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

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1 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS MISHA PEREPELITSA ABSTRACT. In these notes we 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 by D. Hoff[13] with the motivation of studying the dynamics of sets of points of discontinuity of solutions. We demonstrate three results. The first is the existence of weak solutions with uniformly bounded density for the typical initial-boundary value problems. The second result improves the regularity of such weak solutions in the case when the density is piecewise Hölder continuous with a jump discontinuity across a C 1+α hypersurface. We show that the flow generated by the velocity u transports the discontinuity surface and preserve its regularity. In the last result, we consider the fluid flows in which the density, at time t =, is piecewise smooth and the set of its points of discontinuity is a surface with a corner-type singularity. We show that solutions in the Hoff s regularity class are well suited for the analysis of this problem and describe the phenomenon of an instantaneous change of the geometry of the interface at the point of the singularity. 1. EQUATIONS The Navier-Stokes equations express the conservation of mass and momentum: { t ρ + div(ρu) =, (N.-S.) t (ρu j ) + div(ρu j u) = µ u j + λ divu x j P(ρ) x j, j = 1,..,n, where (x,t) R +, is an open subset of R n, n = 2,3; ρ and u = (u 1,..,u n ) are the unknown functions of x and t representing the density and the velocity; P = κρ γ, γ 1, κ >, is the isentropic pressure; µ and λ are the viscosity coefficients, verifying ( ) 2 (1.1) µ >, n 1 µ + λ. The system (N.-S.) is solved subject to the initial conditions (1.2) (ρ(,),u(,)) = (ρ ( ),u ( )) and one of the following boundary conditions. Flows in R n : = R n, and for any t, (1.3) ρ(x,t) ρ, u(x,t), as x +. No-slip boundary conditions: for any (x,t) R +, (1.4) u =. Date: June 2,

2 2 MISHA PEREPELITSA Navier boundary conditions: for any (x,t) R + and n = n(x) the unit outer normal, (1.5) 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. The last condition is typically imposed on the common boundary between a fluid flow and a porous media. A nonnegative coefficient K describes the porous material, see for example Beavers-Joseph[1]. The case K = corresponds to a perfectly lubricated boundary, Joseph[24]. We will assume in this notes that K is a constant. 2. PROPERTIES OF THE SOLUTIONS AT THE INTERFACE OF DISCONTINUITY OF THE DENSITY 2.1. Rankine-Hugoniot conditions. In these notes we focus on discontinuous solutions of the Navier-Stokes equations. We start by considering the Rankine-Hugoniot conditions at the surface of the jump discontinuity of a weak solution (ρ,u). The following analysis can be found in Hoff[12] and Serre[3]. [ ] Let S be a smooth hypersurface in R n+1 nx and n = the normal vector to S, with the space n t and time components n x and n t, and n x = 1. Let O be an open subset of R n+1, such that S divides O into two disjoint open sets O ±. Let (ρ,u) be a 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 at z. We assume that on S, ρ > and u =. The later condition holds since we expect u to be continuous, due to the presence of viscosity in the momentum equation. If we denote by S t = {x : (x,t) S} the section of S by a plane t = const., then from the continuity of u we conclude that u τ =, for every τ in the tangent plane to S t at x S t. Since S t is n 1 dimensional, u is a rank-1 matrix: there is a R n such that (2.1) u = an t x, where n t x stands for the transpose of the column vector n x. The Rankine-Hugoniot condition for the balance of mass equation is expressed as (2.2) n t + u n x =, meaning 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, i.e., the solution of the problem (2.3) dx t dt = u(x t,t), (X s (x,s),s) S O, for some fixed s R, then (X t (x,s),t) S for all times t 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.

3 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS 3 Due to (2.2), the Rankine-Hugoniot conditions for the momentum equations are expressed as the continuity of the normal stress: S n x =. Using the expression for the jump of u from (2.1), we obtain the 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 from this we get u = An xn t x. Taking the trace in the last expression we find that (2.4) (λ + µ)divu P =, and the formula for the jump of u : (2.5) u = P λ + µ n xn t x. Note, that the last formula shows that u is symmetric, meaning that (2.6) curlu =. Condition (2.4) is the first hint to a special role played by the function F = (λ + µ)divu P, called the effective viscous flux. It has some extra regularity, compared to the regularity of u and ρ. We use (2.4) in the balance of mass equation, written as t lnρ + u lnρ + divu =, to find that (2.7) lnρ + 1 λ + µ P =, = t + u, due to the fact that S consists of the trajectories of the flow. For a typical pressure law P = κρ γ, γ 1, there are C = C(sup x,t ρ) and c = c(inf x,t ρ) such that c lnρ P C lnρ, and from the above equation we deduce the bounds for the decay of ρ : (2.8) lnρ (x,s)e C λ+µ (t s) lnρ (X t (x,s),t) lnρ (x,s)e c λ+µ (t s). 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, or when S is not smooth (n is discontinuous at some points of S) S t intersects the no-slip boundary of. Let x be a point of intersection of S t and, and ñ x be a normal to. On, u =, and thus at x, u (x,t) = bñ t x, for some vector b R n. On the other hand, at the same point x we have conditions (2.5). It is compatible with (2.5) (for the transversal type of contact ñ x n x ) if and only if (2.9) P (x,t) = S t intersect the Navier boundary of. Under the boundary conditions (1.5) on can see that u ñ x is normal to the boundary (= Añ x ) and this is compatible with (2.5) iff P (x,t) = (when ñ x n x ).

