NEW METHODS FOR LOCAL SOLVABILITY OF QUASILINEAR SYMMETRIC HYPERBOLIC SYSTEMS

. In this work we establish the local solvability of quasilinear symmetric hyperbolic system using local monotonicity method and frequency trun- cation method. The existence of an optimal control is also proved as an application of these methods.


1.
Introduction. Quasilinear symmetric and symmetrizable hyperbolic systems arise in a wide range of problems in engineering and physics. Some examples include unsteady Euler and potential equations of gas dynamics, inviscid magnetohydrodynamic (MHD) equations, shallow water equations, and Einstein field equations of general relativity to name a few (see for example [22], [13], [3]). The Cauchy problem of smooth solutions for these systems has been studied in the past using semigroup approach and fixed point arguments (see [6], [11], [8], [9], [21]). In this work, we establish the solvability of such system using two different methods, viz. local monotonicity method, which was first used in [15] to establish the solvability of stochastic Navier-Stokes equations, and a frequency truncation method ( [14], [4]). The new methods we present here are motivated by applications to control theory and stochastic analysis (see for example [18], [15], [20], where such methods are used). We also formulate a simple optimal control problem and demonstrate the utility of the new methods in proving the existence of optimal control. Stochastic analysis aspects will be presented in a separate paper.
The construction of the paper is as follows. In Section 2, we describe the quasilinear symmetric hyperbolic system A j (·, ·, ·) is an m × m symmetric matrix with u 0 ∈ H s (R n ) for s > n/2 + 1. We also obtain certain conditions satisfied by the linear operator A (t, u). By proving that the nonlinear term A (t, u)u is locally monotone, a local in time existence and uniqueness of smooth solutions of (1) is obtained in Section 3, using a generalization of the Minty-Browder technique. By considering a truncated quasilinear symmetric hyperbolic system, the local solvability of (1) is established in Section 4 using the fact that the frequency truncated sequence of solutions is Cauchy. As an application of both of these methods, the existence of an optimal control is obtained in Section 5 for a typical control problem.
In the sequel L(X, Y) denotes the space of all bounded linear operators from X to Y and D(A) denotes the domain of an operator A. The main theorem of this paper is A j (t, x, u) ∂ ∂x j , where A j (·, ·, ·)'s are m×m symmetric matrices for j = 1, · · · , n, satisfies and a j ik (·, ·, ·) is an entry of A j (·, ·, ·).
Also u(·) satisfies the energy estimate C( u(t) L ∞ )(1 + ∇u(t) L ∞ ), and 0 < T * < T is the maximal time for which the left hand side of (2) is finite.
2. Quasilinear symmetric hyperbolic system. The main ideas of this section are due to Kato ([8], [9]) and we elaborate here, since several of these results are used in subsequent sections. For the symmetric hyperbolic system (1), in order to compute the basic energy identity of Friedrichs ( [13]), we use the symmetry of A j (·, ·, ·) to find where a j ik (·, ·, ·) is an entry of the matrix A j (·, ·, ·). Hence from (3), we get Finally by using (4), for u, v ∈ H s (R n ), s > n/2 + 1, we have where C is a constant independent of u and we have for u, v, w ∈ H s (R n ). Hence from (7), for s > n/2 + 1, we obtain where From (8), it can be seen that We have J s := (I − ∆) s/2 so that f H s = J s f L 2 , for f ∈ H s (R n ). Let us now recall the commutator estimates ( [10]) and Moser type estimates ( [21]) used in this paper.
Let us define the operator B(t, u) ( [8]) by , which is the Frobenius norm. An application of (13) in (12) yields By using Moser estimates (Theorem 2.2), we have From (14), we get Hence for the symmetric hyperbolic system (1), we obtain the following conditions under which we prove the local solvability of (1).
