NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS

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1 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS MISHA PEREPELITSA 1. ABSTRACT In those notes we will review the theory of weak solutions of the Navier-Stokes equations for compressible flows in near equilibrium flow regime. The study of solutions of this type was initiated and developed by D. Hoff with the motivation to study the dynamics of the interfaces of discontinuity of solutions. The approach utilizes the Lagrangian structure of the equations inside the Eulerian framework, which allows for sharp characterization of the regularity properties of the solutions. An example of this is cancellation of singularities in the viscous pressure, the observation that was crucial in the recent developments of the mathematical theory of the Navier-Stokes equations. In this notes we demonstrate three results. The first is the basic existence of weak solutions with density ρ L. The second result, improves the regularity of such weak solutions, in the case that density has a jump discontinuity across a C 1+α interface. We show that the flow transports the interface without breaking its regularity. In the last, we study a problem of a dynamics of a density discontinuity interface with a corner singularity. We show that solutions in the Hoff s regularity class are well suited for the analysis of the problem and we demonstrate that interface instantaneously changes its geometry. 2. EQUATIONS The Navier-Stokes equations express the conservation of mass and the balance of momentum: { ρt + div(ρu) = (N.-S.) (ρu j ) t + div(ρu j u) = µ u j + λ divu x j P(ρ) x j, j = 1,..,n, where t is time, x open subset of R n, n = 2,3, ρ and u = (u 1,..,u n ) are the unknown functions of x and t representing density and velocity, P = κρ γ, γ 1, is the isentropic pressure, µ and λ are viscosity constants, such that ( ) 2 µ >, n 1 µ + λ, and div and are the usual spatial divergence and Laplace operators. The system (N.-S.) is solved subject to initial conditions (2.1) (ρ(,),u(,)) = (ρ,u ) Date: May 18,

2 2 MISHA PEREPELITSA and one of the following boundary conditions. Flows in R n : = R n, ρ and (2.2) ρ(x,t) ρ, u(x,t), as x +. No-slip boundary conditions: for x (2.3) u(x,t) =. Navier boundary conditions: for x and n = n(x) the external normal vector, (2.4) u n =, (( u + u t )n + Ku) tan =, K, where v tan = v n(v n) is the component of vector v tangent to the boundary. This condition is typically imposed on the common boundary between a fluid flow and a porous media and the positive coefficient K describes the porous material, see for example Beavers-Joseph[1]. The case K = corresponds to a perfectly lubricated boundary, Joseph[29]. We will assume in this notes that K is a constant Short historical overview. The initial-boundary value problem for (N.-S.), and more generally for Navier-Stokes-Fourier equations, has, locally in time, a unique classical or strong solution, as was proved by Nash[3], Itaya[22, 23], Solonnikov[35], Tani[39]. In dimension one the problem is uniquely solvable for the initial data (ρ,u ) H 1 ([a,b]), with strictly positive density. This was proved by Kanel[24], for isentropic flows, and by Kazhikhov-Shelukhin[25] for the Navier- Stokes-Fourier system. Classical solutoins do exists globally in time if they are close to a static equilibrium. This was proved by Masumura-Nishida[31, 32]. Weak solutions with discontinues data in dimension 1 we considered by Hoff-Smoller[21], Hoff[13, 14], Serre[36]. In Hoff-Smoller[21] the authors established that the discontinuities that present initially in the velocity field are being instantaneously smooth out due diffusion, but the discontinuities in the density persist for all times, being propagated by the velocity. An existence theory for solutions with discontinuous (BV) initial data was developed in Hoff[13] and Serre[36] for isentropic flows and in Hoff[14] for the non-isentropic case. The authors prove the global in time existence of weak solutions, with large data, for the corresponding problems. The existence of weak solutions for multi-dimensional problem for (N.-S.) was established in Hoff[15, 17, 16], for isothermal flows near equilibrium and for isentropic flows with sufficiently large γ in P.L.Lions[26]. The approach of P.L.Lions was extended in many different way, most notably for different values of γ by Feireisl-Novotny-Petzeltova[11] and to Navier-Stokes-Fourier system by Feireisl[12]. An extensive exposition of the mathematical theory of compressible Navier-Stokes equations can be found in the monographs P.L.Lions[27], Feireisl[1], Novotny-Straskraba[33]. 3. PROPERTIES OF THE FLOW AT THE INTERFACE OF DISCONTINUITY 3.1. Rankine-Hugoniot conditions. In these lectures we will focus on discontinuous solutions. We start by considering the Rankine-Hugoniot conditions at the surface of the jump discontinuity of a weak solution (ρ,u). The analysis that we present here was done by Hoff[13] for flows in dimension 1 and by Serre[37] for multi-dimensional flows. We mostly follow the presentation from the last reference.

3 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 3 Let S by a smooth hypersurface in R n+1 and n = (n x,n t ) the normal vector with the space and time components n x and n t, with n x = 1. Let O be an open subset of R n+1, such that S divides O into two disjoint open sets O ±. We take a pair (ρ,u) to be the weak solution of equations (N.-S.) in O with a piece-wise smooth structure: ρ C 1,1 x,t (O ± ), u C 2,1 x,t (O ± ). We denote by f (z) = f + (z) f (z), z = (x,t) S, the jump of a function f across S. We assume that S is interface of discontinuity of the density: ρ >, while u C x,t(o) is continuous across S, which we expect due to the diffusivity of the momentum equation. The last condition implies that u (x,t)τ x =, for every τ x R n in the tangent plane to S t = {x : (x,t) S}. Since it is n 1 dimensional, u is rank-1 matrix: there is a R n such that (3.1) u = an t x(= a n x ). The Rankine-Hugoniot condition for the the continuity equation is expressed as (3.2) n t + u n x =, meaning the the velocity (u(z),1) lies in the tangent plane to S at z. This implies that if X t (x,s) is a flow trajectory, (3.3) dx t dt = u(x t,t), (X s (x,s),s) S O, then (X t,t) S for all times while it remains in O. Thus, the interface of jump discontinuities S t is transported by the flow. The right-hand side of the momentum equation in (N.-S.) can be expressed as divs, where S is the Stokes stress tensor: S = µ( u + u t ) + ((λ µ)divu P)I. The Rankine-Hugonoit conditions for the momentum equations are expressed as the continuity of the normal stress at S t : S n x =. Using the expression for the jump of u from (3.1), we obtain an equation µa + µ(a n x )n x + (λ µ)divu P n x =, which implies that a = An x for A = 1 2µ (λ µ)divu P and u = An xn t x. Taking the trace in the last expression we find that (3.4) (λ + µ)divu P =, and (3.5) u = P λ + µ n xn t x. Note, that the last formula shows that u is symmetric, meaning that (3.6) curlu =.

