On universality of homogeneous Euler equation
aa r X i v : . [ n li n . S I] J a n On universality of homogeneous Euler equation
B.G.Konopelchenko and G.Ortenzi ∗ January 14, 2021
Abstract
Master character of the multidimensional homogeneous Euler equation is discussed. It is shownthat under restrictions to the lower dimensions certain subclasses of its solutions provide us withthe solutions of various hydrodynamic type equations. Integrable one dimensional systems in termsof Riemann invariants and its extensions, multidimensional equations describing isoenthalpic andpolytropic motions and shallow water type equations are among them.
Homogeneous Euler equation (also called pressureless Euler equation) ∂u i ∂t + n X k =1 u k ∂u i ∂x k = 0 , i = 1 , . . . , n , (1.1)is one of the basic equations in the theories of fluids, gas and other media at n = 3 (see e.g.[9, 8, 17]). In spite of the fact that it represents the most simplified version (no pressure, noviscosity etc. [9, 8]) of the full equations, it arises in number of studies in many branches ofphysics.Euler equation (1.1) has the remarkable property to be solvable by the straightforward multi-dimensional extension of the classical hodograph equations method [2, 3]. This fact and reductionsto lower number of dependent variables has been used to establish the interrelations betweenequations (1.1) and multi-dimensional Monge-Amp`ere equations and Bateman equations [3, 4, 5,10].In the present paper we will study the restrictions of the n -dimensional Euler type equationto the lower dimensional spaces R m ( m < n ). We start with the slightly modified equation (1.1),namely, with the system ∂u i ∂t + n X k =1 β k α k λ k ( u ) ∂u i ∂x k = 0 , i = 1 , . . . , n , (1.2) ∗ Corresponding author. E-mail: [email protected], Phone: +39(0)264485725 here λ k ( u ) = λ k ( u , . . . , u n ) are arbitrary real-valued functions and α k , β k are arbitrary realconstants. Solutions of the system (1.2) are provided by the hodograph equations α i x i − β i λ i ( u ) t + f i ( u ) = 0 , i = 1 , . . . , n , (1.3)where f k ( u ) = f k ( u , . . . , u n ) are arbitrary real-valued functions (in the case λ i ( u ) = u i see [2, 3]).Constants α i , β i are, obviously, transformable away except the cases when some of them vanish.Exactly such cases are related with restrictions of the system (1.2).It is shown that restrictions on independent variables x i , functions f i and parameters α i , β i in (1.3) gives rise to the various hydrodynamical type systems in the spaces R m ( m ≤ n ). Inparticular, under the restriction x = x = · · · = x n ≡ x plus certain restrictions on g i , thehodograph equations (1.3) provides us with the solutions of the system ∂u k ∂t + λ i ( u ) ∂u i ∂x = 0 , i = 1 , . . . , n , (1.4)that is the classical diagonalized one-dimensional system in terms of Riemann invariants solvableby Tsarev’s generalized hodograph method [16, 1].Under the restriction to the ( n − x n = 0, with constraintson functions f i , the hodograph equations (1.3) provide us with solutions of ( n − ∂u i ∂t + n − X k =1 u k ∂u i ∂x k = − ∂v∂x i , i = 1 , . . . , n − ,∂v∂t + n − X k =1 u k ∂v∂x k = 0 , (1.5)which describes the adiabatic and isoenthalpic motion where v = T S and T is temperature and S is entropy. In the particular case f i = ∂W∂u i , i = 1 , . . . , n , this system describes the potential motion (cid:16) u i = ∂φ∂x i (cid:17) . Second case describes the polytropic motion, namely, ∂u i ∂t + n − X k =1 u k ∂u i ∂x k = − ρ ∂p∂x i , i = 1 , . . . , n − ,∂ρ∂t + n − X k =1 ∂∂x k ( ρu k ) = 0 , (1.6)where the density ρ = u n and the pressure p = ρ γ . In this case the functions f i = ∂W∂u i i = 1 , . . . , n and the function W obeys a determinant type PDE.Natural two and higher dimensional extensions of the system (1.4) are considered too.It is noted that solutions of the systems (1.4)-(1.6) and others are obtained in our approach aregiven by subclasses of solutions of the original system (1.2) which are characterized by the specificchoices of functions f i and restrictions on