4 4 MISHA PEREPELITSA 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 (2.5) with two different values of n x. This is again an overdetermined arrangement, if P (x,t). This discussion shows that with the geometry of one of the above the type, u is not continuous in O + and O. However, this analysis is insufficient to determine the type of the singularity. For that we have to look at the PDEs near the interface S t ; see section 2.2 below. Note here that, if we insisted on piecewise smoothness of the solutions (this is sometimes assumed in so-called two-phase problems), then we would need to assume the compatibility condition (2.9) on the initial data for the solutions to exist. Otherwise, and this is the approach we take in these notes, we should consider the weak solutions for which the derivatives of u are only integrable functions Flows with integrable inertia force. The structure of the surface S, discussed above, implies also that the jump of the material acceleration u =. Since u is discontinuous at S, with the jump given by (2.5), we expect u to be more regular than the derivatives in (x,t). This statement can be formalized by means of the energy estimates obtained by considering the convective derivative of the momentum equations, see section 4.2. As we will see in that section, the energy estimates and the Sobolev embedding lemmas ensure that for any time t > the inertia term is integrable with sufficiently high exponent: (2.1) ρ u(,t) L p (), for some p > n. To obtain the information on u, we consider the momentum equations as Lamé equations for u: (2.11) λ divu + µ u = P + ρ u. Let us also assume that u(,t), u(,t) L 2 (), P(ρ(,t)) L () Interior regularity. Consider equations (2.11) with the boundary conditions (1.3), (1.4) or (1.5). This problem can be reduced to boundary value problems for the Poisson s equations. Setting F = (λ + µ)divu P, and taking divergence and curl of (2.11), we find that (2.12) F = div(ρ u), curlu = curl(ρ u). Since F, curlu L 2 (), ρ u L p (), p > n, by the elliptic regularity and embeddings, we get (2.13) F, curlu L p loc (), F, curlu Cα loc (), α = 1 np 1. This, in particular, shows that divu(,t) is in L. To get the information on other derivatives of u, we re-write equations (2.11) as the Poisson s equations (2.14) 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 on R n, we can write (2.15) u(x,t) = 1 x Γ (x,y)p(y,t)dy + R(x,t), λ + µ B r (x ) for an arbitrary ball B r (x ) and the remainder term R(,t) Cloc 1+α ().

5 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS 5 From (2.15) we can estimate that with P L (), u(,t) is log-lipschitz continuous in x: sup x y u(x,t) u(y,t) x y log + x y < +, with log + r = log(1 + r 1 ). This estimate is sharp: for example, in dimension 2, if ρ(,t) is piecewise C α with a jump discontinuity across a piecewise C 1+α curve with a corner singularity at x of the angle θ {, π/2, π}, then for points x near x, u(x,t) C log + x x. This regularity can be improved for some special geometries. If in the above example θ = {, π/2, π}, then u(,t) L loc (). These cases are considered in more details in sections 5 and Boundary regularity: Navier conditions (1.5). Let us assume that = R n + the half space, and K = in (1.5). We will show that the local regularity (2.13) holds up the boundary of. Indeed, in this case function F solves the same Poisson s equation (2.14) in. We multiply the momentum equation (2.11) by the normal n x and use the boundary conditions (1.5) (with K = ), to obtain that (2.16) nx F =, on. The elliptic regularity and embeddings imply (2.17) 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 (2.14) for u, with the boundary conditions (1.5). We obtain 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 continuous in x Boundary regularity: no-slip conditions (1.4). Unlike the two previous cases, div u and curlu are not uniformly bounded and may blow up at the boundary of. In this case the theory of the Lamé equations (2.11) produces the following formula for divu. where (λ + µ)divu(x,t) = P(x,t) 2µA(P)(x,t) R(ρ u)(x,t) + (n 1)λ + 2µ (n 1)λ + 2µ, A(P)(x,t) = div y x Γ (x y )P(y,t)dy, Γ (x) is the fundamental solution of the Laplace s equation in R n. In the above formula for divu, R(ρ u)(,t), defined in section 7.3, is C 1+α () and the integral A(P) is unbounded at the boundary

6 6 MISHA PEREPELITSA of the domain. For example for = R 2 +, this happens when the 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 the hypothesis ρ L. One might conjecture that the density ρ(x,t) of a weak solution with such arrangement of the discontinuity becomes unbounded or rarefies to vacuum. To prevent such singular behavior we consider density ρ(,t) to be from the space C,α (), defined in (3.1). This restricts jumps in the values the density ρ (x,t) to be of the order dist(x, ) α. The corresponding velocity u has uniformly bounded divergence and is Lipschitz continuous at the boundary of ; see (3.5). 3. EXISTENCE RESULTS In this section we state and give a proof of a theorem establishing the existence of weak solutions of the Navier-Stokes equations in near equilibrium flow regime. The assumptions on the initial data leading to such solutions depend on the type of the initial-boundary value problem we consider. For the problems with boundary conditions (1.4) or (1.5) we assume = R n +. Moreover, in (1.5), for the simplicity of presentation, we set K =. In what follows, H 1 () (or H 1 ()) is the standard notation for the space of square integrable, together with their first derivatives, functions (with a zero trace in the case H 1()), with the norm u L 2 + u L2. We use the notation u LogLip for the space of bounded and log-lipschitz continuous functions and by C,α we denote the space of bounded functions with the finite norm ρ C,α = ρ L + [ρ] α, (3.1) [ρ] ρ(x) ρ(y) α = x y α L ( ), where y = (y 1,...,y n 1, y n ) is the reflection of point y = (y 1,...,y n 1,y n ) across the boundary = {y n = }, and α ],1[. Let ρ be a positive constant and P = P( ρ). For the initial-boundary value problems (1.3) or (1.5) we assume that the initial data (ρ,u ) verify and denote by ρ ρ L () L 2 (), ess inf ρ >,u H 1 (), C = ρ ρ 2 L + ρ ρ 2 L 2 + u 2 H 1. For the initial-boundary problem with no-slip conditions (1.4) we assume that ρ ρ L () L 2 (), ess inf for some α ],1 n/4[, and denote by In this case we also require that ρ >, ρ C,α (),u H 1 (), C = ρ ρ 2 L + ρ ρ 2 L 2 + ([ρ ] α) 2 + u 2 H 1. (3.2) ε = 1 µ max{c 1(4),C 2 (α)} λ + µ >, where C 1 (4) and C 2 (α) appear in (7.12) (with p = 4) and (7.13), respectively.

7 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS 7 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 (3.3) C, then there exists a weak solutions (ρ,u) of the Navier-Stokes equations with the initial and boundary conditions (1.2) and (1.3) ((1.4) or (1.5)), with the following properties. i. For some q = q(n) > 1 and σ(t) = min{1,t}, and ρ ρ L (,+ ;L 2 () L ()), ess inf ρ >, (,+ ) u, u L (,+ ;L 2 ()), u L q (,T ;LogLip()), T >, σ(t)ut L (,T ;L 2 ()), T >, ess sup ( ρ(,t) ρ L 2 + ρ(,t) ρ L ) CC θ. t [,+ ) For the problem with the no-slip boundary conditions, ρ L (,+ ;C,α ()), and ess sup ( ρ(,t) ρ L 2 + ρ(,t) ρ L + [ρ(,t)] α) CC θ. t [,+ ) ii. In the notation u = u t + u u, (3.4) ess sup t [,+ ) u(,t) 2 + u(,t) 2 + σ(t) u(,t) 2 dx + + u 2 + u 2 + σ(t) u 2 dxdt CC θ. iii. Function F = (λ + µ)divu (P P), verifies 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 iv. For all β ],1[, if n = 2, 3(p 2) for n = 3. u β,β/(2β+2) C x,t ( [τ,+ )) C(β,τ)Cθ, τ >, and if n = 3, u 1/2,1/8 C x,t ( [τ,+ )) C(τ)Cθ, τ >. v. For the problem with the no-slip boundary conditions, the velocity is Lipschitz continuous at the boundary of the domain : u(x,t) u(y,t) (3.5) sup L q (,T ), T >, x,y x y for some q = q(n) > 1.

