div x u ε ϕ dx dt B(ϱ 0 )ϕ(0, ) dx (42) where b L C[0, ), B(ϱ) := B(1) + ϱ b(z) 1 z dz. 2 The scaled momentum balance holds in the sense of distributions: T ( ε 2 (ϱ ε u ε t v + ϱ ε u ε u ε : x v) + p F (ϱ ε ) + D ) ζ η ε div x v dx dt 0 T = 0 ε 2 ε 2 (µ x u ε x v + λ div x u ε div x v) ε(βϱ ε + η ε ) x Φ v dx dt m 0 v(0, ) dx (43)

42 Low Stratification System Weak Formulation II η ε 0 is a weak solution of the scaled Smoluchowski equation: T 0 = η ε t ϕ + η ε u ε x ϕ ζη ε x Φ x ϕ D x η ε x ϕ dx dt η 0 ϕ(0, ) dx The energy inequality is satisfied: ε 2 2 ϱ ε u ε 2 + a γ 1 ϱγ ε + D ζ η ε ln η ε + ε(βϱ ε + η ε )Φ dx(t ) ε 2 (µ x u ε 2 + λ div x u ε 2 ) dx dt T T 0 2 D x ηε + ε ζη ε x Φ ζ 2 dx dt (44) ε 2 2 ϱ 0 u a γ 1 ϱγ 0 + D ζ η 0 ln η 0 + ε(βϱ 0 + η 0 )Φ dx (45)

43 Target System Definition (Low Stratification Target System) We say that {u, r, s} solve the low stratification target system if = T 0 T 0 div x u = 0 weakly on (0, T ), ϱu t v + ϱu u : x v dx dt (µ x u (βr + s) x Φ) v dx dt for any divergence-free test function v and r = 1 [(βϱ aγϱ γ 1 + η)φ + Dζ ] s weakly. ϱu v(0, ) dx,

44 Main Result I Theorem (Low Stratification Limit) Let (, Φ) satisfy the confinement hypothesis and for each ε > 0, {ϱ ε, u ε, η ε } solves (42)-(45). Assume the initial data can be expressed as ϱ ε (0, ) = ϱ ε,0 = ϱ + εϱ (1) ε,0, u ε(0, ) = u ε,0, and η ε (0, ) = η ε,0 = η + εη (1) ε,0. where ϱ, η are the spatially uniform densities on. Assume also that as ε 0, ϱ (1) ε,0 ϱ(1) 0, u ε,0 u 0, η (1) ε,0 η(1) 0 weakly- in L () or L (; R 3 ). Then up to a subsequence and letting q := min{γ, 2}, ϱ ε ϱ in C([0, T ]; L 1 ()) L (0, T ; L q ()) η ε η in L 2 (0, T ; L 2 ()) u ε u weakly in L 2 (0, T ; W 1,2 (; R 3 ))

45 Main Result II and ϱ (1) ε η (1) ε = ϱ ε ϱ ε = η ε η ε ϱ (1) weakly- in L (0, T ; L q ()) η (1) weakly in L 2 (0, T ; L 2 ()) where {u, ϱ (1), η (1) } solve the target system mentioned previously.

46 Free Energy Inequality Recasting the energy inequality using the free energy E F (ϱ) + E P (η) := a γ 1 ϱγ (ϱ ϱ) aγ γ 1 ϱγ 1 a γ 1 ϱγ + D ζ η ln η D ζ (η η)(ln η + 1) D η ln η, ζ we obtain 1 2 ϱ ε u ε ε 2 (E F (ϱ ε ) + E P (η ε )) + 1 ε (βϱ ε + η ε )Φ dx(t ) T + µ x u ε 2 + λ div x u ε D x ηε 0 ε 2 + ε 2 ζη ε x Φ ζ dx dt 1 2 ϱ 0 u ε 2 (E F (ϱ 0 ) + E P (η 0 )) + 1 ε (βϱ 0 + η 0 )Φ dx (46)

47 Momentum Equation By using the uniform bounds and Sobolev embeddings, ϱ ε u ε u ε converges to a limit ϱu u. Thus, the momentum equation converges to becomes T ϱu t v + ϱu u : x v dx dt = T 0 0 µ x u : x v (βϱ (1) + η (1) ) x Φ v dx dt By dividing (43) by ε and taking ε 0 +, we have weakly ϱ (1) = 1 [(βϱ aγϱ γ 1 + η)φ + Dζ ] η(1) ϱ 0 u 0 v dx

48 Helmholtz Decomposition Consider a vector v R 3. We can decompose the vector into a gradient part H [v] := x 1 x div x v and a divergence-free part H[v] := v H [v] Note that the Helmholtz projections are continuous and linear.