(C3) For s > n/2 for u, v, w ∈ H s (R n ) and where 3. Existence and uniqueness-local monotonicity method. In this section, we establish the unique solvability of the symmetric hyperbolic system (1) by exploiting the local monotonicity property of A (·, ·) and using the Minty-Browder type technique.
3.1. Energy estimates and local monotonicity. Let {e 1 , e 2 , · · · } be a complete orthonormal system in L 2 (R n ) belonging to H s (R n ) and let L 2 n (R n ) be the n−dimensional subspace of L 2 (R n ). Let us now consider the following system of ODE in L 2 n (R n ): where u n 0 is the orthogonal projection of u 0 into L 2 n (R n ) and for simplicity we take f n = f . Since the system (22) is finite-dimensional and having locally Lipschitz coefficient, by Picard's theorem, the system has a unique solution in some interval [0, T ]. Let us now find the L 2 and H s energy estimates for the system (22). Proposition 1 (L 2 −energy estimate). Let u n (·) be the unique solution of the system of ODE's (22) with u 0 ∈ L 2 (R n ). Then, there exists a time 0 < T * < T such that, for f ∈ L 2 (0, T * ; L 2 (R n )) and 0 < ε ≤ 1, we have the following a-priori energy estimate: for 0 ≤ t ≤ T * , and where M = sup ∇A(t) L ∞ and the left hand side of the inequality (24) is finite whenever M is finite.
Proof. Let us find the L 2 −energy estimate starting with the energy equality By using (17), Cauchy-Schwartz inequality, and Young's inequality in (25), we get for 0 < ε ≤ 1. Integrating (26) from 0 to t, we obtain An application of Gronwall's inequality in (27) yields for 0 ≤ t ≤ T . Let us take the supremum from 0 to T in the inequality (27) to get Once again applying Gronwall's inequality in (29), we find sup 0≤t≤T u n (t) 2 for 0 < ε ≤ 1, and the left hand side of the inequality (31) is finite whenever M < ∞.
Proposition 2 (H s −energy estimate). Let u n (·) be the unique solution of the system of ODE's (22) with u 0 ∈ H s (R n ), for s > n/2 + 1. Then, there exists a time 0 < T * < T such that, for f ∈ L 2 (0, T * ; H s (R n )) and 0 < ε ≤ 1, we have the following a-priori energy estimate: where M = sup and the left hand side of the inequality (33) is finite whenever M, M are finite.
Proof. Let us take J s := (I − ∆) s/2 on the equation (22) to get Let us now find the H s −energy estimate by considering the energy equality By using Cauchy-Schwartz inequality, Young's inequality, (18), (17), and (19) in for 0 < ε ≤ 1. Integrating (36) from 0 to t, we obtain An application of Gronwall's inequality in (37) yields for 0 ≤ t ≤ T . Let us take supremum from 0 to T on both sides of the inequality (38) to obtain Once again an application of Gronwall's inequality in (39) yields It is clear from the inequality (40) that the left hand side of (40) is finite whenever Hence, there exists a time 0 < T * < T , such that and hence T * is the maximal time for which the left hand side of the inequality (40) is finite. Let us take M = sup and the left hand side of the inequality (41) is finite whenever M, M < ∞.
Let us now prove that the nonlinear term A (t, u)u is locally monotone.
Theorem 3.1 (Local Monotonicity). For any given N > 0, we consider the following (closed) ball: then for any u, v ∈ B N and each t ∈ (0, T * ), we have Similarly, if N (t) is a positive and measurable real valued function and B N (t) is the following (closed) time-variable ball: then for any u(·), v(·) ∈ B N (t), and any measurable function ρ(t), we have where T * is the time up to which the energy estimates in Proposition 1 and Proposition 2 are finite.
Proof. Let us consider ( Let us take the term (( (46) and use (8) to get By using (47) in (46), we get Since u, v ∈ B N , from (48), we find The inequality (45) can be easily obtained from (43).