8 8 MISHA PEREPELITSA vi. Additionally, ρ ρ, u C([,+ );L 2 ()) and for a.a. t >, u(,t) is the weak solution of Poisson s equations (3.6) µ u = µ λ + µ P λ F + ρ u. λ + µ Remark 1. The theorem combines results of Hoff[13, 15], for the problems with boundary conditions (1.3), (1.5) and Perepelitsa[28] for the case of the no-slip boundary conditions. Remark 2. 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 in L p (), with p = 2 n, if an additional restriction on the range of λ,µ is made. Remark 3. One can show that under the same assumptions on the data of the problem, curlu has the same regularity as F : curlu L (,+ ;L 2 ()), curlu L q (,T ;L p ()), T >, p with p > 2, 1 q < p 2, for n = 2, and p (2,6), 1 q < 2p 3(p 2) for n = 3. Remark 4 A combined effect of the diffusion for the values of the velocity and the viscous damping of the density oscillations results a long time asymptotics: as t approaches, the weak solution from theorem 1 converges to the equilibrium. For any q (2,+ ), ) lim ρ(,t) ρ L t ( () + u(,t) L q () =. The corresponding estimates can be found in Hoff[13, 15]. Remark 5. The assumption (3.2), for the problem with the no-slip boundary conditions, ensures that the rate of viscous damping of the density oscillations dominates the rate of compression (and expansion) in the motion due to reflection of the waves from the no-slip boundary; see the equation (4.16) and the following estimates. It also happens that for this problem the rate of compression (and expansion) in the reflected component of the motion is unbounded, unless the density is continuous at the boundary of. For this reason, to measure ρ(,t), we use space C,α instead of L. Remark 6. It is not known if the solutions of theorem 1 are unique. The weak solutions are unique (and depend continuously on the initial data in certain topologies) under additional regularity assumptions ρ ρ L (,T ;H 1 ()), u L 1 (,T ;W 1, ()) L ( (,T )), in the case when = R n, n = 2,3 and P = κρ, or for a general pressure law P = P(ρ), with further regularity assumptions on ρ; see Hoff[16]. The density for the solutions in this uniqueness class might have discontinuities across a hypersurface. Remark 7. The weak solutions of this type for the Navier-Stokes-Fourier equations and the equations the magnetohydrodynamics were considered in Chen-Hoff-Trivisa[5], Hoff-Jenssen[17], Hoff- Suen[2]. 4. PROOF OF THEOREM Outline of the proof. To prove theorem 1 we approximate the data (ρ ρ,u ) in the topology of L 2 () H 1 () with a sequence functions (ρ ε ρ, uε ) H3 () H 3 (), which verify,

9 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS 9 uniformly in ε, the conditions of theorem 1. For H 3 initial data the results of Matsumura- Nishida[26, 27] can be used to prove the following Theorem. Let (ρ ε,uε ) be as described above and uε verifies one of the conditions (1.3), (1.4) or (1.5). There is time T > up to which the corresponding initial-boundary value problem for (N.-S.) with the initial data (ρ ε,uε ) has a unique strong solution. The solution belongs to ρ 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 ()). Moreover, if T is the maximal time of the existence of the strong solution, then sup ( ρ(,t) L + u(,t) L 2) = +. t [,T ) Given a local strong solution (ρ ε,u ε ) we will establish uniform in ε and T estimates of the norms appearing in the statement of theorem 1. It will follow that the local solution can be extended for all times, and then, the weak solution (ρ,u) from theorem 1 will be obtained as a strong (L 2 loc ( ],+ [) limit of the approximate solutions (ρε,u ε ). This step follows from the compactness theory of P.L.Lions[22] or, in the case of the pressure P = κρ, from the compactness result of Hoff[13]. We will omit this step from the presentation Energy estimates. To simplify the notation we abbreviate the local strong solution (ρ ε,u ε ), with the properties described above, as (ρ,u). Denote the potential energy by Φ(ρ, ρ) = ρ ρ ρ P(s) P( ρ) s 2 We set ˇρ = ρ/2, ˆρ = 2 ρ, for the problems with boundary conditions (1.3), (1.5), and for the problem with the no-slip boundary conditions we choose ˇρ = ρ ρ 2 ds. 1 (1 ε) 1/(γ+1) 1 + (1 ε) 1/(γ+1), ˆρ = ρ + ρ 1 (1 ε) 1/(γ+1) (1 ε) 1/(γ+1), where ε is defined in (3.2). With these conditions we have ( ) ˆρ γ+1 (4.1) (1 ε) <. ˇρ Assume that ρ ρ L and T are so small that (4.2) ρ(x,t) ] ˇρ, ˆρ[, (x,t) [,T ]. Function Φ is quadratic in (ρ ρ) for moderate values of ρ : there is C such that C 1 (ρ ρ) 2 Φ(ρ, ρ) C(ρ ρ) 2, ρ ] ˇρ, ˆρ[. 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, ρ) and we adopt shorthand notation: u j k = x k u j, u j = t u j + u k xk u j, assuming the summation over repeated indices.