the coordinates x , . . . , x n .Interrelations between infinite-dimensional Euler equation (1.2) and Burgers and Korteweg-deVries equations is considered. We have also discussed the phenomenon of the gradient catastrophefor homogeneous Euler equation. It is shown that it first happens at a point on the n -dimensionalhypersurface.The paper is organized as follows. In section 2 hodograph equations for the generalized homoge-neous Euler equations and its one-dimensional reductions are considered. The ( n − eductions of the homogeneous Euler equations to the ( n − We start with hodograph equations α i x i − β i λ i ( u ) t + f i ( u ) = 0 , i = 1 , . . . , n , (2.1)where λ i ( u ) and f i ( u ) are arbitrary real valued functions and α , β i are arbitrary constants. Thesystem (2.1) is an obvious extension of the hodograph equations considered in [2, 3, 4].Differentiating (2.1) w.r.t. x k and t , one obtains α i δ ik + n X l =1 ∂g i ∂u l ∂u l ∂x k = 0 , i, k = 1 , . . . , n. (2.2)and − β i λ i + n X l =1 ∂g i ∂u l ∂u l ∂t = 0 , i, k = 1 , . . . , n. (2.3)where g i ≡ α i x i − β i λ i ( u ) t + f i ( u ).Relations (2.2) and (2.3) imply that ∂u l ∂x k = − ( A − ) lk α k , i, k = 1 , . . . , n. (2.4)and ∂u l ∂t = n X k =1 ( A − ) lk β k λ k , l = 1 , . . . , n. (2.5)where the matrix A has elements A lk = ∂g l ∂u k = − t ∂λ l ∂u k + ∂f l ∂u k , l.k = 1 , . . . , n (2.6)and it is assumed that det( A ) = 0.Combining (2.4) and (2.2), one gets ∂u l ∂t + n X k =1 β k α k λ k ∂u l ∂x k = 0 , l = 1 , . . . , n . (2.7)For λ k = u k , α k = β k = 1 these calculations has been done in [2, 3, 4].Relations (2.2) and (2.3) also imply that n X l =1 A il ∂u l ∂t + n X k =1 β k α k ∂u l ∂x k ! = 0 , i = 1 , . . . , n . (2.8) t is noted that functions f i ( u ) are arbitrary one in this construction. They are related to initialdata at t = 0 via α i x i + f i ( u ( t = 0)) = 0 . (2.9)Hence, hodograph equations (2.1) provide us with the general solutions of the system (2.7).We would like to note that the system (2.7) and the original Euler equation (1.1), in fact, areequivalent. Indeed it is easy to see that if u i obey the system (2.7) then λ k obey the system (1.1)and viceversa ( u i → λ i ( u )). It is noted also that the systems (2.7) with different λ k ( u ) pairwisecommute. So one has an infinite hierarchy of equations of the form (2.7).Now let us consider the simplest reductions of the system (2.7) for which the matrix A isdiagonal one, i.e. A lk = − t ∂λ l ∂u k + ∂f l ∂u k = 0 , l = k . (2.10)A way to satisfy this condition is to impose the constraints ∂λ l ∂u k = 0 , ∂f l ∂u k = 0 , l = k . (2.11)In this case the relations (2.2) imply that ∂u l ∂u k = 0 , l = k , (2.12)and the n -dimensional system (2.7) (with α k = β k = 1) is decomposed into n decoupled one-dimensional Burgers-Hopf type equations ∂u l ∂t + λ l ( u l ) ∂u l ∂x l = 0 , l = 1 , . . . , n . (2.13)Less trivial reduction with the diagonal matrix A arises if one considers the restriction to theone-dimensional subspace given by the condition x = x = · · · = x n ≡ x . In this reduction, thehodograph system (2.1) assumes the form x − λ i ( u ) t + f i ( u ) = 0 , i = 1 , . . . , n , (2.14)and one obtains ∂u l ∂x = − ∂f l ∂u l , ∂u l ∂t = λ l∂f l ∂u l , l = 1 , . . . , n . (2.15)Hence, one has the system ∂u l ∂t + λ l ( u ) ∂u l ∂x = 0 , l = 1 , . . . , n . (2.16)The equation (2.14) implies that t = f i − f l λ i − λ l , i = l . (2.17)Consequently the condition (2.10) is equivalent to the following one ∂f l ∂u k f l − f k = ∂λ l ∂u k λ l − λ k , l = k . (2.18) quations (2.16) represent the well known form of the one-dimensional multi-component hydrody-namic type systems in terms of Riemann invariants (see e.g. [9, 17]). Hodograph equation (2.14)and condition (2.18) are exactly those of the Tsarev generalized hodograph method [16, 1].So, solutions of the homogeneous Euler equations (2.7), for which functions f l , l = 1 , . . . , n in(2.1) are selected according to the condition (2.18), after the restriction to the one-dimensionalsubspace x = · · · = x n become