Blowup for Hyperbolic Equations. Helge Kristian Jenssen and Carlo Sinestrari

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1 Blowup for Hyperbolic Equations Helge Kristian Jenssen and Carlo Sinestrari Abstract. We consider dierent situations of blowup in sup-norm for hyperbolic equations. For scalar conservation laws with a source the asymptotic prole of the solution close to a blowup point is described in detail. Based on an example of Jerey we next show how blowup for ordinary dierential equations can be used to construct examples of blowup for systems of hyperbolic equations. Finally we outline the construction of solutions to certain strictly hyperbolic 3 3-systems of conservation laws which blow up in either sup-norm or total variation norm in nite time. 1. Scalar Equations 1.1. Introduction In this rst section we consider blowup in sup-norm for scalar equations. For a more complete discussion including proofs see [6]. We study blowup of positive solutions of the scalar conservation law with t u(x t)+@ x f(u(x t)) = g(u(x t)) u(x 0) = u 0 (x): (1) When the source term g(u) grows fast enough the solution may blow up at a nite point in the sense that there is a positive T<+1 and an X 2 (;1 1) such that lim sup u(x t) =+1 x!x t"t We refer to (X T) asablowup point. Our goal is to provide detailed information on how the solution behaves near such a point. We consider this as a rst step towards the understanding of blowup for physically relevant systems with sources. The present work is inspired by the similar analysis by Bressan [2] for a semilinear reactive{euler system. The problems of nding conditions under which the solution of (1) blows up, of determining whether the blowup is \localized", and to which extent the solution can be continued beyond the blowup time T have been studied by several authors, see [1], [10], and [11]. In the present work we bring the analysis one step further, i.e. we give a general procedure to study the asymptotic prole of solutions of (1) near a blowup point. For simplicity we assume that the initial value changes monotonicity only once, having a single maximum point with strictly negative

2 294 Helge Kristian Jenssen and Carlo Sinestrari second derivative. The analysis is complicated by the fact that solutions of (1) may develop discontinuities (shocks) in nite time. We consequently divide the analysis into dierent cases according to whether the solution blows up along a shock curve or in a region of smoothness. We refer to these cases as the shock case and smooth case, respectively. The main tool in our analysis is Dafermos' theory of generalized characteristics [3]. Our results can be summed up as follows. In the shock case the solution converges to an unbounded traveling wave on one side of the shock, while it remains bounded on the other side. In the smooth case the solution tends asymptotically to a function constant on parabolas about the blowup point. We also establish that the smooth case can occur only if the source term grows fast enough. As an application of our results we discuss two specic examples where the source term g is either exponential or powerlike Preliminaries We assume that the following conditions are satised for some k 3. (H1): f is strictly convex, f 0 (0)=0,andf 2 C 1 ([0 1)) \ C k+1 ((0 1)). (H2): g 2 C 1 ([0 1)) \ C k ((0 1)) and g(u) > 0 for any u>0. In addition Z 1 1 f 0 (v) dv < 1: (2) g(v) (H3): u 0 is nonnegative and of class C k, is increasing in (;1 x ] and decreasing in [x 1) for some x 2 R. (H4): For any x<x such that u 0 (x) > 0wehave u 0 0(x) > 0 moreover u 00 0(x ) < 0. These assumptions ensure the existence and uniqueness of an entropy solution of problem (1), at least locally in time. The solution is nonnegative and the solution has a maximal interval of existence [0 T), with 0 <T +1. We assume in the following that T<1. Let () denote the forward characteristic starting at (x 0). The following results form the basis for the detailed analysis of the blowup prole. Theorem 1.1. The curve (t) tends to a limit X<+1 as t " T.Moreover, there is a classical characteristic ending at (X T) and u! +1 along. Theorem 1.2. If the characteristic curve is a shock in some interval [T 0 T), then u is uniformly bounded in the region f(x t) : t<t x>(t)g. Theorem 1.3. If N is a neighbourhood of(x T), then u is uniformly bounded in (R [0 T))n N. Corollary 1.4. There is a unique way to dene u( T):R! [0 +1], such that u( T) is continuous from the left, u(x T) =+1, u(x T ) < +1 for x 6= X, and such that u( t)! u( T) in the L 1 loc (RnfXg)-norm as t " T.