Remark 1.
Let us denote F(u) = A (t, u)u. From the local monotonicity theorem (Theorem 3.1), one can deduce that F(·) + N I is a monotone operator in B N ⊂ H s (R n ) and by an application of Theorem 1.3, Chapter 2 of Barbu [2], one can also prove that the operator F(·) + N I is in fact a maximal monotone operator in B N . From the local monotonicity condition (43), we know that so that F(·) + N I is a monotone operator in B N . For the contrary, let us assume that F(·) + N I is not a maximal monotone operator in B N . Then, there exists For any x ∈ B N , we set and put u = u λ so that from (50), we get Thus, we have Letting λ → 1 in (51), one gets In (52), by taking x 0 − x = λw, for λ > 0, dividing by λ, letting λ → 0, and then using hemicontinuity property of F(·), we obtain and hence y 0 = F(x 0 )+N x 0 , which is a contradiction. Thus, F(·)+N I is a maximal monotone operator in the ball B N ⊂ H s (R n ).

Existence and uniqueness of local solution.
Let us now prove that the system (1) has a unique solution by exploiting the local monotonicity theorem (Theorem 3.1). The similar existence results for 2 − D Navier-Stokes equations can be found in [5].
Theorem 3.2 (Local Existence and Uniqueness). Let f ∈ L 2 (0, T * ; H s (R n )) and u 0 ∈ H s (R n ) with s > n/2 + 1, where T * is the maximal time for which the energy estimates given in Proposition 1 and Proposition 2 are finite. Then, there exists a unique solution u ∈ L ∞ (0, T * ; H s (R n )) to the problem (1).
Proof. Let us prove Theorem 3.2 by using the Minty-Browder technique of local monotonicity in the following steps: Step (1). Finite-dimensional Galerkin approximation of the symmetric hyperbolic system (1): Let {e 1 , e 2 , · · · } be a fixed complete orthonormal system in L 2 (R n ) belonging to H s (R n ). Let L 2 n (R n ) := span{e 1 , e 2 , · · · , e n } be the n−dimensional subspace of L 2 (R n ). Let us now consider the following finite-dimensional ODE in L 2 n (R n ): Also (54) satisfies the energy equality for any t ∈ (0, T * ).
Step (2). Weak convergence of the sequences u n (·) and F(u n (·)): By using Proposition 2, we can extract subsequences {u n (·)} and {F(u n (·))} such that The second convergence (57) is obtained by using the Moser estimates (11) and the algebra property of H s−1 (R n ) as and the right hand side of (58) is finite, since u n ∈ L ∞ (0, T * ; H s (R n )). We know that where r (t) is the derivative of r(t).
Let us consider ∂ ∂t e −r(t) u n (t) , e −r(t) u n (t) Hence from (60), we have 1 2 and satisfies the energy equality for any t ∈ (0, T * ). Also on passing to limit in (54), the limit u(·) satisfies ∂ ∂t and the energy equality for any t ∈ (0, T * ). A similar calculation of (62) yields for any t ∈ (0, T * ). Also, it should be noted that the initial value u n (0) converges to u(0) strongly, i.e., Step (3). Local Minty-Browder Technique: For any v ∈ L ∞ (0, T * ; L 2 m (R n )) with m < n, let us define so that r (t) = C ( ∇A(t) L ∞ + ∇v(t) L ∞ ∇ u A(t) L ∞ ), a.e. From the local monotonicity theorem (Theorem 3.1), by using (48), we have In (68), we use the energy equality (62) to get On taking liminf on both sides of (69), we obtain By using the lower semicontinuity property of the L 2 −norm and the strong convergence of the initial data u n (0) (see 66), the second term on the right hand side of the inequality satisfies the following inequality: Hence by using (65) and (71) in (70), we find where in the second step, we used the energy equality (65). The estimate (72) holds for any v ∈ L ∞ (0, T * ; L 2 m (R n )) for any m ∈ N, since the estimate (72) is independent of m and n. It can be easily seen by a density argument that the inequality (72) remains true for any v ∈ L ∞ (0, T * ; H s (R n )) for s > n/2 + 1. Indeed, for any v ∈ L ∞ (0, T * ; H s (R n )), there exists a strongly convergent subsequence v m ∈ L ∞ (0, T * ; H s (R n )) that satisfies the inequality (72).