10 1 MISHA PEREPELITSA The first energy estimate, obtained by multiplying the balance of mass equation by Φ ρ (ρ(x,t), ρ), adding it to the balance of momentum multiplied by u, and integrating states that t (4.3) sup u 2 + ρ ρ 2 dx + u 2 dxdτ CC, [,t] for some C as described above. We write the equation of conservation of momentum as (4.4) ρ u i λu k k,i µui l,l + P i =. We multiply this equation by u i and integrate over : (4.5) ρ u 2 dx λu k k,i ui + µu i l,l ui dx + P i u i dx =. We set P = P( ρ) = κ( ρ) γ, and consider P i u i dx = P i ut i + P i u k u i k dx = d P i u i dx P i,t u i + (P i u k ) k u i dx 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 integration by parts vanish, for all three types of the boundary conditions (1.3), (1.4) and (1.5) (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[15] for a detailed exposition for this case. It follows from the above computation that t P i u i t dxdt C sup ( ρ ρ L 2 + u L 2) +C u 2 dxdt. [,t] Consider now 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 dx u i l 2 dt uk l ui k + ui l uk u i l,k dx = 1 d 1 u 2 dx + 2 dt 2 u 2 u k k ui l uk l ui k dx. Similarly, 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. Combining all these computations we obtain: t t (4.6) u 2 dx + u 2 dxdt CC +C u 3 dxdt, t [,T ]. sup [,t]

11 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS11 Next we apply operator t ( ) + div(u ) to equation (4.4): 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 by 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 2 ρ u 2 σ (t)dx. The crucial term is the pressure; to estimate it we use the balance of mass equation to substitute the derivative Ṗ with ρp ρ 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 ρ P)u l l ui iσ(t) Pu k i u i k σ(t)dx. Then, Also, (P i,t + (P i u k ) k ) u i σ(t)dx 1 8 (u i l,l,t + (ui l,l uk ) k ) u i σ(t)dx = u 2 σ(t)dx +C u 2 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 estimates we arrive at t (4.7) sup u 2 σ(t)dx + u 2 σ(τ)dxdt CC + C [,t] t u 4 σ(τ)dxdτ, t [,T ] Closing the energy estimates. We need the next lemma to obtain estimates of theorem 1. Lemma 1. There are 1,θ,C, depending on ( ρ,λ,µ,κ,γ,n,α) such that if C 1, then, T (4.8) sup u 2 + σ(t) u 2 dx + u 2 + u 2 + σ(t) u 2 dxdt CC, t [,T ] and for the boundary value problems (1.3) and (1.5), (4.9) sup ρ(,t) ρ L CC θ, t [,T ]

12 12 MISHA PEREPELITSA or, for the boundary problem with the no-slip conditions, (4.1) sup ( ρ(,t) ρ L + [ρ(,t)] α) CC θ. t [,T ] Assuming the lemma holds we can conclude the proof of theorem 1. The estimates of this lemma are obtained under assumption (4.2). Selecting = min{ 1, ρ/4}, or, for the problem with the no-slip boundary conditions, { = min 1, ρ } 1 (1 ε) 1/(γ+1) (1 ε) 1/(γ+1), and the initial data such that C, we obtain a priori estimates (4.8) (4.1) that hold on any time interval [,T ] of the existence of the strong solution. It follows from the theorem cited at the beginning of section 4.1 that the solution can be continued for any time T <. The remaining estimates on (ρ,u) that appear in the statement of theorem 1 are obtained from the elliptic regularity of solutions of the corresponding Poisson s equations, concluding the proof of theorem 1. Proof. (Lemma 1) We give a detailed proof for the case n = 3. Let us consider Poisson s equations (2.14). We split velocity u = v w, where µ w = ρ u and v and w satisfy the same boundary conditions as u. Using the momentum equation in (N.-S.) and the elliptic regularity of section 7.1 we obtain (for a fixed t [,T ]): w L 2 C( u L 2 + ρ ρ L 2). Moreover, by the embedding (7.6) 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 (2.12) (for boundary problems (1.3) and (1.5) with K = ) with the help of the embedding and the elliptic regularity we get (4.11) F L 4 C( u L 2 + ρ ρ L 2) 1/4 u 3/4 L 2. For the no-slip boundary conditions we use the representation (7.9) with g = P P and f = ρ u. In this case, using (7.12) 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 (7.6), elliptic regularity for v and the above estimates for w and F, we can write u 3 L 3 C u L 2 u 2 L 4 C u L 2( v L 4 + w L 4) 2 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) 1/2 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) 1/2 u 2 L 2.

13 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS13 Combining this with the estimate (4.6) we obtain: t (4.12) sup u 2 dx + u 2 dxdτ τ [,t] ( t C σ(τ) ρ ρ 4 L 4 dτ + Using similar arguments we obtain: and Combining this with (4.7): (4.13) sup τ [,t] u 2 σ(τ)dx + t ) u 6 L 2 + u 4 L 2 + σ(τ) 1/2 u 2 L 2 dτ. 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(t) ρ ρ 4 L 4 +CC u 2 L 2 ( u L 2 t u 2 σ(τ)dxdτ CC t +C σ(τ) ρ ρ 4 L 4 dτ +CC t σ(t) ). u 2 ) L ( 2 u L 2 σ(τ) dτ. Next, we obtain the estimates on t σ(τ) ρ ρ 4 L 4 dτ. Consider first the problems with (1.3) or (1.5). We write the balance of mass equation using the viscous flux F : (4.14) (ρ ρ) t + u (ρ ρ) + ρ P P λ + µ = ρ λ + µ F, From the estimate (4.11) we obtain: t F 4 t L 4 σ(τ)dτ C sup ( u(,τ) L 2 + ρ(,τ) ρ L 2) sup u(,τ) L 2 σ(τ) τ [,t] τ [,t] u 2 L 2 dτ. We multiply equation (4.14) by 4ρ(ρ ρ) 3 σ(t), integrate over (,t), and estimate the righthand by Hölder inequalities to obtain: ρ(,t) ρ(,t) ρ 4 L 4 σ(t) + 1 t 4ρ 2 (P P) (λ + µ)(ρ ρ) (ρ ρ)4 σ(τ)dxdτ ρ ρ ρ 4 L 4 dτ +C(δ) t σ(τ) F 4 L 4 dτ + δ t σ(τ) ρ ρ 4 L 4 dτ, for any δ > and a suitable C(δ). Since the ratio (P P)/(ρ ρ) is strictly positive we can choose δ small enough to obtain (4.15) t σ(τ) ρ ρ 4 L 4 dτ CC t +C sup ( u(,τ) L 2 + ρ(,τ) ρ L 2) sup u(,τ) L 2 σ(τ) τ [,t] τ [,t] u 2 L 2 dτ. Combining (4.3), (4.12), (4.13) and (4.15) and using the Gronwall type argument we conclude the proof (4.8) for the boundary problems (1.3) and (1.5).