49 Convective Term I We decompose the tensor ϱ ε u ε u ε using the Helmholtz projections into ϱ ε u ε u ε = H[ϱ ε u ε ] u ε + H [ϱ ε u ε ] H[u ε ] + H [ϱ ε u ε ] H [u ε ] Using the properities of the Helmholtz projections and the convergence results earlier, H[ϱ ε u ε ] H[ϱu] = ϱu in C weak ([0, T ]; L 2q/q+1 (; R 3 )), and H[u ε ] u in L 2 (0, T ; L 2 (; R 3 )), so H[ϱ ε u ε ] u ε ϱu u H [ϱ ε u ε ] H[u ε ] 0 weakly in L 2 (0, T ; L 6q/4q+3 (; R 3 3 )).

50 Convective Term II After some manipulations, the scaled NSS system becomes T 0 T 0 T = where 0 εr ε t φ + V ε x φ dx dt = εv ε t v + ωr ε div x v dx dt T 0 h 2 ε x φ dx dt (47) [β(ϱ ϱ ε ) + (η η ε )] x Φ v + h 1 ε : x v h 3 ε div x v dx dt V ε := ϱ ε u ε r ε := ϱ (1) D (βϱ + η)φ ε + aγϱ γ 1 ζ η(1) ε + aγϱ γ 1 ω := aγϱ γ 1 and h 1 ε, h 2 ε, and h 3 ε are quantities converging to zero. (48)

51 Convective Term III In light of (47)-(48), we consider the eigenvalue problem x q = Λq x q n = 0 Λ = λ2 ω with a countable system of eigenvalues 0 = Λ 0 < Λ 1 Λ 2 Λ 3... and eigenvectors {q n } n=0.

52 Decomposition of L 2 (; R 3 ) Defining ω w ±n := ±i x q n Λ n where q n, Λ n are defined from the previous eigenvalue problem. Thus, we decompose the space L 2 (; R 3 ) = L 2 σ(; R 3 ) L 2 g (; R 3 ) where { { } } i L 2 g (; R 3 ) := closure L 2 span ω w n n=1 L 2 σ(; R 3 ) := closure L 2{v Cc (; R 3 ) div x v = 0} and define the projection P M : L 2 (; R 3 ) span { i ω w n } n M Note that we define H M := P M H = H P M since the operators H and P M commute.

53 Return to the Singular Term Rewriting the singular term and noting convergences, the problem of showing the singular term converges weakly to a gradient reduces to showing T H M[ϱ ε u ε ] H M[ϱ ε u ε ] : x v dx dt 0 0 for each fixed M N as ε 0.

54 Concluding Remarks The mechanical relative entropy for the NSS system can be used to obtain a weak-strong uniqueness result by finding the relative entropy between a weakly-dissipative solution and a smooth solution. A modification of the mechanical relative entropy between a weak solution and a solution to a given target system is used to find uniform bounds to show the convergence of the weak solutions to the target system as the Mach number becomes small. Current work is investigating the use of the relative entropy to show the existence of measure-valued solutions to a corresponding model for inviscid fluids.

55 Acknowledgments Thanks for your attention. Research described in this talk was supported by NSF grants DMS and DMS

56 References 1) J. Ballew and K. Trivisa, Suitable weak solutions and low stratification singular limit for a fluid particle interaction model. Quart. Appl. Math. 70: , ) J. Ballew and K. Trivisa, Weakly dissipative solutions and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system. Nonlinear Analysis Series A: Theory, Methods & Applications. 91:1-19, ) J. A. Carrillo, T. Karper, and K. Trivisa. On the dynamics of a fluid-particle interaction model: The bubbling regime. Nonlinear Analysis, 74: , ) G. -Q. Chen and M. Perepelitsa. Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow. Comm. Pure Appl. Math. Vol. LXIII: , ) C. M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, 3rd ed. Springer, Berlin, 2010.