the solutions of the system (2.16).It is noted that reduction of the homogeneous Euler equation to the system (2.16) arises alsofor other one-dimensional restrictions of the n − dimensional space ( x , . . . , x n ), for instance, givenby x = x = · · · = x n = 0. In these cases the characterizations of functions f l are quite differentfrom (2.18). ( n − -dimensional reductions: Jordan system Here we will consider reductions of the Euler system (2.1) with λ k ( u ) = u k to the ( n − x n = 0. It is equivalent to require α n = 0 in the hodographequations (2.1). The relation (2.4) implies that ∂u l ∂x n = 0, l = 1 , . . . , n under this restriction, howeverdue to (2.4) 1 α n ∂u l ∂x n = − ( A − ) ln = 0 , l = 1 , . . . , n . (3.1)Using this relation, one rewrites equation (2.7) as ( α k = β k = 1, k = 1 , . . . , n − β n = 1) ∂u l ∂t + n − X k =1 u k ∂u l ∂x k − u n ( A − ) ln = 0 , l = 1 , . . . , n − ∂u n ∂t + n − X k =1 u k ∂u n ∂x k − u n ( A − ) nn = 0 . (3.2)Under the requirements ( A − ) ln = − ∂u n ∂x l , l = 1 , . . . , n − A − ) nn = 0 (3.4)the system (3.2) assumes the form ∂u l ∂t + n − X k =1 u k ∂u l ∂x k + u n ∂u n ∂x l = 0 , l = 1 , . . . , n − ∂u n ∂t + n − X k =1 u k ∂u n ∂x k = 0 . (3.5)In terms of variables u i , i = 1 , . . . , n − v = u n /
2, the system looks like ∂u l ∂t + n − X k =1 u k ∂u l ∂x k + ∂v∂x l = 0 , l = 1 , . . . , n − ∂v∂t + n − X k =1 u k ∂v∂x k = 0 . (3.6) he system (3.6) represents the ( n − − dimensional generalization of the one-dimensional ( n = 2)Jordan system introduced in [6].The system (3.6) at n = 4 arises also in physics. Indeed, hydrodynamical equations describingadiabatic flow of an ideal fluid are of the form [8, 9] ∂u l ∂t + X k =1 u k ∂u l ∂x k + 1 ρ ∂P∂x l = 0 , l = 1 , , ∂S∂t + X k =1 u k ∂S∂x k = 0 . (3.7)where ρ is the fluid density, P stands for pressure and S is the entropy. The variation of enthalpy W is given by (see e.g. [9]) ∂W∂x i = T ∂S∂x i + 1 ρ ∂P∂x i , i = 1 , , . (3.8)So for the isoenthalpic motion with constant temperature one has1 ρ ∂P∂x i = ∂∂x i ( T S ) , i = 1 , , n = 4 and v = − T S describes the adiabatic and isoenthalpic motion of a fluid at constant temperature.Now let us analyze the conditions (3.3) and (3.4). The relation (2.4) says that ∂u n ∂x l = − ( A − ) nl , l = 1 , . . . , n − , (3.10)and so the condition (3.3) is satisfied if( A − ) nl = ( A − ) ln , l = 1 , . . . , n − . (3.11)Thus, equations (2.7) are reducible to (3.6) if the matrix A lk = ∂g l ∂u k obeys the constraints (3.11),(3.4) or, equivalently ˜ A ln = ˜ A nl , l = 1 , . . . , n − A nn = 0 , (3.12)where ˜ A is the matrix adjugate to A (i.e. A ˜ A = det( A ) I n ).Using the known formula for the adjugate matrix ˜ A , one can obtain more explicit form of theconditions (3.12). We instead will use an explicit form of the matrix A − . Indeed, since A lk = ∂g l ∂u k ,one has ( A − ) lk = ∂u l ∂g k , l, k = 1 , . . . n . (3.13)Using (3.13), one rewrites (3.12) as ∂u l ∂g n = ∂u n ∂g l , l = 1 , . . . , n − ∂u n ∂g n = 0 . (3.14) onditions (3.14) imply that u l = ∂∂g l φ ( g , . . . , g n ) + A l ( g , . . . , g n − ) , l = 1 , . . . , n − ,u n = ∂∂g n φ ( g , . . . , g n ) , (3.15)where φ ( g , . . . , g n ) and A l ( g , . . . , g n − ) are arbitrary functions.Consider now 1 − form ( x i , i = 1 , . . . , n and t are fixed)Ω = n X l =1 u l d g l = d φ + n − X l =1 A l ( g , . . . , g n − )d g l (3.16)and perform the Legendre transformation defined by n X l =1 g l d u l = d n X l =1 u l g l ! − Ω . (3.17)Due to (3.16) one has n X l =1 g l d u l = d W − n − X l =1 A l ( g , . . . , g n − )d g l , (3.18)where W = n X l =1 u l g l ! − φ . (3.19)Equation (3.18) rewritten as n X l =1 g l − ∂W∂u l + n − X k =1 A k ∂g k ∂u l ! d u l = 0 , (3.20)implies that g l = ∂W∂u l − n − X k =1 A k ∂g k ∂u l , l = 1 , . . . , n . (3.21)The compatibility condition for (3.21) (i.e. ∂ W/∂u l ∂u k = ∂ W/∂u k ∂u l ) is given by ∂g l ∂u m − ∂g m ∂u l + n − X k,i =1 (cid:18) ∂ A k ∂g i − ∂ A i ∂g k (cid:19) ∂g i ∂u m ∂g k ∂u l = 0 , l, m = 1 , . . . , n . (3.22)Correspondingly for f l one has f l = ∂ ˜ W∂u l − n − X k =1 ˜ A k ∂f k ∂u l , l = 1 , . . . , n . (3.23)and ∂f l ∂u m − ∂f m ∂u l + n − X k,i =1 ∂ ˜ A k ∂g i − ∂ ˜ A i ∂g k ! ∂f i ∂u m ∂f k ∂u l = 0 , l, m = 1 , . . . , n , (3.24)where ˜ W ( u , . . . , u n ) and ˜ A l ( f , . . . , f n − ), l = 1 , . . . , n − n − W ( u , . . . , u n ) (or W ) on n variables and( n −