4 4 MISHA PEREPELITSA Condition (3.4) is the first hint to a special role played by the function F = (λ + µ)divu P. It gives the formula for the jump of divu in terms of the jump of P. Since u is continuous across S, which consists of trajectores, the acceleration u = u t + u u is continuous as well, u =. We can use this in the continuity equation at the interface S t to find that (3.7) d dt lnρ + 1 d P =, λ + µ dt = t + u, thanks to the fact that S consists of the trajectories of the flow. For a typical pressure law P = κρ γ, γ 1, P C(inf x,t ρ) lnρ, and from the above equation we deduce (for inf x,t ρ > ) the exponential decay of the discontinuities of ρ : (3.8) lnρ (X t (x,s),t) e C λ+µ (t s) lnρ (x,s). This analysis gives a good picture of the dynamics of a jump discontinuity. In multi dimensions we would also like to know the regularity and other geometric properties of the interface S. This however requires additional analysis and can not be deduced only from the Rankine-Hugoniot conditions. Let us now consider the situation when the interface S intersects with another hypersurface on which some information on (ρ,u) is given a priori, such as the boundary of the flow domain. S t intersects the no-slip boundary. Let x be the intersection point of S t and, and by ñ x we denote the normal to. On, u =, and thus at x, u = bñ t x, for some vector b R n. On the other hand, at the same point x we have conditions (??). It is compatible with (3.5), for the transversal type of contact (ñ x n x ), only if P (x,t) =. S t intersect the Navier boundary of. Under the boundary conditions (2.4) on can see that the jump of the normal stress u ñ x is normal to the boundary (= Añ x ) and this again is not compatible with (3.5) if n x ñ x and P. S t is piece-wise smooth. Let x S t be a point of a jump discontinuity of n x. By approaching x from different directions, we obtain at x two conditions (3.5) with two different values of n x. This is again an overdetermined arrangement. Our conclusion from this analysis is that u must be discontinuous at contact points in a flow domain O + or O. However, this analysis is not sufficient to determine the type of the singularity; it is not clear if u is unbounded or merely L. For that we have to look at the solution near the interface S t Flows with integrable inertia force. Let us fix time t > and assume that the inertia term is integrable with sufficiently high exponent: (3.9) ρ u(,t) L p (),

5 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 5 for some p > n, where u = t u + u u. In section 6 we show that condition (3.9) is verified for a generic weak solution near a static equilibrium. To obtain the information for u, we consider the momentum equations as Lamé equations for u: (3.1) λ divu + µ u = P + ρ u. Let us also assume that u(,t), u(,t) L 2 (), P(ρ(,t)) L (). In section 6 we will show that the solutions with such properties do exist Interior regularity. Given a bounded function ρ and ρ u L p () we d like to obtain the information on u by solving (3.1) with the boundary conditions (2.2), (2.3) or (2.4). We will pay special attention to function divu, since it controls the growth of the density. A standard way to solve (3.1) (and to obtain suitable estimate) is to reduce it to Poisson s equation by solving for divu first. It is particularly easy to do if = R n, or is a domain with the Navier boundary conditions. Setting F = (λ + µ)divu P, and taking divergence and curl of (3.1), we find that (3.11) F = div(ρ u), curlu = curl(ρ u). Since F, curlu L 2 (), by the elliptic regularity and embeddings, we get (3.12) F, curlu L p loc (), F, curlu Cα loc (), α = 1 np 1. This, in particular, shows that divu is a L function for a generic bounded P(ρ), not necessarily with a piece-wise continuous structure. To get the information on other derivatives of u, we re-write equations (3.1) as Poisson s equations (3.13) µ u = µ λ + µ P λ F + ρ u, λ + µ and using Γ (x,y) the fundamental solution of the Laplace s equation in R n write the integral representation for (3.14) u(x,t) = 1 x Γ (x,y)p(y,t)dy + R(x,t), λ + µ sup x y B r (x ) for an arbitrary ball B r (x ) and the error term R(,t) Cloc 1+α (). With P L (), u(,t) is utmost Log-Lipschitz: u(x,t) u(y,t) x y log(1 + x y 1 ) < +. For example, if ρ(,t) is piecewise C α (), R 2, with a jump discontinuity across a piecewise C 1+α curve with a corner singularity at x of the angle θ π/2, than for points x near x, u(x,t) C ln + x x. For θ = π/2 and for interfaces of class C 1+α, on the other hand u(,t) L loc (). These cases are considered in more details in sections 9 and 1. When = R n the regalarity is uniform throughout the domain.

6 6 MISHA PEREPELITSA Boundary regularity: Navier conditions. To consider effects of the boundaries, we will make some simplifying assumptions that will ease the presentation without hurting the generality. We will assume that = R n + the half space and K = in the the boundary conditions (2.4). Let us show that the local regularity (3.12) holds all the way up the boundary in this case. Indeed, in this case function F solves the same Poisson s equation (3.13) in. We mutliptiply the momentum equation (3.1) by the normal n x and use the boundary conditions (2.4) (with K =,) we obtain that (3.15) n F =, on. The elliptic regularity and embeddings imply (3.16) F L p () C(p) ρ u L p (), F C α () C(p)( F L 2 () + ρ u L p ()), with α = 1 np 1. The similar estimate can be obtained for curlu as well. Thus, divu is uniformly bounded, provided that ρ is bounded and the inertia is in L p. The information on other derivatives of u can be obtained, just as before, by solving three Poisson s equations (3.13) for u, with the boundary conditions (2.4). We obtian a representation formulas u i (x,t) = 1 λ + µ yi G(x,y)P(y,t)dy 1 µ G(x, y) ( λ λ + µ yf(y,t) ρ u(y,t) ) dy, where G(x, y), the Green s function for the Laplace s equation with the Neumann conditions for i = 1,..,n 1, and G(x,y) the Green s function for the Dirichlet conditions, for i = n. The second integral is in C 1+α, while the first is less regular; it is only Log-Lipschitz in x Boundary regularity: no-slip conditions. Unlike the two previous cases divu and curlu are in general not bounded at the boundary with the no slip conditions (2.3). This case is a bit more complicated as there is no a priori given values of F on. There is however a way to get around this difficulty. One can obtain the following formulas using the Lichtenstein[28] method, that we expose in section 5. where (λ + µ)divu(x,t) = P(x,t) A(P)(x,t) = µ A(P)(x,t) + R(x,t), (n 1)λ + 2µ div y x G(x,y)P(y,t)dy, G(x,y) is the Green s function for the Laplace s equation with the Dirichlet data on R n + and R(,t), defined in section 5 with f = ρ u, is C 1+α (). The integral A(P) is in general unbounded at the boundary of the domain. For example for = R 2 +, this happens when the pressure (density) is piecewise C α, with an interface of jump discontinuity intersecting the boundary at the angle θ π/2. The infinite compression(expansion) rate is not consistent with our hypothesis that ρ L. Most likely, (however there is no rigorous prove of this) the density ρ(x,t) of a weak solution (ρ,u) with the bounded initial density ρ having the piecewise smooth structure at the boundary becomes unbounded instantaneously. It also indicates that weak solutions with bounded density are more regular at the no slip boundary than in the interior of the flow domain. In this notes