Remark 2.
If we use the Galerkin approximation and if the problem is defined on a bounded domain Ω ⊂ R n or if we use the frequency truncation method (see Section 4) in the whole space R n , then we can have the following simplified proof for step (3) of Theorem 3.2. This can be done by making use of the strong convergence of Galerkin approximated sequence in L 2 (Ω) due to compactness and strong convergence of frequency truncated sequence in L 2 (R n ) due to Proposition 5. When the problem is defined on a bounded domain Ω and use Galerkin approximations, in step (3) of Theorem 3.2, the following simplification can be adapted. In (55), if we take limit supremum, and use the lower semicontinuity property of the L 2 −norm, the strong convergence of the initial data u n (0) and (56), we obtain lim sup n→∞ t 0 (F(u n (s)), u n (s)) L 2 ds = lim sup for all t ∈ (0, T * ). Hence, we have lim sup for all t ∈ (0, T * ). Since, F(·) + N I is a monotone operator (in fact F(·) + N I is maximal monotone in B N ⊂ H s (R n ), Remark 1), by using (43), we get On taking lim sup in (86), and using (85), (56) and (57), we get Since Ω ⊂ R n is bounded, the embedding of H s (Ω) in L 2 (Ω) is compact and hence we can pass to limit in the final term of (87). Thus, we have In (88), by taking u(t) − v(t) = λw(t), for λ > 0, dividing by λ, letting λ → 0 and then by using the hemicontinuity property of F(·), we get F 0 (t) = F(u(t)).
Proof. Using the energy estimate from Proposition 2, and (56), it can be easily seen that u ∈ C w (0, T * ; H s (R n )). Here, C w means continuity on the interval (0, T * ) with values in the weak topology of H s , i.e., u ∈ C w (0, T * ; H s (R n )) means that for any fixed φ ∈ H −s , φ, u is a continuous scalar function on (0, T * ), where ·, · denote This shows that u(·) H s is continuous from the right at time t = τ . Also the symmetric hyperbolic system given by (1) is time-reversible and so we get u(·) H s is continuous from the left at time t = τ . Since τ is arbitrary, we find that u(·) H s is continuous. Since u ∈ C w (0, T * ; H s (R n )) and the continuity of u(·) H s in times implies u ∈ C(0, T * ; H s (R n )). Now, let us prove that u ∈ Lip(0, T * ; H s−1 (R n )), where Lip(0, T * ; H s−1 (R n )) denotes the Lipschitz continuous functions on (0, T * ) with values in the norm topology of H s−1 . Let us consider for u, v ∈ C(0, T * ; H s (R n )). The first term in the right hand of the inequality (93) can be simplified using the algebra property of H s−1 −norm and (11) as For the second term in the right hand of the inequality (93), we use the algebra property of H s−1 −norm to find 2 H s−1 using the identity (6), Fubini's theorem, and algebra property of H s−1 −norm as From (95), it can be seen that Note that ∇ u A H s−1 is bounded, whenever u, v ∈ C(0, T * ; H s (R n )). Combining (94) and (97), and substituting it in (93), we obtain Since u, v ∈ C(0, T * ; H s (R n )), we get u ∈ Lip(0, T * ; H s−1 (R n )) and an application of Theorem 2.1(b), [13] yield u ∈ C(0, T * ; H s (R n )) ∩ C 1 (0, T * ; H s−1 (R n )).