1) functions ˜ A l ( f , . . . , f n − ) of n − Potential flows
The formulae presented in the previous section are simplified drastically in the particular case when ∂ A k ∂g i = ∂ A i ∂g k , i = 1 , . . . , n − , (4.1)or, consequently A i = ∂ψ∂g i , i = 1 , . . . , n − , (4.2)where ψ is an arbitrary function. So g l = ∂∂u l ( W − ψ ). Equivalently without loose of generalitiesone can put directly A i ≡
0. In this case g l = ∂W∂u l , l = 1 , . . . , n , (4.3)and the matrix A is of the form A lk = ∂ W∂u l ∂u k , l, k = 1 , . . . , n . (4.4)Then, one has ( A − ) nn = det B det A , (4.5)where ( n − × ( n −
1) matrix B is the algebraic complement to the element A nn , i.e. B lk = ∂ W∂u l ∂u k , l, k = 1 , . . . , n − . (4.6)So, the condition (3.4) assumes the form det( B ) = 0 . (4.7)The form (4.4) of the matrix A leads to a constraint of the variables u i , i = 1 , . . . , n . Indeed, sincethe matrix A (4.4) is symmetric one, then the matrix A − is symmetric too. In such a case therelations (2.4) imply that ∂u l ∂x k = ∂u k ∂x l , k, l = 1 , . . . , n − . (4.8)So u i = ∂φ∂x i , i = 1 , . . . , n − , (4.9)where φ ( x , . . . , x n − ) is some function. Thus in this case the equations (3.6) or (3.7) describe thepotential adiabatic isoenthalpic flows.Due to (4.9) equations (3.6) are equivalent to the following (assuming that all constants ofintegration vanish) ∂φ∂t + 12 n − X k =1 (cid:18) ∂φ∂x k (cid:19) + v = 0 ,∂v∂t + n − X k =1 ∂φ∂x k ∂v∂x k = 0 . (4.10) he first equation (4.10) is well known in hydrodynamics, e.g. for the isoentropic potential motion(see [9], § v from the system (4.10) gives us the following equation for thevelocity potential φ ∂ φ∂t + 2 n − X k =1 ∂φ∂x k ∂ φ∂x k ∂t + n − X i,k =1 ∂φ∂x k ∂φ∂x i ∂ φ∂x k ∂x i = 0 . (4.11)Solutions of this equation provide us the solutions of the system (4.10) via v = − ∂φ∂t − P n − k =1 (cid:16) ∂φ∂x k (cid:17) .The system (4.10) can be viewed also as the Hamilton-Jacobi equation given by the first ofequations (4.10) for the action φ with the time-dependent potential v ( x , . . . , x n − ; t ) obeyingthe second equation (4.10). So the solutions of the system (4.10) provide us with a solvable( n − − dimensional system of classical mechanics.In the one dimensional case n = 2 the equation (4.11) is of the form ∂ φ∂t + 2 ∂φ∂x ∂ φ∂x∂t + (cid:18) ∂φ∂x (cid:19) ∂ φ∂x = 0 . (4.12)or ∂∂t ∂φ∂t + (cid:18) ∂φ∂x (cid:19) ! + ∂∂x (cid:18) ∂φ∂x (cid:19) ! = 0 . (4.13)and it is of parabolic type.Solutions of the system (4.10) and equation (4.11) are provided by hodograph equations (2.1)with function g l obeying (4.3) and (4.6), (4.7).One can obtain the corresponding expression also for the functions f l , l = 1 , . . . , n . Indeedsince g l = x l − u l t + f l , the 1-form˜Ω = n X l =1 f l d u l = − Ω ∗ − d n X l =1 x l u l − t n X l =1 u l ! (4.14)is closed due to the fact that Ω ∗ = d W . Consequently one has f l = ∂ ˜ W∂u l , l = 1 , . . . , n (4.15)for some function ˜ W ( u , . . . , u n ). In terms of the function ˜ W the condition (4.7) looks likedet( ˜ B ) = 0 . (4.16)where ˜ B lk = ∂ ˜ W∂u l ∂u k − tδ lk , l, k = 1 , . . . , n − . (4.17)Thus, for the ( n − λ i = u i ) are the equations ∂W∂u i = 0, i = 1 , . . . , n defining the critical points offunctions W = W ( u, x, t ) of the form ( v = u n / W = n − X i =1 x i u i − t v + 12 n − X i =1 u i ! + ˜ W ( u, v ) (4.18) ith functions ˜ W obeying the constraint (4.16) at the critical points. The potential Jordan system(3.6) describes the dynamics of the critical points of such functions W .In other words, solutions of the potential Jordan system (3.6) or equations (3.7) and (3.9) arethose solutions of the homogeneous Euler equation (2.7) which corresponds to the functions f i