7 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 7 we will show the existence of solutions with the density ρ(x,t) verifying the following boundary regularity property. (3.17) [ρ(,t)] ρ(x,t) ρ(y,t) α := x y α L ( ) <, where y is the reflection of point y through the boundary. We denote the space of such functions with the norm ρ C,α = ρ L + [ρ] α, by C,α. In such setting we will show that F C α () C( ρ C,α () + ρ u L p ()), which will allow us to control pointwise behavior of divu. Whit that, one can consider u as a solution of (3.13), and obtain the integral representations for u and u. 4. SOME FACTS ABOUT SOLUTIONS OF THE POISSON S EQUATION. We let = R n or R n +, n = 2,3. Consider the Poisson s equation (4.1) v = x j g, x, for j {1..n}, with one of the following boundary conditions. If = R n, v(x), as x. If = R n +, then on = {x : x n = }, B(v) =, where B(v) = v, n v, or n v+av, a >, for the no-slip, perfect slip and Navier boundary conditions respectively. Suppose that g L 2 (). We define function v Lloc 1 (), v L2 () to be a weak solution of Poisson s equation with the above boundary conditions if v wdx = x j wgdx for any test function w with w L 2 (), for = R n, or R n + with the perfect slip condition. For = R n + with the no-slip boundary conditions we require in addition that w has zero trace on. For the Navier slip condition, we require v wdx + avwds x = x j wgdx to hold for any test function with w L 2 (). We will record below the standard elliptic regularity estimates. The proofs can be found, for example, in Gildbarg-Trudinger[7]. Lemma 1. There is a unique weak solution v to all of the above problems. If in addition g L p (), then there is C = C(p) such that (4.2) v L p () C g L p (). Proof. The solution can be constructed in the limit of approximate solutions v ε that solve the same problem as v with g ε C (), and such that g ε g in L 2 () L p (). The v ε is expressed through the appropriate Green s function. v ε (x) = y j G(x,y)g ε (y)dy,

8 8 MISHA PEREPELITSA where G(x,y) = Γ (x y), = R n, G(x,y) = Γ (x y) Γ (x y ), G(x,y) = Γ (x y) +Γ (x y ), = R n +, v =, on, = R n +, n v =, on, with Γ (x) being the fundamental solution of the Laplace s equation in R n. y refers to the reflection of y across the boundary. For the Navier condition, G(x,y) = Γ (x y) Γ (x y ) 2a + e as Γ (x y + s)ds, see for example Gildbarg-Trudinger[7]. Writing G(x, y) = Γ (x y) + H(x, y) for an appropriate H, we can obtain the integral representation for the derivatives of v ε : (4.3) xi v ε (x) = p.v. x 2 i x j Γ (x y)g ε (y)dy + c i j g ε (x) x 2 i y j H(x,y)g ε (y)dy, with c i j = x =1 From this representation one obtains estimates xi Γ (x)x j dx. v ε L p () C(p) g ε L p (), by a direct application of the Calderon-Zygmund theory of singular integrals. Passing to the limit one recovers the a (unique) solution verifying the same estimate. For the second derivative estimates we will use the next lemma. Lemma 2. Let v Lloc 1 (), v L2 () be a solution of the problem { v = f, x, (4.4) B(v) =, x, with f L p (), for some p (1,+ ). There is C = C(p), such that (4.5) 2 v L p () C f L p (). The proof uses the Green s representations for the solutions and the estimates of singular integrals as in the previous lemma. We will also make a frequent use of the following embedding theorems, see for example Zimmer[43]. Lemma 3. Let = R n or R n +, n = 2,3. Let u H 1 (). For n = 2, u L p (), p > 2 and there is C = C(p), for which 2 p 2 (4.6) u L p () C u p L 2 () u p L 2 (). For n = 3, u L p (), p [2,6] and there is C = C(p), for which 6 p 3p 6 (4.7) u L p 2p () C u L 2 () u 2p L 2 ().

9 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 9 5. LICHTENSTEIN METHOD FOR LAMÉ EQUATIONS Let us recall a method of Lichtenstein for solving Lamé equations. Consider the Lamé equations λ divu + µ u = g + f, with u = on. Setting F = (λ + µ)divu g, and taking div of the equations we obtain Poinsson s equation F = div f, and if G(x,y) is the Green s function for the domain, then (5.1) F(x) = G(x,y)div y f dy + ny G(x,y)F(y)ds y. One can also write F = 2 1 (xf) 2 1 xdiv f, and ( ) λ λ xf + µu = ( 2(λ + µ) 2(λ + µ) xdiv f + µ λ + µ g + f. This can be solved: [ ] λ 2(λ + µ) xf(x) + µu(x) = λ G(x, y) 2(λ + µ) ydiv µ y f + ( λ + µ yg + f dy λ + ny G(x,y)yF(y)ds y. 2(λ + µ) Take div of this equation and use (5.1) to obtain: nλ + 2µ 2(λ + µ) F(x) + µ λ + µ g(x) = λ 2(λ + µ) λ + 2(λ + µ) + x G(x,y) (y x)div y f dy ny x G(x,y) (y x)f(y)ds y [ ] µ x G(x,y) λ + µ yg(y) + f (y) dy. Restricting x to one obtains an integral equation for F with a weakly singular integral operator (the second integral on the right). In the case = R n +, the equation provides a formula for F for any x, because for all (x,y), Using once again (5.1) we solve for F : (n 1)λ + 2µ F(x) = µ λ + µ λ + µ λ (5.2) + 2(λ + µ) λ + 2(λ + µ) ny x G(x,y) (y x) = ny G(x,y). div y x G(x,y)g(y)dy x G(x,y) (y x)div y f dy y G(x,y) f dy + x G(x,y) f dy. The first integral on the right is a regular integral (not a p.v.) for x. Let us denote by R(x) the sum on integrals involving f. Then if f L p (), for some p (1,+ ) and is such that the

10 1 MISHA PEREPELITSA integrals are well-defined then, there is C = C(λ,µ, p), for which (5.3) R L p () C f L p (). This statement if verified by Calderon-Zygmund theory for singular integral operators, see for example p.289 of Torchinsky[41]. We denote by A(g)(x) = div y x G(x,y)g(y)dy = div y x (Γ (x y ))g(y)dy, where Γ (x) is the fundamental solution of the Laplace s equation on R n, where y is the reflection of y = R n +, across the boundary = {y n = }. We set 1 x1 2 x2 2 π, n = 2, x (5.4) K(x) = 4 Then 3 2π x 2 1 +x2 2 2x2 3 x 5, n = 3. div y x (Γ (x y )) = K(x y ). In both cases they are the kernels of the type K(x) = (x) x n, with smooth, homogeneous of degree (x), that has zero mean on the sphere x = 1. It follows, that for any p (1,+ ), (5.5) A(g) L p () C(p) g L p (). Standard potential estimates can be used to show that for any x y, (5.6) A(g)(x) + A(g)(x) A(g)(y) x y α c (α)( g L α () + g L 2 ()), α (,1). It follows that A(g) can be extended by continuity to as a C α () function with the norm verifying bound (5.6). In what follows we use the same symbol A(g) to denote such an extension. 6. EXISTENCE RESULTS Let ρ be a positive constant, P = P( ρ), and (ρ,u ) be initial data with ρ ρ L () L 2 (), ess inf ρ >,u H 1 (), for the problem with boundary conditions (2.2) or (2.4). For the no-slip conditions, additionally, let ρ C,α (), u H 1 (), for some α > (,1). Denote for the cases (2.2) or (2.4), and for (2.3). In the last case we also assume that C 2 = ρ ρ 2 L + ρ ρ 2 L 2 + u 2 H 1, C 2 = ρ ρ 2 L + ρ ρ 2 L 2 + ([ρ ] α) 2 + u 2 H 1, (6.1) ε = 1 µ max{c (α),(c 4 ) 4 } 2λ + 2µ >