14 14 MISHA PEREPELITSA For the case of the no slip boundary conditions we write the balance of mass equation using the representation of divu from (7.9) with g = P P and f = ρ u : (4.16) ρ t + u ρ + ρ P P λ + µ The estimate on R(ρ u) is analogous to F above: t R(ρ u) 4 L 4 σ(τ)dτ µρa(p P) ρr(ρ u) = (λ + µ) 2 2(λ + µ) 2. t C sup ( u(,τ) L 2 + ρ(,τ) ρ L 2) sup u(,τ) L 2 σ(τ) τ [,t] τ [,t] u 2 L 2 dτ. To obtain an estimates independent of T, we need to make sure that the damping effect of the µ pressure dominates the possible growth of λ+µ A(P P). This is the place where we need to assume that µc 1 (4) λ + µ < 1, where C 1 (4) comes from the estimate (7.12) with p = 4. Using condition (4.1), (4.8) follows by the same argument as for the other boundary problems. Now we will prove (4.9) and (4.1). Consider first the problem with (1.3) or (1.5). We use again equation (4.14). By integrating it along a trajectory we obtain (4.9) using the estimate (4.17) F L C( F L 2 + F L 4) 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 2 L 2 σ(t) +CC 2/5 σ(t) 4/5. It remains to consider the case of the no-slip boundary conditions. We need to estimate both ρ L and [ρ] α. We use formula F = µa(p P) R(ρ u) λ+µ + λ+µ, where A(P P) and R(ρ u) are defined in (7.9). R(ρ u) verifies the estimate analogous to (4.17) for F (recall that α ],1 n/4[): (4.18) R(ρ u) L + [R(ρ u)] α CC 1/2 + u 2 L 2 σ(t) +CC 2/5 σ(t) 4/5, here [ ] α is the standard Hölder seminorm. The supremum norm of A(P P) is estimated in (7.13). Thus, integrating equation (4.16) we obtain: (4.19) sup ρ ρ L C sup[ρ] α +CC θ. [,T ] [,T ] To get the estimate on [ρ] α we first notice that it is equivalent to ρ(x) ρ(y) sup x,y x y α,

15 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS15 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 (4.2) d (logρ ) Xt + κ dt Y t λ + µ ργ Xt µa(p P) = Xt R(ρ u) Y t (λ + µ) 2 Xt Y t 2(λ + µ) 2. Y t In the next lemma we show that the distance between the trajectories is of the order x y. Lemma 2. There are C and θ such that for it holds: a(t) = C(C θ +Cθ σ(t) 4/5 + u(,t) 2 L 2 σ(t) + [ρ(,t)] α), u(x,t) u(y,t) a(t) x y, (4.21) e t τ a(s)ds X τ Y τ X t Y t e t τ a(s)ds X τ Y τ, for any (x,y), and t τ >. Moreover, for any τ t, (with a possibly different choice of C,θ), t a(s)ds C(C θ + sup [ρ(,τ)] α)(t τ) +CC θ. τ τ [,t] Proof. The first estimate of the lemma is obtained by splitting u = v + w, where the singular part, v, solves and the regular part w : µ v = µ λ µ (P P) A(P P), x, λ + µ (λ + µ) 2 v =, x, λ µ w = R(ρ u) + ρ u, x, 2(λ + µ) 2 w =, x. w L is estimated exactly as F in (4.17). On the other hand ( ) µ λ µ µv(x,t) = y G(x,y) (P(y,t) P) A(P(,t) P)(y) dy. λ + µ (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). d The lemma follows by combining the above estimates on w, v, using the ODEs: dt (X t Y t ) = u(x t,t) u(y t,t) and energy estimates (4.3), (4.8).

16 16 MISHA PEREPELITSA Let us return to the proof of lemma 1. We integrate equation (4.2) in time, divide the result by x y α, and using the estimate (7.13) for A(P P), (4.18) for R(ρ u), the estimate on X t Y t from the last lemma and energy estimates (4.3), (4.8) to derive that for any t [,T ] : (4.22) [ρ(,t)] α C[ρ ( )] αe t κγ ˇργ a(s)ds λ+µ t +CC θ + µc 2(α)κγ ˆρ γ t (λ + µ) 2 ([ρ(,τ)] α +CC )e tτ κγ ˇργ a(s)ds λ+µ (t τ) dτ, Under the condition (4.1), we can choose C small enough so that the last estimate and (4.19) provide a priori bounds finishing the proof of lemma 1. sup [,T ] ρ ρ L, sup[ρ] α CC θ, [,T ] 5. DYNAMICS OF SURFACES OF DISCONTINUITY The density ρ of the weak solution of theorem 1 is a discontinuous function. This puts a limit on the regularity of u; from the Poisson s equations (2.14), with ρ, F and ρ u as in theorem 1, u(,t) is not better than log-lipschitz. The flow generated by such velocity is a well defined homeomorphism but its only Hölder continuous in (x,t). Such flow does not preserve the differential structure of the manifolds transported by it. More regularity on u can be obtained if the set of points of discontinuity of ρ(,t) is regular and contained inside the flow domain. For example, in the domain = R 2, if u is the solution of (2.14), where ρ(,t) piecewise C α function with a jump discontinuity across C 1+α hypersurface S t, and F, ρ u as in theorem 1 then u(,t) is bounded and u(γ(,t))γ s (,t) C α (R), where γ(,t) : R R 2 is a parametrization of S t. Assuming that S t is transported by u and using the evolution equations for γ(s,t) and γ s (s,t), t γ(s,t) = u(γ(s,t),t), t γ s (s,t) = u(γ(s,t),t)γ s (s,t), one can obtain a priori bounds on the C α norm of the tangent to S t vector γ s (,t) and on u L. We illustrate this analysis on a model problem which captures the main features of the corresponding problem for the Navier-Stokes equations. For the reference, we state the result for the later problem at the end of this section. Thrughout this section = R 2. We consider weak solutions (ρ,u) of the problem: (5.1) ρ t + div(ρu) =, u = ρ, u(x,t), x +, ρ(x,) = ρ (x),