3 Blowup for Hyperbolic Equations 295 Any set of the form U \ (R [0 T] nf(x T)g) where U is a neighbourhood of (X T) isreferredtoasabackward neighbourhood of (X T). We nowlistthe possible behaviours of the solution near the blowup point. (I): The characteristic isashock for t close enough to T, i.e. u((t); t) > u((t)+ t) for all t greater than some T 0 <T. (II): The solution u is smooth in some backward neighbourhood of the blowup point (X T) and lim u(x t) =+1: (3) x!x t"t (III): The characteristic is classical in [0 T) but either any neighbourhood of (X T) contains points where u is discontinuous, or u tends to dierent values along dierent characteristics ending at (X T). We next give some results and examples for the two rst cases. For further details and a discussion of case (III) we refer to [6] Shock Case Let us set u = u 0 (x). The problem (1) has a local C k (;" ") solution both forward and backward in time. Hence there exists ">0 and a function 2 C k (;" ") such that (0) = 0 and u(x + (t) t) u for any t 2 (;" "). Also let c = 0 (0) = f 0 (u) ; g(u)p ;1. Theorem 1.5. Assume that (H1){(H4) are satised and suppose that the solution u of problem (1) blows up at time T and that the forward characteristic x = (t) from the point (x 0) is a shock for t close to T. Then Z +1 f 0 (v) ; c dv = X ; x ; c(t ; t)+o(jx ; x ; c(t ; t)j 2 ) (4) g(v) u(x t) for all (x t) with x (t), while u(x t) is uniformly bounded forx>(t). Observe that, up to lower order terms, formula (4) denes u implicitly as a traveling wave with speed c. Example 1.6. Exponential case. Consider the equation (1) with f(u) =u m =m for some m>1 and g(u) =e u.inthiscase we have Z 1 Z 1 u f 0 (v) ; c g(v) dv = u v m;1 ; c e v dv = e ;u u m;1 (1 + o(1)): Using (4) and setting = X ; x ; c(t ; t) we nd the following expression for the asymptotic prole of u(x t) as (x t)! (X T) with x (t): u(x t) =; ln +(m ; 1) ln(; ln )+o(1): (5)

4 296 Helge Kristian Jenssen and Carlo Sinestrari 1.4. Smooth Case The possibility of having smooth blowup depends on the relative strengths of the source and convection terms. We rst give a denition. Denition 1.7. Assume that f and g satisfy assumptions (H1){(H2) and let p = lim u!1 f 0 (u). Wesaythatg has subcritical growth with respect to f if p =+1 and f lim 00 (v)g(v) v!+1 f 0 =0: (6) (v) 3 If the limit in (6) is +1 or if p < +1 we say that g has supercritical growth. One can show that if the source has subcritical growth, then the blowup is necessarily of type (I). However for supercritical sources one can have smooth blowup. We restrict ourselves to case (II). Lemma 1.8. If g is supercritical and the solution u of (1) has the behaviour (II), then the translates of the level curve of u through (x 0) along are again level curves of u and cover univalently N. Theorem 1.9. Let assumptions (H1){(H4) be satised and let g have supercritical growth. Suppose that the solution u of problem (1) has the behaviour (II). Then, as (x t)! (X T), Z +1 dv g(v) =(T ; t + 2 (X ; x)2 )(1 + o(1)) (7) u(x t) where = ;u 00 0(x )=g(u). Example Power case. Consider equation (1) with f(u) =u m =m and g(u) = u p (m>1). Using the theorem above one can show that the solution can have smooth blowup only if p 2m ; 1 in addition, if p>2m ; 1, then where w =(p ; 1) 1 p;1. u(x t)=w (T ; t)+ 2 (X ; x)2 1 1;p (1 + o(1)) 2. Systems of Equations 2.1. Introduction The above results show that one has very detailed information about the blowup pattern for scalar hyperbolic equations. Much less is known for systems of equations and the situation here is far from fully understood. Even simple examples give insight about the possible mechanisms of blowup. The phenomenon of resonance and magnication for systems of hyperbolic conservation laws has recently been studied by several authors [8, 9, 12]. Young [12] constructs exact solutions to 3 3-systems of conservation laws. Depending on the choice of initial data and the interaction coecients one can construct