11 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 11 where c (α) and c 4 appear in estimates (5.6) and (5.5). We say that u LogLip, if it is bounded and Log-Lipschitz. To measure it, we use the norm with log + r = log(1 + r 1 ). u LL = u L + sup x y u(x) u(y) x y log + x y, Theorem 1. There are C, θ, depending on ( ρ,λ,µ,κ,γ,n) and α (for the no-slip boundary conditions), such that if the initial data (ρ,u ) verify (6.2) C, then there exists a weak solutions (ρ,u) of the Navier-Stokes equations with the following properties. ρ ρ L (,+,L 2 ()) ess inf (,+ ) ρ >, sup ( ρ ρ L 2 + ρ ρ L + [ρ] α) CC θ, [,+ ) and some q = q(n) > 1. u, u L (,+,L 2 ()), u L q (,T ;LogLip), T >, σ(t)ut L (,T ;L 2 ), T >. With u = u t + u u, (6.3) sup [,+ ) u 2 + u 2 + σ(t) u 2 dx + F = (λ + µ)divu (P P), verifies + u 2 + u 2 + u 2 σ(t)dxdt CC θ. F L (,+ ;L 2 ()), F L q (,T ;L p ()), T >, p p 2 with p > 2, 1 q <, for n = 2, and p (2,6), 1 q < 2p For all β (,1), if n = 2, and if n = 3, 3(p 2) for n = 3. u β,β/(2β+2) C x,t ( [τ,+ ) C(β,τ)Cθ, τ >, u 1/2,1/8 C x,t ( [τ,+ ) C(τ)Cθ, τ >. ρ ρ, u C([,+ );L 2 ()) and for a.a. t >, u(,t) is the weak solution of Poisson s equations (6.4) µ u = µ λ + µ P λ F + ρ u. λ + µ

12 12 MISHA PEREPELITSA For the problem with the no-slip boundary conditions, the velocity is Lipschitz continuous at the boundary: u(x,t) u(y,t) sup L q (,T ), T >, x,y x y and some q > 1. The theorem combines results of Hoff[15, 17], for the problems with boundary conditions (2.2), (2.4) and Perepelitsa[34] for the case of the no-slip boundary conditions. The conditions in the above theorem can be relaxed in several ways while keeping the result essentially the same. The initial velocity u can be from L p with p = 2 n, and some restriction on λ,µ. One can also express the smallness assumption (6.2) only in term of the total energy ρ u 2 + Φ(ρ, ρ)dx, where Φ is the potential energy, see section 7. By making an additional energy estimate one can show that under the same assumptions, curlu has the same regularity as F. The most compete account of this theorem can be found in Hoff[15, 17]. 7. ESTIMATES ON THE INERTIA ρ u. We will assume that in addition to the properties stated in the conditions of theorem 1, ρ ρ,u H 3 (). For such data there is a classical solution (ρ(x,t),u(x,t)) to the Navier-Stokes equations (N.-S.) on [,T ] for some T >, with the regularity: ρ C([,T ];H 3 ()) C 1 ([,T ];H 2 ()), u C([,T ];H 3 ()) C 1 ([,T ];H 1 ()) L 2 (,T ;H 4 ()). Solutions with such properties can be obtained by methods of Matsumura-Nishida[31, 32]. We will use the fact that such local solutions can be continued on any time interval while ρ(x,t), u(x,t) are uniformly bounded. For the classical solution we will obtain estimates showing that under assumptions of the theorem 1 there is an uniform bound on ρ,u independent on the interval of existence, and and the classical solution can be continued for all times. The general case in obtained by approximating initial data ρ,u from the theorem by a sequence of smooth (H 3 ) initial data and passing to the limit of the corresponding smooth solutions. This last step can be proved using the P.L.Lions[26] compactness result and we will omit it from the presentation. In this section by the means of energy estimates we obtain a priori bounds on ρ u in L q (,T ;L p ()) spaces with some (p,q), p > n, q > 1. We introduce the potential energy Φ(ρ, ρ) = ρ ρ ρ P(s) P( ρ) s 2 Let ˇρ = inf ρ, ˆρ = sup ρ. Time interval [,T ] can be restricted further in such a way that (7.1) ρ(x,t) [2 ˇρ, ˆρ/2]. With this property there is C such that ds. C 1 (ρ ρ) 2 Φ(ρ, ρ) C(ρ ρ) 2

13 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 13 In the following estimates C will stand for a generic quantity depending only on the parameters of the problem λ,µ,κ,γ,α,n and ˇρ, ˆρ. The first energy estimate states that (7.2) sup [,t] u 2 + ρ ρ 2 dx + t u 2 dxdt CC, for some C as described above. In the following energy estimates we adopt the shorthand notation of Hoff[15]: u j k = x k u j, u j = ut j + u k u j k, and we always assume the summation over a repeated index. We write the equation of conservation of momentum as (7.3) ρ u λu k k,i µui l,l + P i =. We multiply this equation by u i and integrage over : (7.4) ρ u 2 dx λu k k,i ui + µu i l,l ui dx + P i u i dx =. Consider P i u i dx = P i ut i + P i u k u i k = d P i u i dx P i,t u i + (P i u k ) k u i dt = d (P P)u i i dx + P t u i i + (Pu k ) k u i i + (Pu k i ) k u i dx dt = d (P P)u i i dx + (P ρp )(u i dt i) 2 Pu k i u i k dx. In this estimate, and the subsequent ones, all boundary terms resulting from the iteration by parts vanish, for all three types of the boundary conditions (2.2), (2.3) and (2.4) (with K = ). For the Navier boundary conditions with K >, the boundary terms are either dissipative or of a lower order and the energy estimates still hold with minor modification, see Hoff[17] for a detailed exposition for this case. It follows from the above computation that t P i u i t dxdt sup ( ρ ρ L 2 + u L 2) +C u 2 dxdt. [,t] Consider now Similarly, u i l,l ui dx = u i l,l ui t + u i l,l uk u i k dx = 1 d u 2 u i l 2 dt uk l ui k + ui l uk u i l,k dx = 1 d 1 u dt 2 u 2 u k k ui l uk l ui k dx. u l l,i ui dx = 1 d u l l 2 dt 2 dx + u l l 2 u k k ul l uk i u i k dx.