in(2.1) being the components of the gradients of functions W of the form (4.18) with obeying theconstraints (4.16).The condition that det( ˜ B ) belongs to the ideal generated by the functions ∂W∂u i , i = 1 , . . . , n is the sufficient one to characterize the above subclasses of solutions. However, more explicitdescription of the class of the class of the functions W would be, definitely, rather convenient.For this purpose we first observe that at the critical point t = ∂ ˜ W∂v . So the matrix ˜ B can beequivalently rewritten as ˜ B lk = ∂ ˜ W∂u l ∂u k − δ lk ∂ ˜ W∂v , l, k = 1 , . . . , n − . (4.19)Then since ∂ ˜ W∂u l ∂u k − δ lk ∂ ˜ W∂v = ∂ W∂u l ∂u k − δ lk ∂W∂v , (4.20)the condition (4.16) is equivalent todet (cid:18) ∂ W∂u l ∂u k − δ lk ∂W∂v (cid:19) = 0 . (4.21)The last step is to extend this condition outside the critical points and to consider (4.21) as theequation defining the function W of the form (4.18). For such functions conditions (4.16) and(4.17) are automatically satisfied at the critical points. Note that the formula (4.18) and (4.21)are natural ( n − − dimensional extensions of the corresponding formulae ∂W∂v = ∂ W∂u for theone-dimensional Jordan system [6].We note also that one gets the same system (3.6) considering the restrictions to the ( n − − dimensional subspaces defined by conditions x i = x k with fixed i and k . In these cases one hascharacterizations of the functions f i similar to those considered above.Finally we note that the subclass of the hodograph equations (2.1) with g i = ∂W∂u , i = 1 , . . . , n and, hence, λ i = ∂F∂u i , i = 1 , ..., n where W and F are some functions give us solutions of thepotential reduction u i = ∂φ∂x , i = 1 , . . . , n of the homogeneous Euler equation (2.7). In this caseequation (2.7) is equivalent to the following ( α i = β i = 1, i = 1 , . . . , n ) ∂φ∂t + F (cid:18) ∂φ∂x , . . . , ∂φ∂x n (cid:19) = 0 . (4.22) ( n − -dimensional reductions: Polytropic gas Now we will consider equations (3.2) with the constaints( A − ) ln = − u an ∂u n ∂x l , l = 1 , . . . , n − , (5.1)and ( A − ) nn = − n − X k =1 ∂u k ∂x k , (5.2) here a is an arbitrary real number. Under these constraints equations (3.2) assume the form ∂u l ∂t + n − X k =1 u k ∂u l ∂x k = − a + 2 ∂∂x l (cid:0) u a +2 n (cid:1) , l = 1 , . . . , n − ,∂u n ∂t + n − X k =1 ∂∂x k ( u n u k ) = 0 . (5.3)For a = − n − − dimension with u n beingthe fluid height h . For arbitrary a it describes the polytropic motion with pressure p = a +3 ρ a +3 and the density ρ = u n (see e.g. [9, 17]).The formula (2.4) implies that ∂u n ∂x l = − ( A − ) nl , l = 1 , . . . , n − , (5.4)and n − X k =1 ∂u k ∂x k = − n − X k =1 ( A − ) kk . (5.5)So, the constraint (5.1) and (5.2) are equivalent to the following( A − ) ln = u an ( A − ) nl , l = 1 , . . . , n − A − ) nn = n − X k =1 ( A − ) kk . (5.6)Using (3.13), one rewrites these constraints as ∂u l ∂g n = u an ∂u n ∂g l ≡ ∂∂g l u a +1 n a + 1 , l = 1 , . . . , n − ∂u n ∂g n = n − X k =1 ∂u k ∂g k . (5.7)The first condition (5.7) imply that u l = ∂φ∂g l + B l ( g , . . . , g n − ) , l = 1 , . . . , n − , a + 1 u a +1 n = ∂φ∂g n , (5.8)where φ ( g , . . . , g n ) and B l ( g , . . . , g n − ) are arbitrary functions. On can find the correspondingformulae and constraints for g l in a way similar to that described in section 3. Here we will considerthe simplest case B l ( g , . . . , g n − ) = 0, l = 1 , . . . , n −
1. In this case one has g l = ∂W∂u l , l = 1 , . . . , n − ,g n = u − an ∂W∂u n , (5.9)for some function W . o the matrix A is of the form A = B V V ∂∂u n (cid:16) u − an ∂W∂u n (cid:17) . (5.10)where B is a ( n − × ( n −
1) matrix with elements ∂ W∂u l ∂u k , V is a column with ( n −
1) elements ∂ W∂u l ∂u n , V is a row with ( n −