17 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS17 with the initial data ρ (x), such that ρ ρ L 1 () L (), for some ρ >. We prove in Appendix lemma 12 that the weak solutions defined on [,T ], are unique in the class ρ ρ L (,T ;L () L 1 ()), u L 1 (,T ;L ()) L (,T ;L q ()), u L (,T ;L 2q 2 q ()), for some q ]1,2[. The problem (5.1) is closely related to the problem of motion of a vortex patch in an ideal fluid and can be dealt with the methods developed for that problem in Chemin[3, 4] and Constantin-Bertozzi[2]. To prove the existence of solutions of (5.1) we adopt the method of [2]. Let us choose the initial density to be piecewise constant: { ρ, x + (5.2) ρ (x) =, ˆρ, x, where ± disjoint, connected, open sets. Let + be bounded and, for some φ C 1+α (), be represented as + = {x : φ (x) > }. The level set S = {x : φ (x) = } is the surface of discontinuity of ρ and we assume that (5.3) inf x S φ (x) >, [ φ ( )] C α () < +, for some α ],1[. Here, [ ] C α denotes the classical Hölder seminorm. The density ρ of the solution of (5.1) is piecewise constant; we can solve for divu = ρ ρ first, and then solve the continuity equation t ρ + u ρ + ρ(ρ ρ) = : { ρ, x (5.4) ρ(x,t) = ˆρ(t) = t, t, ρ ˆρ ˆρ ( ˆρ ρ)e ρt, x + where t ± = X t [ ± ], Xt ( ) is the flow generated by u. Thus, the set of points of discontinuity is transported by the flow and the amount of the jump of values of ρ decays exponentially to zero. This is the viscous damping effect. It follows from the formula for the density that C 1 + t + C +, for some C >, independent of t. Let ρ(t) = ˆρ(t) ρ. The set of discontinuities of ρ, S t, coincides with the zero level set of function φ(x,t) which solves the transport equation t φ + u φ =, φ(,) = φ ( ). The vector field φ = ( x2 φ, x1 φ) t is tangent to S t and is a weak solution of (5.5) t ( φ) + (u ) φ + (ρ ρ) φ = u φ, where we used the fact that divu = ρ ρ. Note, that since ( ) divu jumps across the interface, φ is a discontinuous function. Assuming that φ C α t ±, the velocity and its gradient can be expressed by singular integrals: ( ) σ(x y) 1 u(x,t) = ρ(t)p.v. 2π x y 2 dy + ρ(t)χ t + (x), 1 + t

18 18 MISHA PEREPELITSA with and σ(x) = x 2 ( x 2 2 x 2 1 2x 1 x 2 2x 1 x 2 x 2 1 x2 2 χ + t (x) = + t 1, x +, x 2, x S 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) σ(x y) φ(x,t) = ρ(t) 2π x y 2 ( φ(x,t) φ(y,t))dy. We set [ φ(,t)] ± α = [ φ(,t)] C α ( t ± ), The proofs of the following lemmas can be found in Bertozzi-Constantin[2]. Lemma 3. There is C = C( + ), such that u(,t) L C ρ(t) 1 t, t, ), ( 1 + log + [ φ(,t)] + ) α inf St, φ(,t) where by φ(x,t), for x S t, we denote the limit of φ(y,t) as y x, for y + t. Lemma 4. There is C = C( + ) for which [ u(,t) φ(,t)] α C ρ(t) u(,t) L [ φ(,t)] + α. These estimates allow us to obtain a priori bounds on φ. Indeed, integrating (5.5) on trajectories X t (x) in t +, we obtain that for any t [,T ], for which the solutions with the assumed above properties exists, (5.6) inf φ(,t) C inf φ e C S t S t ρ(τ) u(,τ) L dτ, for some C depending only on the data of the problem, but not T. Integrating (5.5) on two trajectories X t (x), X t (y) in + t, and using the inequalities: we obtain: X τ (x) X τ (y) e t τ u(,s) L ds X t (x) X t (y) X τ (x) X τ (y) e t τ u(,s) L ds, (5.7) [ φ(,t)] ± α e α t u(,τ) L dτ C[ φ ( )] ± α t ( +C ρ(t) u(,τ) L [ φ(,τ)] ± α e α ) τ u(,s) L ds dτ. The last inequality implies: (5.8) [ φ(,t)] ± α C[ φ ( )] ± α ( e α t u(,τ) L dτ+c t ρ(τ) u(,τ) L dτ ),

19 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS19 for some C independent of T. Combining lemma 3, estimates (5.6) and (5.8), we obtain a priori bounds: sup u(,t) L C u(,) L, t [,T ] sup [ φ(,t)] ± α C[ φ ( )] ± α, t [,T ] inf inf φ(,t) C 1 inf φ ( ), t [,T ] S S t for some C independent of T. Based on the above estimates we can prove Theorem 2. There exists a unique soltion of (5.1), (5.2) on [,+ ), with the following properties. u L (,+ ;L p ()), p ]1,+ ], u L (,+ ;L ()), ρ(x,t) is given by (5.4), where t + = X t [ + ], is an open, connected set with the boundary of the class C 1+α given by a parametrization γ(s,t), s [,L], with γ s (s,t) = 1, and sup [γ s (,t)] C α ([,L]) < +. t [,+ ) Proof. The solution can be obtained by showing first the existence of approximate solutions obtained by cutting off the singularity in the integral representation for u, and verifying the estimates of lemma 3 and lemma 4 for such approximate solutions. Specifically, 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. The approximate solution u a (x,t) is determined from the problem: x y u a (x,t) = ρ(t) η a ( x y ) 2π x y 2 dy, a,t + = Xa( t + ), + a,t where Xa t is the flow produced by the velocity field u a (x,t). Solving this problem (using a fixed point theorem) and passing to the limit a we recover the solution of theorem 2. We left the details of the proof to the reader. Concerning the problem of motion of the interface of discontinuity of ρ for the Navier-Stokes equations, the following result was established in Hoff[14]. Let = R 2, and P = κρ. ( Let )(ρ,u ) be the initial data verifying the assumptions of theorem 1. Additionally, let ρ C α ±, for some α ],1[, with ± open disjoint sets separated by a C 1+α hypersurface S. Let γ (s), s R, be a parametrization of S, with γ,s = 1, [γ,s ( )] C α (R) < +, and such that either (i) there is K > with the property that for any s 1,s 2 R, γ (s 1 ) γ (s 2 ) K 1 s 1 s 2, or (ii) γ is periodic, with period L, and there is K max{1,l} such that γ (s 1 ) γ (s 2 ) K 1 min{ s L 1 sl 2, sl 1 L sl 2 }, where sl [,L[, s L = s mod L. Constant K controls the amount of folding of curve γ. By ρ we denote the jump of ρ on S.