14 14 MISHA PEREPELITSA Combining all these computations we obtain: t (7.5) u 2 dx + u 2 dxdt CC +C sup [,t] T u 3 dxdt. Next we hit the equation (7.3) with operator t ( ) + div(u ) : ρ( u i t + u k u i k ) λ(ul l,i,t + (ul l,i uk ) k ) µ(u i l,l,t + (ui l,l uk ) k ) (P i,t + (P i u k ) k ) =. We multiply it with u i σ(t), σ = min{1,t}, and integrate over. The first term ρ( u t i + u k u i k ) ui σ(t)dx = d 1 dt 2 ρ u 2 σ(t)dx 1 ρ u 2 σ (t)dx. 2 The crucial term is the pressure; to estimate it we use the continuity equation to substitute the derivative Ṗ with P u l l : (P i,t + (P i u k ) k ) u i σ(t)dx = (P t + (Pu k ) k ) u i iσ(t) Pu k i u i k σ(t)dx = (ρp P)u l l ui iσ(t) Pu k i u i k σ(t)dx. It can be estimated as (P i,t + (P i u k ) k ) u i σ(t)dx 1 8 u 2 σ(t)dx +C u 2 dx. Also, (u i l,l,t + (ui l,l uk ) k u i σ(t)dx = = = (u i l,t + (ui l uk ) k ) u i l ui l uk l ui k (σ(t)dx) (u i l,t + uk u i k,l + ui l uk k ) ui l ui l uk l ui k (σ(t)dx) u 2 σ(t)dx + (u k l ui k ui l uk k ) ui l + ui l uk l ui k (σ(t)dx). Similarly, (u l l,i,t + (ul l,i uk ) k ) u i σ(t)dx = u l l 2 σ(t)dx + u k l ul k ui i u l l uk k ui i u l l uk i u i k (σ(t)dx). Collecting all computations we arrive at t (7.6) u 2 σ(t)dx + u 2 σ(t)dxdt CC +C sup [,T ] T u 4 σ(t)dxdt. 8. CLOSING THE ENERGY ESTIMATES We will need the next lemma to a priori estimates from theorem 1.

15 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 15 Lemma 4. There is 1 = 1 ( ˇρ, ˆρ) such that if T (8.1) σ(t) ρ ρ 4 dxdt C, C 1, then, (8.2) sup t [,T ] for some C = C( ˇρ, ˆρ). u 2 + u 2 σ(t)dx + T Lemma 5. There is 2 = 2 ( ˇρ, ˆρ) such that if for the boundary problems (2.2), (2.4), or for the no-slip conditions (2.3), then T Lemma 6. There is 3 = 3 ( ˇρ, ˆρ) such that if then u 2 + u 2 + u 2 σ(t)dxdt CC, sup ρ ρ L +C 2, t [,T ] sup ( ρ ρ L + [ρ] α) +C 2, t [,T ] σ(t) ρ ρ 4 dxdt C. C 3, ˇρ/2 ρ(x,t) 2 ˆρ, sup [,T ] ρ ρ 2, or, for the problems with the no slip boundary conditions, sup( ρ ρ L + [ρ] α) 2. [,T ] Proof. (Lemma (4)). We give a detailed proof for the case n = 3. Let us consider Poisson s equations (3.13). We will split velocity u = v w, where µ w = ρ u. Note, that from the momentum equation in (N.-S.) and the elliptic regularity of section 4: w L 2 C( u L 2 + ρ ρ L 2). Moreover, by the embedding (4.7) and elliptic regularity w L 4 C w 1/4 L 2 2 w 3/4 L 6 C( u L 2 + ρ ρ L 2) 1/4 u 3/4 L 2. In the similar way from the equation (3.11) (for boundary problems (2.2) and (2.4) with K = ) with the help of the embedding and the elliptic regularity we get F L 4 C( u L 2 + ρ ρ L 2) 1/4 u 3/4 L 2.

16 16 MISHA PEREPELITSA For the no-slip condition we use the representation (5.2) with g = P P and f = ρ u. In this case we get the estimate F L 4 C( u L 2 + ρ ρ L 2) 1/4 u 3/4 L 2 +C ρ ρ L 4, where the extra term comes from estimating A(P P). Using the embedding (4.7) and elliptic regularity for v and the above estimates for w and F, we can write u 3 L 3 u L 2 u 2 L 4 C u L 2( ρ ρ L 4 + F L 4 + w L 4) 2 C u L 2( ρ ρ 2 L 4 + ( u L 2 + ρ ρ L 2) u 3/2 L 2 ) 1 8 u 2 L 2 +C( u 6 L 2 + ρ ρ 2 L 2 u 4 L 2 + σ(t) ρ ρ 4 L 4 + σ(t) 3/2 u 2 L 2 ). Combining this with the estimate (7.5) we obtain: sup τ [,t] u 2 dx + C t ( t u 2 dxdτ t ) σ(τ) ρ ρ 4 L 4 dτ + u 2 L 2 ( u 4 L 2 + u 2 L 2 + σ(t) 3/2 )dτ. From this, the first energy estimate (7.2) and since C controls u L 2, one can see that there is 2 such that if t σ(t) ρ ρ 4 L 4 dτ C and C 2, then (8.3) sup t [,T ] and u 2 + Using the same estimates we also obtain: T u 2 dxdt CC. u L 4 C( ρ ρ L 4 + ( u L 2 + ρ ρ L 2) 1/4 u 3/4 L 2, σ(t) u 4 L 4 Cσ ρ ρ 4 L 4 +CC u 2 L 2 ( u L 2 σ(t)). Combining this (7.6) and (8.3) we obtain: sup τ [,t] u 2 σ(τ)dx + t u 2 σ(τ)dxdτ CC which (use (8.3) one more time) impies: (8.4) sup t [,T ] u 2 σ(t) + under the assumptions of lemma (4). t +C T t ρ ρ 4 L 4 +CC u 2 L 2 ( u L 2 σ(τ))dτ, u 2 σ(t)dxdt CC,

17 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 17 Proof. (Lemma 5) Consider first the cases of = R 3 and R 3 + with conditions (2.4) (K = ). We write the continuity equation using the viscous flux F : (8.5) (ρ ρ) t + u (ρ ρ) + ρ P P λ + µ = ρ λ + µ F, From the estimate F(,t) 4 C( u(,t) L 4 L 2 + ρ(,t) ρ L 2) u(,t) 3, using (8.3) and (8.4) L we obtain: 2 T T F 4 L 4 σ(t)dt CC u 2 L 2 dt CC. 2 We multiply equation (8.5) by 4ρ(ρ ρ) 3 σ(t), integrate in (,T ), using the continuity equation, and estimating the right-hand by Hölder inequalities we obtain: ρ(,t ) ρ(,t ) ρ 4 L 4 σ(t ) + T 1 4ρ 2 (P P) (λ + µ)(ρ ρ) ρ ρ 4 L 4 σ dxdt ρ ρ ρ 4 L 4 dτ +CC 2 C sup ρ ρ 2 L 2 +CC 2 CC. [,T ] The statement of the lemma follows because the ratio (P P)/(ρ ρ) is positive and lower bounded. For the case of the no slip boundary conditions we write the continuity equation using the representation of the divu from (5.2) with g = P P and f = ρ u. ρ t + u ρ + ρ P P λ + µ = 1 µ ρa(p P) ρr, λ + µ 2λ + 2µ The estimates on R are analogous to F above: T R 4 σ(t)dt CC L 4. To obtain an estimate on T σ(t) ρ ρ 4, independent of T, we need make sure that the damping effect of the pressure L 4 µ dominates the possible growth of 2λ+2µ A(P P). This is the place where we need to assume that µ(c 4 ) 4 2λ + 2µ < 1, where c 4 comes from the estimate (5.5) with p = 4. With this, the lemma follows by the same argument as for the other cases. Proof. (Lemma 6) Consider first the cases of = R 3 and R 3 + with conditions (2.4) (K = ). We use again equation (8.5). By integrating it along a trajectory we obtain the statement of the lemma if we show that 1 F L dt CC θ, and T 1 ( F L CC θ 1 )dt CCθ,