1) elements u − an ∂ W∂u n ∂u l . Hence, the second condition (5.6) assumesthe form det B − n − X k =1 det C k = 0 , (5.11)where C k are algebraic complements of the elements A kk .So, the solutions of the system (5.3) describing polytropic motion are those solutions of thehomogeneous n − dimensional Euler equation which correspond to the choice (5.9) of functions g l with W obeying the constraint (5.11) on the manifold g i = 0, i = 1 , . . . n .Analogously to the previous section one can show that the functions f l are given by f l = ∂ ˜ W∂u l , l = 1 , . . . , n − ,f n = u − an ∂ ˜ W∂u n , (5.12)for some function ˜ W . So the function W is of the form W = n − X i =1 x i u i − t n − X k =1 u k + 1 a + 2 u a +2 n ! + ˜ W . (5.13)Since ∂ W∂u k ∂u l = − tδ lk + ∂ ˜ W∂u k ∂u l , l, k = 1 . . . , n − t = u − − an ∂ ˜ W∂u n , (5.15)the condition (5.11) is equivalent todet ∂ ˜ W∂u l ∂u k − u − − an δ lk ∂ ˜ W∂u n ! − n − X k =1 det ˜ C k = 0 , (5.16)where ˜ C k are principal minors of the matrix˜ A lk = ∂ ˜ W∂u l ∂u k − u − − an δ lk ∂ ˜ W∂u n . (5.17)Finally using the relation ∂ W∂u l ∂u k − u − − an δ lk ∂W∂u n = ∂ ˜ W∂u l ∂u k − u − − an δ lk ∂ ˜ W∂u n , l, k = 1 , . . . , n − , (5.18) ne can extend the equation (5.16) outside the critical points ∂W∂u i = 0 to obtain the equationcharacterizing the function W , i.e.det (cid:18) ∂ W∂u l ∂u k − u − − an δ lk ∂W∂u n (cid:19) − n − X k =1 det C k = 0 , (5.19)where C k are principal minors of the matrix (5.17) with the substitution ˜ W → W .In the simplest case n = 2 all above formulae become rather compact. For arbitrary a thefunction W is of the form ( u = u, u = v ). W = xu − t (cid:18) u + 1 a + 2 v a +2 (cid:19) + ˜ W (5.20)while the equation (5.19) becomes ∂ W∂u − v − a ∂ W∂v + av − − a ∂W∂v = 0 . (5.21)For a = − § ∂ Γ ± ∂t = λ ± ∂ Γ ± ∂x (5.22)with the Riemann invariants Γ ± and the characteristics velocities λ ± given by [17]Γ ± = − a + 22 u ± v a +22 , λ ± = − u ± v a +22 . (5.23)In terms of the Riemann invariants, the equation (5.21) becomes the classical Euler-Poisson-Darboux equation (Γ + − Γ − ) ∂ W∂u∂v = − aa + 2 (cid:18) ∂W∂ Γ + − ∂W∂ Γ − (cid:19) . (5.24)For the classical shallow water equation a = − W are nonlinear for multi-dimensional systems(3.6) and (5.3), in contrast to the one dimensional situation with the linear equation ∂W∂v = ∂ W∂u and equation (5.24). Such situation seems to be typical in applications of the hodograph equationto multi-dimensional PDEs [11, 12], except, of course, the master homogeneous Euler equation(2.7).We note also that one can study in a similar manner the dimensional reductions of the gener-alised equation (2.7) with arbitrary function λ i ( u ). In this section we consider two particular examples of the Euler equation in three dimensions.First, let us start with the two-dimensional restriction of the hodograph equation given by( x = x = x, x = y ), x − λ t + f = 0 ,x − λ t + f = 0 ,y − λ t + f = 0 . (6.1) ifferentiation of (6.1) w.r.t. x, y and t gives ∂u l ∂x = − ( A − ) l − ( A − ) l ,∂u l ∂y = − ( A − ) l ,∂u l ∂t = X k =1 ( A − ) lk λ k , l = 1 , , . (6.2)Combining expressions (6.2), one obtains ∂u ∂t + λ ∂u ∂x + λ ∂u ∂y = ( λ − λ )( A − ) ,∂u ∂t + λ ∂u ∂x + λ ∂u ∂y = ( λ − λ )( A − ) ,∂u ∂t + λ ∂u ∂x + λ ∂u ∂y = ( λ − λ )( A − ) . (6.3)Imposing the constraint ( A − ) = ( A − ) = ( A − ) = 0 , (6.4)one gets the system ∂u ∂t + λ ∂u ∂x + λ ∂u ∂y = 0 ,∂u ∂t + λ ∂u ∂x + λ ∂u ∂y = 0 ,∂u ∂t + λ ∂u ∂x + λ ∂u ∂y = 0 , (6.5)which is two dimensional extension of the one-dimensional system for Riemann invariant u and u . Constraints (6.4) are equivalent to the following three equations for three functions g , g , g ∂g ∂u ∂g ∂u − ∂g ∂u ∂g ∂u = 0 ∂g ∂u ∂g ∂u − ∂g ∂u ∂g ∂u = 0 ∂g ∂u ∂g ∂u − ∂g ∂u ∂g ∂u = 0 (6.6)In terms of the functions f i one has the equations (6.6) with the substitution ( t = ( f − f ) / ( λ − λ )) ∂g l ∂u k = ∂f l ∂u k − f − f λ − λ ∂λ l ∂u k , l, k = 1 , , . (6.7)So any solution of the three-dimensional homogeneous Euler equation, constructed using the func-tions f i , i = 1 , , x = y . The reason is that the constraint (6.4) represent only the part of the constraint(2.10). n order to recover this system we combine the expressions (6.2) into another system of equations(equivalent to (6.4)), namely, ∂u ∂t + λ ∂u ∂x = ( λ − λ )( A − ) + λ ( A − ) ,∂u ∂t + λ ∂u ∂x = ( λ − λ )( A − ) + λ ( A − ) ,∂u ∂t + λ ∂u ∂x = λ ( A − ) + λ ( A − ) . (6.8)Now, requiring that x = y and ( A − ) lk = 0, l = k , l, k = 1 , ,