20 2 MISHA PEREPELITSA Theorem 3. There are numbers,θ,c depending on ρ,κ,λ,µ,α,b,m, such that if and K, [γ,s ( )] C α (R) M, C = ρ ρ C α ( ± ) + ρ γ L 2 L (R) + u H 1 () + ( ρ ρ 2 + u 2 )(1 + x 2 ) b dx, where b > 2, then there is a weak solution (ρ,u), of the Navier-Stokes equations with the initial data (ρ,u ). (ρ,u) verifies all properties of theorem 1 with the corresponding constants θ,c,c. Additionally, ρ(,t) is a piece-wise C α function with a jump across C 1+α interface S t, obtained by evolving S with the velocity u. There is a parametrization γ(,t) of S t, such that γ s (s,t) = 1, and For any x S t, inf lnρ ( ) e κ sup x,t ρ λ+µ S [γ s (,t)] C α (R) Ce Ct. lnρ(x,t) sup lnρ ( ) e κ infx,t ρ λ+µ. S Remark 1 The assumption u H 1 () can be relaxed to u H β (), β (,1). This property, as well as the detailed exposition of theorem 3 and its proof are contained in Hoff[14]. 6. DYNAMICS OF SURFACES OF DISCONTINUITY WITH CORNER SINGULARITIES As was explained in section 2, when the surface of discontinuity of the density is not smooth (the normal vector is discontinuous) or when the surface of discontinuity is attached to the boundary of the flow domain (with the boundary conditions (1.4) or (1.5)), the velocity u(,t) is only log- Lipschitz continuous in x. As the result one expects that the differential structure of geometric objects transported by such velocity may change drastically. 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 S = {x : x 1, x 2 = }. 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 (6.1) u(x) = ( Cx 1 log x 1, Cx 2 log x 2 ), C >, and S is a boundary of a wedge shaped region: it consists of {x : x 1 (s,) = s, x 2 (s,) = µs, s > } {x : x 1, x 2 = }, for some µ >.

21 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS21 The image of S by the flow map is given by (6.2) S t = {x 1 (s,t), x 2 (s,t)) = (s e Ct,(µs) ect ) : s } {x : x 1, x 2 = } = {x : (x 1, µ ect (x 1 ) e2ct ), x 1 } {x : x 1, x 2 = }. For any t >, S t has a cusp singularity at x =. We show in this section that the motion of the discontinuity surface of the density is also characterized by the instantaneous change of the geometry, when the surface is not smooth, or when it is attached to the boundary of the flow domain. This phenomenon is of the type described in second example above. However, the thickness of the cusp is different from that in (6.2). Note, that the functional framework of theorem 1 is general enough to cover the weak solutions with singularities of this type. Thus, the properties of transported interfaces, for the solutions of the Navier-Stokes equations, are similar to those of the solutions of the model problem (5.1). As in previous section, we will discuss in details problem (5.1). The corresponding results for the Navier-Stokes and incompressible Euler equations are contained in Hoff-Perepelitsa[18, 19]. In the context of incompressible Euler equations, the problem of dynamics of vortex patches with singularities was considered by Danchin[7, 8] and it was shown that i. 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; ii. sufficiently narrow cusps are stable. Moreover, numerical investigation of vortices with corner singularities were performed by Cohen- Danchin[6]. The results suggest that the interface around the singularity point very rapidly forms a cusp (for acute corners) or flattens (for obtuse corners). The result of Hoff-Perepelitsa[18] gives a formal verification of this phenomenon for a pair of symmetric vortex patches, rotating in opposite directions. Consider now problem (5.1), which we supply with the piecewise constant initial data { ρ, x (6.3) ρ (x) =, ˆρ, x +, with ˆρ < ρ. We assume that + is open, bounded, symmetric with respect to the line x 2 =, domain with the boundary S = + defined by a continuous map γ(s) = (γ 1(s), γ 2 (s)) : [ L,L] R 2, such that γ() = (,), γ( L) = γ(l), γ 1 (s), γ 2 (s), s [,L], γ 1 is even, γ 2 is odd and such that S is a corner-like at x, i.e.: γ 2 (s) (6.4) lim s + γ 1 (s) = µ ],1[. The following theorem holds. Theorem 4. There is a weak solution (ρ,u) of (5.1), (6.3) on R 2 [,T ] for some T >, with the following property: for any t [,T ], X t (γ( )) is a parametrization of the set of the points of discontinuity of ρ(,t), and (6.5) lim s + Here, X t ( ) the unique flow generated by u. X2 t(γ(s)) X1 t(γ(s)) =. Xt 1 (γ())

22 22 MISHA PEREPELITSA The theorem describes the instantaneous collapse of a corner to a cusp. Remark 1 It is not known if the weak solutions of theorem 4 are unique. However, given a solution (ρ,u), the flow X t is determined uniquely. 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,st n ), t ](n 1)τ, nτ], constructed in the following way. For n =, we set Xn(x) t = x, u n (x,t) =, St n = S, t +, = +. Given the values of the functions for t t n 1, for t ](n 1)τ, nτ] we define Xn(x) t as the solution of dx t n dt With this flow map we define +,n t = u n 1 (X t n(x),t τ), X t n 1 n (x) = x. u i n(x,t) = ρ(t) 2π = Xn[ t t +,n 1 n 1 ], St n = Xn[S t t n 1 ] and x i y i dy, i = 1,2. x y 2 X t n[ + t n 1 ] By X t (x), u(x,t) we will refer to the composite functions equal to Xn t... X t 1 1 (x),u n(x,t) for t ]t n 1,t n ]. We set t + = X t [ + ], and let x (t) denote the law of motion of the tip of the corner located initially at x =. One computes divu = ˆρ(t)χ +,n(x) for t ]t n 1,t n ]. This shows that the t size of t + is bounded: t +,n C( ˆρ, ρ) +. An easy potential estimate shows that there is C depending only on the initial data such that u(x,t) C, u(x,t) u(y,t) C x y log + x y. Define a curve S t = {(γ 1 (s,t),γ 2 (s,t)} by solving ODEs: d dt (γ 1 x,1 (t)) = C γ 1 x,1 (t) log + γ 1 x,1 (t), d dt γ 2 = C γ 2 log + γ 2, where (γ 1 (s),γ 2 (s)) is the parametrization of S. Lemma 5. For all t [,T ], S t lies above S t near the contact point x (t). γ 1 (s,) = γ 1 (s), γ 2 (s,) = γ 2 (s), Proof. Lemma follows from the definition of the curve S t. The curve is obtained from S by moving it with the velocity that strictly dominates the actual relative velocity u(x,t) u(x (t),t) in x 1 and x 2 directions. Now we will show that the kernels in the integral representation of the relative velocity have a definite sign when the interface lies in a suitable wedge. Lemma 6. If S t lies below a line {(x 1 x,1, µ 1 (x 1 x,1 ) : x 1 )} with µ 1 < 1, then for E c x = {y : y 2 x 2, y 1 x,1 + M(x 1 x,1 )},