18 18 MISHA PEREPELITSA for some θ, θ 1 >. Since F L 4 C u L 4, we verify both inequalities using the estimate: (8.6) F L C( F L 2 + u L 4) C( u L 2 + ρ ρ L 2) + u 1/4 u 3/4 L 2 L 2 CC 1/2 +CC 1/4 ( u L 2 σ(t)) 3/4 σ(t) 1/2 CC 1/2 + u L 2σ(t) +CC 2/5 σ(t) 4/5. Now, we consider the case of the no-slip boundary conditions. It is slightly more involved since we need to work additionally with the semi-norm [ρ] α. Since F = µ 2λ+2µ A(P P) + R, where R is defined in (5.2) with f = ρ u, R verifies the estimate (8.6) as F. The supremum norm of A(P P) is estimated in (5.6). Thus, integrating the continuity equation (8.5) we obtain: (8.7) sup ρ ρ L C sup[ρ] α +CC θ. [,T ] [,T ] To get the estimate on [ρ] ρ(x) ρ(y) α we first notice that it is equivalent to sup x,y x y α, which we denote by the same symbol. Lets take two trajectories X t = X t (x) and Y t = X t (y) with x, y. Using the continuity equation we get (8.8) d dt (logρ Xt Y t ) + 1 λ + µ ρ Xt Y t = 1 λ + µ F Xt Y t. Note, that the integration is along trajectories and the distance between them may change significantly because the maps X t are only Hölder continuous. We will show now that this doesn t happen at the boundary, i.e. there is C(t) such that C 1 (t) x y X t Y t C(t) x y, with x,y, as above. C(t) has the property logc(t) is utmost linear in C t with the coefficient C that can be made arbitrary small by restricting suitably the initial data. Lemma 7. For x and y, u(x,t) u(y,t) x y for some θ >. Moreover, a(t), a(t) = C(C + max{t 1/5,t}([ρ] α +C θ )), e a(t) Xt Y t x y e a(t). Proof. The first estimate of the lemma is obtained by splitting u = v + w, where the singular part, v, solves { µ v = µ λ µ λ+µ (P P) A(P P), x, 2(λ+µ) 2 v =, x and the regular part w : { µ w = λ λ+µ R + ρ u, x w =, x.

19 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 19 w L is estimated exactly as F in (8.6). On the other hand ( ) µ µv(x,t) = y G(x,y) λ + µ (P(y,t) P) λ µ A(P(,t) P)(y) dy + const. 2(λ + µ) 2 A standard potential estimate applies (x, y ): v(x,t) v(y,t) x y ([ρ] α + [A(P P)] α + ρ ρ L 2 + A(P P) L 2) C([ρ] α + ρ ρ L 2). The lemma follows from the above estimates on w and v. We integrate equation (8.8) in time, divide the result by x y α, and using the estimate (5.6) for A(P P) and the estimate on X t Y t from the last lemma to derive sup[ρ] α [ρ ] αe 1 λ+µ t+a(t) +CC θ + c (α)µ [,t] 2λ + 2µ sup [ρ] α [,t] t 1 λ + µ e 1 λ+µ (t τ)+a(t) a(τ) dτ. Under the condition c (α)µ 2λ + 2µ < 1, we can restrict initial data by taking sufficiently small C so that the last estimate and (8.7) provide a priori bounds sup ρ ρ L, sup[ρ] α CC θ. [,T ] [,T ] 9. DYNAMICS OF INTERFACES In this section we construct weak solutions in which density has a piecewise smooth (C α ) structure with a smooth interface (C 1+α ) separating the phases. We will not be concerned here with the boundary effects, so we assume that = R n, and for simplicity of presentation we take n = 2. The solution with such structure is an instance of solutions of two-phase flows, in which other parameters of the problem, such as viscosities, are discontinuous across the same interface. The local in time wellposedness for two phase flows was proved by Tani[4], for the full system of the Navier-Stokes-Fourier equations with the data ρ C 1+α, u, T C 2+α, and interface Γ C 2+α, by considering the linearized problem in Lagrangian coordinates. We will work in Eulerian coordinates and this will have two advantages; the solutions and interface can be significantly less regular than needed for the Lagrangian formulation in [4], and one can trace interface globally in time. ( ) The hypotheses on the initial density are: ρ C α ±, with ± open disjoint sets separated by a C 1+α hypersurface Γ ; = + Γ. On Γ, ρ >. We assume that ρ and u verify all requirements of theorem 1. By theorem 1, there is a weak solution (ρ,u) of the Navier-Stokes equations, which also is the solution of the system (9.1) ρ t + div(ρu) =, µ u = µ λ+µ P + f ρ(x,) = ρ (x).

20 2 MISHA PEREPELITSA with f = λ λ+µ F + ρ u Lq (,T ;L p ()), for some p > 2, q = q(p) > 1. The regularity of this solution is however too low to conclude that the discontinuity interface Γ t = X t (Γ ) C 1+α. To go around this difficulty, we will solve the interface problem for (9.1) first, with f as above, and obtain a more regular solution (ρ 1,u 1 ). Then, using a weak-strong uniqueness for (9.1) we conclude that the weak solution of the Navier-Stokes system coincided with (ρ 1,u 1 ). The reason for using this indirect argument is that the system (9.1) is much simpler to solve than the Navier- Stokes equations. This approach was developed in Hoff[16]. It comes at the price of restricting pressure to be a linear function of ρ; the condition under which uniqueness is proved, see p.142 of [16]. Lemma 8. Let (ρ,u) and (ρ 1,u 1 ) be two solutions of (9.1) on [,T ], with the following properties: for (ρ 1,u 1 ) : ρ 1 ρ L (,T ;L q ()), u 1 L 1 (,T ;L ()) L (,T ;L q ()), u 1 (,t) =, and for (ρ,u) : for some q (1,2). Then a.e. (x,t) (,T ), Proof. Denote F(t) = ρ ρ L (,T ;L q ()), ρ L ( (,T )), u L (,T ;L q ()), u(,t) =, sup q φ L q 1, φ L q 1 (ρ 1,u 1 ) = (ρ,u). (ρ 1 ρ)φ dx φ L q 1, 1 1 q = 2q 1 2 q 1. Let us fix φ L q q 1 with φ L q 1, and solve the transport problem { ψτ + u (9.2) 1 ψ =, ψ(x,t) = φ(x). The solution clearly exists and for all τ [,t] we can estimate it as ψ(,τ) L q 1 C( u 1 L 1 (,T ;L ) ) φ L q 1. We use ψ as a test function in the weak formulation of the equation (ρ 1 ρ) τ + div((ρ 1 ρ)u 1 ) = div(ρ(u u 1 )), to get that for a.e. τ (,t), (9.3) (ρ 1 ρ)(x,t)φ(x)dx = t (u 1 u)ρ ψ dxdτ Note that since (u 1 u) L (,T ;L q ), u 1 u L 2q 2 q, and the integral on the right in (9.5) is well-defined. Define now w as a solution of a vector Poisson s equation µ w = ρ ψ.