5 Blowup for Hyperbolic Equations 297 dierent types of behaviour such as arbitrary large magnication of total variation and p-norms (1 p 1) in nite time. The eigenvalues are constant so that these systems are linearly degenerate in each family. In[8] Joly, Metivier, and Rauch consider examples of systems which are genuinely nonlinear in all three elds. Building on the work by Rosales and Majda [9] and using the theory of weakly nonlinear geometric optics [7] they give examples of solutions for which the variation grows arbitrarily large and the sup-norm is amplied by arbitrarily large factors in nite time. Common to these two works is the use of data with small sup-norm. The results are local in the sense that to have large amplication of the data one has to prescribe data with correspondingly small sup-norm. The aim of this section is to present two other types of blowup for systems. The rst is a generalization of an example given by Jerey [4] and illustrates how ordinary dierential equations can be used to construct blowup patterns for sysetms of hyperbolic equations. In the nal section we give a class of 3 3- systems of conservation laws with solutions whose sup-norm or total variation become innite in nite time Degenerate Hyperbolic Systems The object of this section is to understand the blowup mechanism for a particular class of systems and establish to which extent it is particular to 3 3-systems of hyperbolic equations. We also consider the possibility of writing the systems in conservative form. In [4] Jerey considered a quasilinear 3 3-system of the form U t + M(U)U x =0 (8) where U = U(x t) =(u(x t) v(x t) w(x t)) T, and M(U) is a strictly hyperbolic matrix (i.e. M(U) has three real and distinct eigenvalues) which depends nonlinearly on U. The system is in nonconservative form, the data is given on a compact interval, and the system is linearly degenerate in each eld (i.e. each eigenvalue is constant along the integral curves of the corresponding eigenvector elds). In Jerey's example the matrix M(U) is given by M(U) = ; cosh(2v) 0 ; sinh(2v) cosh(v) 0 sinh(v) sinh(2v) 0 cosh(2v) 1 A (9) where is a non-negative parameter. The eigenvalues of M(U) are 1 and 0. With initial data U(x 0) = (x=(h) 0 ;x=(h)) on the compact interval [;h h], an explicit solution U (x t) isgiven by u (x t) = ;1 1 1;t + x ; 1 v (x t) =lnj1 ; tj w (x t) = ;1 1 1;t ; x ; 1 :

6 298 Helge Kristian Jenssen and Carlo Sinestrari Here >0 is a constant and = =(h). As observed in [4], the vector U represents the solution only in the domain of determinacy D given by D = f(x t) :0 t h ;jxjg For the set D intersects the critical time-line t = t c = ;1 where the solution becomes unbounded. Thus, when, this provides an example of a strictly hyperbolic 3 3-system for which the sup-norm (and consequently also the total variation) tends to innity in nite time. To analyze the example given by Jerey, we note some key properties of the system given above. The matrix M(U) depends on v only and the entries along the middle column are identically zero. Also, the rst and third components of the solution are linear in x (with opposite coecients) while the middle component v of the solution is a function of t alone. Starting with these properties we see what kind of systems we can construct. Thus, suppose the rst and third components of the solution are of the form u(x t)=x +~u(t) w(x t) =;x +~w(t) where is a real constant. Observe that u(x 0) and w(x 0) are then necessarily linear ane functions of x. Thus if the initial data are to have nite total variation or sup-norm, then they must be prescribed on a compact interval. Next let the matrix M(U) have the form M(U) = The system (8) then takes the form a(v) 0 A(v) b(v) 0 B(v) c(v) 0 C(v) ~u 0 + (a(v) ; A(v)) = 0 v t + (b(v) ; B(v)) = 0 ~w 0 + (c(v) ; C(v))=0 1 A : (10) where prime denotes dierentiation with respect to t. If we in addition assume that v depends only on t, then this is a system of ordinary dierential equations where the second equation is decoupled from the other two. One can thus construct blowup for the system (12) simply by giving coecients, b(v), and B(v) such that the second equation blows up in nite time. However, we also want the matrix M(U) tobehyperbolic and that the blowup occurs within the domain of determinacy. A simple way of doing this is to prescribe constant eigenvalues (as in Jerey's example) and then adjusting the compact interval where the initial data are given so that the domain of determinacy intersects the critical timeline t = t c where the solution of the second equation blows up. Notice that 0 is always an eigenvalue of M. Also, M has eigenvalues 0 ( 0 being a positive constant) if