3, i.e. A lk = 0, l = k , one obtainsthe 3-component system (2.16).As second example we consider the one-dimensional reduction of the Euler equation with λ k = u k and α = 1, α = α = 0 and β = β = 1, β = 0. So we start with the hodograph system x − u t + f = 0 , − t + f = 0 ,f = 0 , (6.9)where x = x , x = x = 0 and we redefine the function f → u f . Differentiating (6.9) w.r.t. x and t , we obtain ∂u l ∂x = − ( A − ) l , l = 1 , , ,∂u l ∂t = ( A − ) l u + ( A − ) l . (6.10)Consequently one has the system ∂u l ∂t + u ∂u l ∂x = ( A − ) l , l = 1 , , . (6.11)Now we impose the constraints( A − ) = − ∂u ∂x , ( A − ) = − ∂u ∂x , ( A − ) = 0 . (6.12)Due to the relation (6.10), these constraints are equivalent to the following( A − ) = ( A − ) , ( A − ) = ( A − ) , ( A − ) = 0 . (6.13)Using the explicit form of the 3 × A , one obtains the following system ofequations ∂g ∂u ∂g ∂u − ∂g ∂u ∂g ∂u + ∂g ∂u (cid:18) ∂g ∂u − ∂g ∂u (cid:19) = 0 ∂g ∂u ∂g ∂u − ∂g ∂u ∂g ∂u + ∂g ∂u (cid:18) ∂g ∂u − ∂g ∂u (cid:19) = 0 ∂g ∂u ∂g ∂u − ∂g ∂u ∂g ∂u = 0 . (6.14)Any solution of this system with the substitution ∂g ∂u l = ∂f ∂u l − f δ l , ∂g ∂u l = ∂f ∂u l , ∂g ∂u l = ∂f ∂u l , (6.15) rovide us with the functions f , f , f for which three dimensional homogeneous Euler equationsis reducible to the system ∂u ∂t + u ∂u ∂x + ∂u ∂x = 0 ,∂u ∂t + u ∂u ∂x + ∂u ∂x = 0 ,∂u ∂t + u ∂u ∂x = 0 , (6.16)which is the 3-component one-dimensional Jordan system described in [6].It is not difficult to show that the system (6.14) has a solution for which g i = ∂W∂u i , i = 1 , , W obeys the equation ∂W∂u = ∂ W∂u , ∂W∂u = ∂ W∂u . (6.18)Hodograph equations (6.9) represent the critical points equations ∂W∂u i = 0, i = 1 , , W = xu − t (cid:18) u + u (cid:19) + ˜ W ( u , u , u ) , (6.19)which obeys the equations (6.18).Equations (6.18) and function W (6.19) are exactly those given in the paper [6]. The above result on the 3-component Jordan system can be extended to the n -component case.Indeed, let us consider the hodograph equations for arbitrary n and α = 1, α = · · · = α n = 0, β = β = 1, β = · · · = β n = 0, i.e. the equations ( x = x ) x − u t + f =0 , − t + f =0 ,f m =0 , m = 3 , . . . , n (7.1)where, for convenience we redefine the function f → u f . Relation (2.4) imply that ∂u l ∂x k = 0, k = 2 , , . . . , n and ∂u l ∂x = − ( A − ) l ,∂u l ∂t = ( A − ) l u + ( A − ) l . (7.2)Combining (7.2), one gets ∂u l ∂t + u ∂u l ∂x = ( A − ) l , l = 1 , . . . , n . (7.3) mposing the constraint ( A − ) l = − ∂u l +1 ∂x , l = 1 , . . . , n − , ( A − ) n = 0 , (7.4)one obtains the n -components system ∂u l ∂t + u ∂u l ∂x + ∂u l +1 ∂x = 0 , l = 1 , . . . , n − ,∂u n ∂t + u ∂u n ∂x = 0 , (7.5)that is the n -component Jordan system introduced in [6].Since ∂u l +1 ∂x = − ( A − ) l +1 , , l = 1 , . . . , n −
1, the constraints (7.4) are equivalent to the following( A − ) l = ( A − ) l +1 , , l = 1 , . . . , n − , ( A − ) n = 0 , (7.6)or ˜ A l = ˜ A l +1 , , l = 1 , . . . , n − , ˜ A n = 0 . (7.7)Using the explicit expression for the elements of the adjugate matrix ˜ A , one rewrites the constraints(7.7) as the system of n differential equations for the functions g i , i = 1 , . . . , n or f i , i = 1 , . . . , n .It is not difficult to show that this system has a solution for which g i = ∂W∂u i , f i = ∂ ˜ W∂u i , i = 1 , . . . , n (7.8)where the functions W and ˜ W obey the equations ∂W∂u k = ∂ k W∂u k , k = 2 , . . . , n , (7.9)and W = xu − t (cid:18) u + u (cid:19) + ˜ W ( u , . . . , u n ) , (7.10)that coincides with those formulae presented in [6]. It is noted that in this one-dimensional reduc-tion the constraint (4.8) is absent.Now, following [6] one can consider the system (7.5) in the formal limit n → ∞ and get theinfinite Jordan chain which has been discussed in different contexts in [7, 14, 6]. So the Jordanchain represent a particular reduction of the infinite-dimensional Homogeneous Euler equation.In the paper [15] it was observed that the Jordan chain admits