21 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 21 For all τ [,t], 2 w(,τ) L q 1 (), w(,τ) L q q 1 (), and there is C = C(q 1 ) such that 2 w(,τ) L q 1 C ψ(,τ) L q 1. Using the fact that 1 is self-adjoint we obtain: t t t (u 1 u)ρ ψ dxdτ = (u 1 u)µ wdxdτ = κ(ρ 1 ρ)divwdxdτ κ t F(τ) divw(,τ) L q 1 dτ C t F(τ)dτ φ L q 1. Thus, for a.e. t (,T ), t F(t) C F(τ),dτ, and F(t) = implying that ρ 1 (x,t) = ρ(x,t) and u 1 (x,t) = u(x,t), as the solutions of the same Poisson s equations. The integrability of ρ ρ with low exponent q, can not be derived directly from theorem 1, because domain is unbounded. This property still can be obtained from the following lemma, see lemma 2.2 of [15]. Lemma 9. Let (ρ,u ) be as in theorem 1. Suppose, that for some β >, (1 + x 2 ) β ( u 2 + ρ ρ 2 )dx < +, The for the weak solution (ρ,u) from theorem 1, for any T >, there is C(T,C ) such that (1 + x 2 ) β ( u 2 + ρ ρ 2 )dx C(T,C ). sup t [,T ] With β > n(2 q) 2q, by Hölder inequality, one obtains the needed integrability ρ(,t) ρ L q (). Let us go back to system (9.1). Notice that because F and ρ u are in L q (,T ;L p ), with a large exponent p > n, their contribution to u is a L q (,T ;C 1+β ), β = 1 2/p, function. The velocity field with such regularity preserves the structure of a C 1+α (α β) interface. Moreover, it also preserves the piecewise Hölder structure of the density. For these reasons, to shorten the presentation, we will drop this term from the equations and focus only on the singular part P. The compete account of this problem is recorded in Hoff[16]. Instead, we consider a model problem: ρ t + div(ρu) =, (9.4) u = ρ, ρ(x,) = ρ (x), u(,t) =, ρ(,t) = ρ. Without loosing much generality we also assume that ρ is piecewise constant: { ρ x + ρ =, ˆρ x,

22 22 MISHA PEREPELITSA Problem (9.4) in 2 dimensions is closely related to the problem of a dynamics of a patch of vorticity ω = u 2 x 1 u 1 x 2, for incompressible Euler equations which can be expressed as ω t + u ω =, (9.5) u = ω, ω(x,) = ω (x) The problem (9.5) wiht ω L 1 () L (), was considered by Yudovich[42] who established its uniquely solvability. Later Chemin[2] showed that a vortex patch that initially has smooth C 1+α boundary, retains this regularity of the boundary for all positive times. His proof is based on a careful potential estimates of u near the boundary of the vortex patch t. The approach applies directly to the problem (9.4) as well. We will sketch this by following an elegant proof of Chemin s result by Bertozzi-Constantin[3]. Let us reformulate problem (9.4). Taking the div of Poisson s equations we find that divu = (ρ ρ) is piecewise constant. Then, we solve the continuity equation, by integrating along the trajectories: { ρ x + ρ(x,t) = t, ˆρ(t) x where ˆρ(t) = ρ + ρ(t), ρ(t) = ( ˆρ ρ)e ρ t. Using this in Poisson s equations we can write (9.6) 1 x y u(x,t) = ρ(t) 2π x y 2, dy, t = X t ( ), t where X t is the flow produced by the velocity field u(x,t). From the formula for divu it also follows that the area of the density patch does not change significantly: (9.7) t = ˆρ. ˆρ(t) It is convenient to work with an interface transported by the flow by introducing a global vector field which is tangent to the interface. This can be done by representing the interface as a level curve of a function. Let φ (x) C 1+α (), with = {φ > } open, bounded and Γ = {φ (x) = }. φ (x) is the vector tangent to Γ at x Γ. The position of the interface Γ t = X t (Γ ) is the same as the zero level set of the function φ(x,t) solution of the transport problem (9.8) φ t + u φ =, with φ(x,) = φ. The equation for φ is obtained by differentiation: (9.9) ( φ) t + u ( φ) = u φ. φ is measured by [ φ(,t)] α Holder seminorm, and inf φ = inf x Γt φ(x,t). We will demonstrate the following theorem. Theorem 2. Let φ (x) be as above. There is a unique solution u(x,t) of (9.6) where t = {x : ρ(x,t) = } and φ(x,t) is the solution of (9.8). Moreover, there is a constant C depending on the t,

23 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 23 initial data only such that sup u(,t) L C u(,) L, [,+ ) sup φ(,t) C α C φ ( ) C α, [,+ ) inf inf φ(,t) [,+ ) C 1 inf φ ( ). Proof. The uniqueness is proved in lemma 8. To show the existence, first, we assume that there a weak solution with u given by (9.6), Γ t = t is simple, closed curve of class C 1+α and we derive a priori estimates on the φ in the norms listed in the theorem. Then, we will construct a sequence of approximate solutions for which the same estimates hold independently of the approximating parameter, and recover the solution of (9.6) with the properties stated in the theorem 2. Let us explain how a priori estimates are obtained. The gradient u can be expressed as singular integral ( ) σ(x y) ρ(t) 1 u(x,t) = ρ(t)p.v. dy + t 2π x y 2 2 χ t (x), 1 with ( x 2 σ(x) = 2 x1 2 ) 2x 1 x 2 2x 1 x 2 x1 2, x2 2 and χ t (x) = 1, x t,, x R 2 \ 1 2, x Γ t. σ(x) is homogeneous of degree, x =1 σ(x)dx =, and σ( x) = σ(x). The last two properties imply that σ(x) has zero mean on half circles. With φ as above, there is integral formula for u(x,t) φ(x,t) = ρ(t)p.v. t t, σ(x y) 2π x y 2 ( φ(x,t) φ(y,t))dy. The argument is based on the following two estimates from Bertozzi-Constantin[3]. Lemma 1. There is C = C( ), such that u(,t) L C ρ(t) Lemma 11. There is C = C( ) for which ( 1 + log + [ φ(,t)] ) α. inf φ(,t) [ u(,t) φ(,t)] α C u(,t) L [ φ(,t)] α By integrating equation (9.9) on trajectories and using estimates of the last two lemmas in a straightforward way one obtains the estimates on φ L, inf φ, [ φ] α as stated in theorem 2. We will construct approximate solutions in the following way. Let η a (x) = η( x /a), a >, be a smooth radial cut-off function: η(s) =, s [, 1], η(s) = 1, s > 2, and η is monotone. Consider a problem (9.1) u a (x,t) = ρ(t) t x y η a ( x y ) 2π x y 2, dy, t = X t ( ),