23 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS23 with M >, (6.6) u 1 (x,t) u 1 (x,1 (t),t) = ρ(t)(x 1 x,1 ) (6.7) u 2 (x,t) = ρ(t)x 2 + t E c x R 2 + t + Ex c R 2 + sin2θ π(r ) 2 dy + O sin2θ π(r ) 2 dy + O( x 1 x,1 ), ( ) x 2 log + x 2, x 1 x,1 where (r,θ) is the polar representation of x y, r = x y, The error terms may depend on M, but are independent of T and τ. Depending on µ 1, M can be chosen so big that sin2θ > in the integrals above. Proof. Note that because of the symmetry of the data, u 1 (x,t) = u 1 (x,t), u 2 (x,t) = u 2 (x,t), and in particular u 2 ((x 1,),t) =. The density patch is symmetric with respect to the reflection as well. We can write u 1 (x,t) = ρ(t) 2π u 2 (x,t) = ρ(t) 2π t + R t R 2 + x 1 y 1 x y 2 + x 1 y 1 x y 2 dy, x 2 y 2 x y 2 x 2 + y 2 x y 2 dy. The elementary computation show that the integration outside set Ex c produces the corresponding error terms. Let M > be chosen large enough, as stated in the previous lemma, so that the kernels in (6.6), (6.7) are positive. Lemma 7. If S t lies below a line {(x 1 x,1, µ 1 (x 1 x,1 ) : x 1 )} with µ 1 < 1, for t [t n 2,t n 1 ], then for t [t n 1,t n ], S t lies below a line {(x 1, µ 2 (x 1 x,1 )} with the slope with C > independent of T and τ. µ 2 = (µ 1 ) e Cτ, Proof. Denote x(s,t) = X t (γ(s)). From the estimate (6.6) d (6.8) dt (x 1(s,t) x,1 (t)) C x 1 (s,t) x,1 (t). At time t = t n 1, the interface is below a line of slope µ 1 and so x 1 (s,t n 1 ) > x,1 (t n 1 ). The above inequality shows that x 1 (s,t) > x,1 (t) holds for t [t n 1,t n ], as well. We estimate using (6.6), (6.7), Setting µ(s,t) = and so ( ) d x (x 2 1 x,1 )O x 2 log + x 2 x 1 x,1 + x 2 O( x 1 x,1 ) dt x 1 x,1 (x 1 x,1 ) 2. x 2 (s,t) x 1 (s,t) x,1 (t), (note that µ < 1): dµ dt Cµ log µ, µ(s,t n 1 ) = µ 1, µ(s,t) (µ 1 ) e Cτ, t [t n 1,t n ].

24 24 MISHA PEREPELITSA Claim 1. For any µ 1 ]µ,1[, there is T > such that the curve S t lies below the line of slope µ 1 through the point x (t), for all t [,T ]. Proof. Fix µ 1 ]µ,1[. Take µ = µ in lemma (7) and T such that (µ ) e CT µ 1. Lemmas 6 and 7 with such µ 1, can be iterated for all time levels with lemma 7 implying that S t is below the line of slope µ 1. We proved that for our approximate solution u(x,t) there is a time interval [,T ] on which the estimates of lemma 6 hold. The kernel in the integrals in (6.6), (6.7) is positive, so we can integrate over a smaller set in R 2 +, + t = + t E c x {y : y lies below S t }, and keep both inequalities. S t is given explicitly as (x 1, (x 1 ) 1+Ct ), and the integrals can be estimated sin2θ (6.9) π(r ) 2 dy C1 (max{m(x 1 x,1 ),(x 2 ) e Ct }) t. t + t E c x R 2 + Now we can show the instantaneous tangency. We set x(s,t) = X t (γ(s)) and µ(s,t) = x 2 (s,t) x 1 (s,t) x,1 (t). Since velocity is log-lipschitz and u 2 (,t) = on R 2 +, x 2 (s,t) (µ s) e Ct. Because x 1 (s,t) x,1 > x 2 (s,t) and by the log-lipschitz continuity of the velocity Using both estimate in (6.9) we obtain: for some C,c > and thus (x 1 (s,t) x,1 (t)) (s) e Ct. log µ(s,t) µ(s,) C t 1 (Cs) c τ dτ, τ [ ] C c µ(s,t) µ t log +, (Cs) for some positive C,c, showing (6.5) for the time-discrete solution (ρ(x,t),u(x,t),x t (x)). This solution depends on parameter τ, that we omitted in formulas for the convenience of notation. In the limit τ the sequence of time-discrete solutions contains a subsequence that converges pointwisely to a weak solution of (5.1), (6.3), for which the same estimates hold, thus establishing theorem 4. We omit the details of this step.

25 NEAR-EQUILIBRIUM WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE FLOWS25 7. APPENDIX 7.1. Some facts about solutions of the Poisson s equation. In this section we collect the first and the second derivative estimates for solutions of the Poisson s equation that are used in the notes. The results stated here can found in Gilbarg-Trudinger[9]. Let i be an integer from 1..n. Consider the Poisson s equation (7.1) u = xi g, x, being supplemented by one of the following conditions. P1: = R n ; P2: = R n + = {x : x n > }, and u(x) =, x ; P3: = R n +, xn u(x) =, x. We assume that g L 2 () for problems P1, P2 and for the problem P3, if i n. When i = n, in the problem P3, we assume that g L 2 () C() and g = on. A function u Lloc 1 (), with u L2 (), is called a weak solution of the problems P1 and P3 if (7.2) u wdx = g xi wdx holds for any test function w L 1 loc (), with w L2 (). A function u L 1 loc (), with u L2 () and u = on, is called a weak solution of the problem P2 if (7.2) holds for any test function w L 1 loc (), with w L2 (), and w = on. Lemma 8. (i) The weak solutions to problems P1 P3 exist. Solutions are unique for the problem P2 and unique modulo a constant for problems P1 and P3. (ii) If, in addition to the conditions on g stated before the lemma, g L p (), for some p ]1,+ [, then there is C = C(p) such that (7.3) u L p () C g L p (). (iii) If, in addition to the conditions on g stated before the lemma, xi g L p (), for some p ]1,+ [, then there is C = C(p) such that (7.4) 2 u L p () C xi g L p (). Proof. The solution can be constructed as a limit of approximate solutions u ε that solve the same problem as u with g ε C (), and such that g ε g in L 2 () L p (). u ε is expressed through the appropriate Green s function. u ε (x) = yi G(x,y)g ε (y)dy, 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 +, u =, on, = R n +, xn u =, on,