differential reductions to variousintegrable partial differential equations, for example, to the Burgers equation and Korteweg-deVries equation. Indeed, if one imposes the constraint u = ∂u ∂x , (7.11) hen the first equation ( l = 1) in (7.5) becomes the Burgers equation ∂u ∂t + u ∂u ∂x + ∂ u ∂u = 0 , (7.12)while the other equations (7.5) with l = 2 , , . . . represent themselves the recursive relations todefine u , u , . . . .If one requires that u = ∂ u ∂x , (7.13)then the Jordan chain is reduced to the Korteweg-de Vries equation ∂u ∂t + u ∂u ∂x + ∂ u ∂u = 0 . (7.14)Constraints (7.11) and (7.13) can be rewritten in terms of the elements ( A − ) lk . Indeed, thedifferential consequence of (7.11), namely, ∂u ∂x = ∂∂x (cid:16) ∂u ∂x (cid:17) after the use of (2.4), assumes the form( A − ) + ∞ X k =1 ∂ ( A − ) ∂u k ( A − ) k = 0 . (7.15)The differential consequence of (7.13) is equivalent to the following( A − ) − ∞ X k,l =1 ∂∂u l (cid:18) ∂ ( A − ) kl ∂u k ( A − ) k (cid:19) ( A − ) l = 0 . (7.16)Though the constraint (7.15), (7.16) are rather cumbersome, one concludes that the solutions ofthe Burgers and Korteweg-de Vries equations represent particular subclasses of solutions of theinfinite-dimensional homogeneous Euler equation. All the results presented in the previous sections are valid under the assumption that det( A ) = 0.If instead det( A ) ≡ det (cid:18) ∂f l ∂u k − t ∂λ l ∂u k (cid:19) = 0 , (8.1)then, according to (2.4), (2.5) solutions of the equation (2.7) and other equations exhibit thegradient catastrophe ∂u l ∂x k → ∞ , ∂u l ∂t → ∞ .Let us consider such a situation for the classical homogeneous Euler equation ( λ k = u k , α k = β k = 1). In this case the equation (8.1) is simplified todet( A ) ≡ det (cid:18) ∂f l ∂u k − tδ lk (cid:19) = 0 , (8.2)i.e. to the characteristic polynomial equation t n + n − X k =0 B k ( u ) t k = 0 (8.3) f the n × n matrix ˜ A lk = ∂f l ∂u k . Due to (2.9) the functions f ( u ) are the local inverse of the initialvalues of u ( t = 0 , x ) and, consequently, coefficients B k depends on u , . . . , u n only.Thus, gradient catastrophe for the homogeneous Euler equation happens in general on the n − dimensional hypersurface in R n +1 given by the equation (8.3). If the polynomial (8.3) has noreal roots then the gradient catastrophe does not happen for given initial data u i ( t = 0 , x ). Let usassume that equation (8.3) has at least one real root t c . So the gradient catastrophe happens onthe hypersurface S given by t c = φ ( u , . . . , u n ) (8.4)where φ ( u ) is a certain function constructed out from the (local) inverse of the u ( t = 0 , x ). Usuallydiscussed first moment of appeareance of gradient catastrophe corresponds to the minimum valueof t c , i.e. to the situation when ∂t c ∂u i = ∂∂u i φ ( u , . . . , u n ) = 0 , i = 1 , . . . , n (8.5)plus a condition on the second derivatives (for generic catastrophes such condition is given by theclassical condition on the Hessian of φ ).For generic initial data the function φ is generic. Consequently n equations (8.5) has genericallya single solution u c , . . . , u cn .Thus, generically, the gradient catastrophe for the homogeneous Euler equation first happensat the time t c min = φ ( u c , . . . , u cn ) (8.6)at the point u c , . . . , u cn on the hypersurface S (8.4). Then it expands on the whole hypersurface(8.4).It is noted that for the first time such property of the gradient catastrophe for multi-dimensionalequations has been observed in [11, 12].In more detail the gradient catastrophes for the homogeneous Euler equation, related equationsand its regularization will be considered in a separate paper. Acknowledgement
The authors are thankful to M. Pavlov for the information about D. B. Fairlie’s papers. This projectthanks the support of the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk lodowska-Curie grant no 778010
IPaDEGAN . We also gratefully acknowledgethe auspices of the GNFM Section of INdAM under which part of this work was carried out.
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