24 24 MISHA PEREPELITSA where X t is the flow produced by the velocity field u a (x,t). The radial structure of the cut-off is used to prove the next claim. Claim 1. The estimates of lemmas (1) and (11) hold for u a (x,t) with constants independent of a >. Now we establish the existence of approximate solutions. Lemma 12. For every a >, there is a unique solution u a (x,t) of problem (9.1) on [,+ ) and for all T > and k >, u a C k,1 x,t ( [,T ]). Proof. We will use a fixed point argument to prove the existence of solution on a small interval [,T ] first, and then extend this solution for all times. Let T >. Define { } B = u Cx,t 1+α, ( [,T ]). Given two positive numbers M i, i = 1,2, we define { C = u B : sup u L M 1, sup u C α M 2 }. [,T ] [,T ] C is a convex closed set in B. Define a map A : C B, by following rule; for u C, let X t (x) be the corresponding flow and set x y v(x,t) = A(u)(x,t) = ρ(t) η a ( x y ) X t ( ) 2π x y 2, dy. Let us show that when T is small A maps C into C. Let R > be such that some x. It follows that π(r + M 1 t) 2, and there is C = C(a), such that t v(x,t) C(a) ˆρ(t)(R + M 1 t) 2, v(,t) C α C(a)(R + M 1 t) 2. It follows that given M i > C(a) ˆρ(t)R 2, i = 1,2, there is T >, such that v(x,t) M 1, v(,t) C α M 2, t [,T ]. Choose such M i, i = 1,2 and T. Then one easily obtains that sup (1 + x )( v(x,t) + v(x,t) + 2 v(x,t) ) C(a,M 1,M 2,T ), [,T ] and for t v(x,t) = ρ(t) η a (x y) x y Γ t x y 2 u(y,t) n y ds y, where Γ t = X t (Γ ) is C 1+α curve with the bounds depending only (a,m 1,M 2,T ), sup t v(x,t) C(a,M 1,M 2,T ). [,T ] B R (x ), for

25 NEAR EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATION FOR COMPRESSIBLE FLOWS 25 The above estimates on the derivatives, by Ascoli-Arzela theorem, imply compactness of map A. Thus, there is a fixed point u = A(u) solution of (9.1). For this solution we compute its divergence: divu(x,t) = ρ(t) t (η a (x y)) x y 2π x y 2 dy, and estimate T divu(x,t) C(a), with C(a), independent of T. With this estimate, by changing to Lagrangian coordinates, u(x,t) = ρ(t) η a (x X t x X t (y) t (y)) 2π x X t (y) 2 e divu(x s (y),s)ds dy, we see that there is C = C(a) such that u(x,t) C, This means t B R +C(a)t(x ) for all t while the solution exists. The argument can be repeated forward in time to an arbitrary large interval. Uniqueness holds trivially. A sequence (u a (x,t),xa(x)), t where Xa(x) t is the flow corresponding to u a, converges in C loc ( (,+ )) to a pair (u(x,t),xa(x)) t solution of the problem (9.6). 1. EVOLUTION OF INTERFACES WITH CORNER SINGULARITIES In this section we consider a model problem from the previous section, (9.4), which we supply with the piecewise constant initial data { ρ x + (1.1) ρ (x) =, ˇρ x, with ˇρ < ρ. We assume that is open, bounded, symmetric with respect line x 2 =, and that its boundary Γ =, is a continuous closed curve, corner-shaped at a point x = (,) with an angle θ < π 2 ; i.e. there is a continuous map γ(s) = (γ 1(s), γ 2 (s)) : [ L,L] R 2, such that γ() = (,), γ( L) = γ(l), γ 2 (s), s [,L], and γ 2 (s) (1.2) lim s + γ 1 (s) = µ (,1). As we saw in the previous section, if (ρ,u) is the solution of (9.4), the density ρ(x,t) is piecewise constant function, equal ρ in t +, and ( ρ ˇρ)t (1.3) ˇρ(t) = ρ + ( ρ ˇρ)e and the measure of t does not grow in time. With this information, we expect for u(x,t) to be Log-Lipschitz: there is C = C(, ρ, ˇρ) such that u(x,t) u(y,t) C x y log + x y, (x,y) R 4. Because of the corner-like shape of the interface, this estimate can not be improved to Lipschitz continuity of u(t,x) even if Γ is piecewise smooth. With such velocity field one can solve for trajectories X t (x). They are Hölder continuous in x : x y ect X t (x) X t (y) x y e Ct. Such X t ( ) defines a homeomorphism R 2 R 2. Denote t = X t ( ). Xt maps Γ to a continuous curve Γ t = t. Curve Γ t at the best is Hölder continuous and the geometric properties of Γ t might

26 26 MISHA PEREPELITSA change rapidly (discontinuously). Consider the following examples of curves evolving in a Log- Lipschitz velocity field. In the first example we take velocity u(x) = ( x 2 log x, x 1 log x ), and Γ = {x : x 2 =, x 1 }. In polar coordinates the dynamics is described by the system dr dt =, dθ dt = logr. The image of the positive x 1 axis in the flow generated by such velocity is a curve with an infinite spiral at x =, for any t >. The velocity field in this example is highly oscillatory. In the second example we take (1.4) u(x) = ( Cx 1 logx 1, Cx 2 logx 2 ), C >, and Γ is described by x 1 (s,) = s, x 2 (s,) = µs, s >, for some µ >. The velocity u(x,) u(x,) determined by solving Poisson s equations in (9.4) with the density ρ, as described above, is very similar to u(x) in (1.4) and this example will give us a hint at the dynamics of the density patch. The image of Γ by the flow map is given by Γ t = {x 1 (s,t), x 2 (s,t)) = (s e Ct,(µs) ect )} = {x : (x 1, µ ect (x 1 ) e2ct ), x 1 }. Γ t has a cusp singularity at x =, that was developed at any t > from a corner. Let µ (, µ). In the context of incompressible Euler equations, for the problem of dynamics of vortex patches with singularities, Danchin[5, 6] proved that If a vortex patch has C 1+α boundary with an isolated singularity, then it evolves as a C 1+α patch with the isolated singularity transported by the flow. Sufficiently narrow cusps are stable. Moreover, numerical investigation of vortices with corner singularities were performed by Cohen- Danchin[4]. The results suggest that the interface around the singularity point very rapidly forms a cusp (for acute corners) or flattens (for obtuse corners). We will demonstrate this phenomenon for the problem (9.4), solutions of which behave essentially (for this type of flows) as the solutions of the Navier-Stokes equations. For the Navier-Stokes equations, the Euler equations similar result was established in Hoff-Perepelitsa[19, 2]. Theorem 3. There is a weak solution (ρ,u) of (9.4), (1.1) on R 2 [,T ] for some T >, with the following property; if γ(s) is the parametrization of Γ as described above and X t is the flow obtained from the velocity u(x,t), then for any t >, (1.5) lim s + X2 t(γ(s)) X1 t(γ(s)) =. Xt 1 (γ()) The theorem describes the instantaneous collapse of a corner to a (geometric) cusp. Proof. We use time discrete scheme to show the existence of a solution with the required properties. Let T > and n N. We set τ = T /n and t n = nτ. Consider a sequence (u n (x,t),xn(x), t t,n,γt n ) for t ((n 1)τ, nτ] constructed in the following way. For n =, we set Xn(x) t = x, u n (x,t) =,