4.
Existence and uniqueness-frequency truncation method. In this section, we establish the unique solvability of the symmetric hyperbolic system (1) using a frequency truncation method ( [4], [14], [16], [17]). Main steps are as follows: Step (i). We first consider a ball B R in the Fourier space, centered at the origin and of radius R > 0 to obtain the Fourier truncation S R f (ξ) = 1 B R (ξ) f (ξ), ξ ∈ R n . We prove that the solution u R (·) of smoothed version of (1) exist and the H s -norm of u R (·) are uniformly bounded up to time T * such that the bound is independent of R.
Step (ii). We show that the solutions u R (·) of the truncated problem is Cauchy in L ∞ (0, T * ; L 2 (R n )) as R → ∞.
4.1. Truncated symmetric hyperbolic system. Let us define the Fourier truncation S R as follows ( [4]): where B R , a ball of radius R centered at the origin and 1 B R (·) is the indicator function. For s ≥ 0, we have ( [4]) where C is a generic constant independent of R. Let us consider the truncated system By taking truncated initial data S R u 0 , the solution u R (·) of (102) lie in the space Since S R A (t, u R )u R is locally Lipschitz in u R (see (21)), by Picard's theorem for infinite-dimensional ODEs (see Theorem 3.1, [14]), there exists a solution u R (·) (102) in H R for some time interval [0, T ], where T depends on R. The solution will exist as long as u R H s is finite.
Proposition 4 (H s −energy estimate). Let u R (·) be the unique solution of the ODE (102) with u 0 ∈ H s (R n ), for s > n/2 + 1. Then, there exists a time 0 < T * < T such that, for f ∈ L 2 (0, T * ; H s (R n )), we have the following a-priori energy estimate for 0 < ε ≤ 1: for 0 ≤ t ≤ T * and where M = sup and the left hand side of the inequality (107) is finite whenever M, M are finite. Both estimates (104) and (107) are uniformly bounded and the bounds are independent of R, since for s > n/2 + 1.
Proof. Let us take J s := (I − ∆) s/2 on the equation (102) to get

MANIL T. MOHAN AND SIVAGURU S. SRITHARAN
The operators J s and S R commute, since for all ξ ∈ R n . The rest of the proof is same as that of Proposition 2 by using the fact that S R u R = u R in H R .

4.2.
Existence and uniqueness of local solution. We will now show that u R (·) is a Cauchy sequence in L ∞ (0, T * ; L 2 (R n )).
Proposition 5. Let u 0 ∈ H s (R n ) and f ∈ L 2 (0, T * ; H s (R n )) for s > n/2 + 1, and T * is the maximal time defined in Proposition 3 and Proposition 4. Then, the family of local solutions u R (·) of (102) is Cauchy (as R → ∞) in L ∞ (0, T * ; L 2 (R n )), i.e., Proof. If u R (·) and u R (·) are two local solutions, then for For R ≥ R, (u R − u R )(·) satisfy the energy equality The second term from the right hand side of the equality (112) can be simplified using Cauchy-Schwartz inequality, (101), and Young's inequality as for 0 < ε < 1. The first term from the right hand side of the equality (112) can be written as

QUASILINEAR SYMMETRIC HYPERBOLIC SYSTEMS 295
First term from the right hand side of the equality (114) can be simplified using Cauchy-Schwartz inequality, (101), algebra property of H s−1 −norm, Moser estimate (11) (with p = 2), and Young's inequality as The second term from the equality (114) can be estimated using Cauchy-Schwartz inequality, and (8) as The final term from the equality (114) can be simplified using (17) as By substituting (115), (116), and (117) in (114), we obtain By applying (113) and (118) in (112), we find d dt Let us integrate the inequality (119) in t, and take supremum from 0 to T * , we find By using (101), for 0 < ε < 1, we have Hence from (120), we get where the constants K = sup  (108)). An application of Gronwall's inequality on (123) yield We know that u 0 ∈ H s (R n ), f ∈ L 2 (0, T * ; H s (R n )), and M, M , K < ∞, up to the maximal time T * . On passing R, R → ∞, one can easily see that the right hand side of the inequality (124) goes to zero and hence the sequence of solution u R (·) of (102) is Cauchy (as R → ∞) in L ∞ (0, T * ; L 2 (R n )).
From Proposition 5, we get u R (·) is a Cauchy sequence in L 2 (0, T * ; L 2 (R n )) and it follows that u R → u strongly in L ∞ (0, T * ; L 2 (R n )). Now we prove that the sequence u R (·) converges to u(·) in L ∞ (0, T * ; H s (R n )), for 0 < s < s by making use of Sobolev interpolation theorem. Proposition 6. Let u 0 ∈ H s (R n ) and f ∈ L 2 (0, T * ; H s (R n )) for s > n/2 + 1, and T * be the maximal time defined in Proposition 3 and Proposition 4. Then, the family of local in time solution u R (·) of (102) is Cauchy (as R → ∞) in L ∞ (0, T * ; H s (R n )), for 0 < s < s.
From Proposition 6, it follows that u R → u strongly in L ∞ (0, T * ; H s (R n )). Now we prove that the nonlinear term S R A (t, u R )u R → A (t, u)u strongly in L ∞ (0, T * ; H s −1 (R n )).
Proposition 7. For s > n/2+1 and 0 < s < s, the nonlinear term S R A (t, u R )u R converges to A (t, u)u strongly in L ∞ (0, T * ; H s −1 (R n )) as R → ∞.
For the first term from the right hand side of the inequality (126), we use (99), (93) and (98) to get

MANIL T. MOHAN AND SIVAGURU S. SRITHARAN
For 0 < ε < 1, the second term from the right hand side of the inequality (126) can be estimated using (100), the algebra property of H s −1+ε −norm, and (11) as A substitution of (127) and (128) in (126) yield The right hand side of the inequality (129) tend to zero as R → ∞, since u R → u in L ∞ (0, T * ; H s (R n )) and u, u R ∈ L ∞ (0, T * ; H s (R n )).

Theorem 4.1 (Local Existence and Uniqueness
). Let f ∈ L 2 (0, T * ; H s (R n )) and u 0 ∈ H s (R n ) with s > n/2 + 1 and let T * be the maximal time for which the energy estimates in Proposition 3 and Proposition 4 are finite. Then, there exists a unique solution u ∈ L ∞ (0, T * ; H s (R n )) to the problem (1).
Proof. Existence: By using Proposition 5, Proposition 6, Proposition 7, Remark 4 and Remark 5, we have u(·) solves (1) as an equality in L ∞ (0, T ; H s −1 (R n )), for s < s and s > n/2 + 1. The uniform bounds in Proposition 4 and Banach-Alaoglu theorem allows us to extract a subsequence u R m such that u R m w * − − → u in L ∞ (0, T * ; H s (R n )).
Hence the limit u satisfies u ∈ L ∞ (0, T * ; H s (R n )). Uniqueness: Uniqueness results for the solution of symmetric hyperbolic system (1) is given in Theorem 3.2.
Remark 6. We can also show that u ∈ C(0, T * ; H s (R n )) ∩ C 1 (0, T * ; H s−1 (R n )) with the same arguments as in Theorem 3.3.

5.
Existence of optimal controls. In this section, we consider a control problem for the symmetric hyperbolic system, where the control appear as a "body force", and prove the existence of an optimal control. The similar ideas for establishing the existence of an optimal control for the Navier-Stokes equations can be found in [19]. The controlled symmetric hyperbolic system is given by ∂t + A (t, u)u(t) = K U(t), 0 ≤ t ≤ T * , where K is a bounded linear operator from L 2 (R n ) to H s (R n ) and U(·) is the control. We formulate the control problem of finding U to minimize the cost functional J(U) = 1 2 where u d (t) is a desired solution.