7 Blowup for Hyperbolic Equations 299 and only if the coecients a(v), A(v), c(v), and C(v) satisfy a(v) =;C(v) A(v)c(v) = 2 0 ; a(v) 2 : These relations are independent of the coecients in the second equation, which makes it easy to produce examples of the same type as that of Jerey. Example 2.1. Let 0 = =1, B(v) =;b(v) =v 2 =2, and assume that v(x 0) = 1 on the compact interval [;h h], where h 1. Inthiscase v satises the ordinary dierential equation Hence _v = v 2 v(0)=1: v(x t) =v(t) = 1 1 ; t which blows up at time t c =1. The choice for h guarantees that the domain of determinacy intersects the critical timeline t = t c. There are many ways to choose the remaining coecients. For example let T (v) =1; v 1 and dene a(v) =;C(v) = cos(t (v)) With B and b as above this gives the system with an explicit solution given by A(v) =c(v) = sin(t (v)): u t + cos(t (v))u x +sin(t (v))w x =0 v t ; v2 2 u x + v2 2 w x =0 w t +sin(t (v))u x ; cos(t (v))w x =0 u(x t) =x ; cos(t) ; sin(t) v(x t) = 1 1;t w(x t) =;x + cos(t) ; sin(t): We observe that in this case only the v-component blows up. Thus we see that it is easy to construct examples of blowup for a large class of degenerate hyperbolic systems. Furthermore the blowup mechanism for these systems is essentially that of an ordinary dierential equation with a superlinear righthand side. As a consequence the blowup does not depend on the fact that the above systems are 3 3-systems. Indeed, using the same technique as above one can easily give examples of 2 2-systems with similar behavior. This is in contrast to the results in [5, 8, 12] for conservation laws where it is essential that one consider systems of three or more equations, see below. Finally we observe that the systems above can in fact be written on conservation form. This follows since

8 300 Helge Kristian Jenssen and Carlo Sinestrari the matrix M depends only on a component of the solution which is independent of x. Hence [M(U)U] x =[M(U)] x U + M(U)U x = M(U)U x such that if U satises the quasilinear system (8), then U also satises the conservative system U t +[M(U)U] x =0: (11) Note however that U is not necessarily a solution of (11) for all times t<t c.the domain of determinacy for (11) will in general be dierent from that of (8). In particular the maximal and minimal characteristics emanating from the left and right endpoints where the initial data are given may meet before the blowup time t = t c Systems of Conservation Laws In this section we present a class of 3 3-systems of conservation laws U t + F (U) x =0 (12) for which one can prescribe initial data such that the solution blows up in nite time. The main result is the following theorem. Theorem 2.2. There exist 33-systems of strictly hyperbolic conservation laws for which the solution U(x t) has one of the following properties. (a) There exists a time T, 0 <T<1, such that limku( t)k 1 =+1: (13) t"t (b) There exist a constant C>0 andatimet, 0 <T<1, such that limt.v. [U( t)]=+1 while ku( t)k 1 <C for all t<t: (14) t"t Outline of Construction The class of systems is a modication of the examples considered by Young [12]. We want to set up an interaction pattern where two 2-shocks approach each other while 1- and 3-waves (which will be contact discontinuities) are reected back and forth between the 2-shocks. Notethatthis requires at least three equations, i.e. it is not possible to get an interaction pattern like this for 2 2-systems. We dothisby constructing solutions to 3 3-systems of the form (12) where the ux function F has the form F (U) = ua(v)+w ;(v) u( 2 0 ; a 2 (v)) ; wa(v) 1 A : (15) Here 0 > 0 is a constant and a(v) will be chosen to obtain the various behaviour stated in Theorem 2.2. To simplify the analysis we assume that ;(v) has the following properties, (i) ;(v) is strictly convex (ii) ; 0 <(v) =; 0 (v) < 0 for all v 2 R

9 Blowup for Hyperbolic Equations 301 (iii) ;(0) = 0 and ;(;v) =;(v) for all v 2 R. It is readily checked that the eigenvalues of the Jacobian DF are 1 = ; 0 2 = (v) 3 =+ 0 : (16) Thus (ii) guarantees that the system is strictly hyperbolic. Note that the second equation in the system is a decoupled scalar conservation law for v with a strictly convex ux. It follows that the second characteristic eld is genuinely nonlinear. The rst and third elds are linearly degenerate so that all 1- and 3-waves are contact discontinuities. It follows that shock and rarefaction curves coincide in the rst and the third families, and these are straight lines in planes with v = constant. The interaction pattern is constructed by prescribing initial data with four constant states l, m, M, and r (ordered from left to right). The states are chosen so that the Riemann problems (l m), (m M), and (M r) are solved by a single 2- shock, a single 1-wave, and a single 2-shock, respectively. At some later time the left 2-shock interacts with the 1-wave producing a transmitted 1-wave, a transmitted 2- shock, and a reected 3-wave. In turn this reected 3-wave interacts with the right 2-shock and gives a reected 1-wave, a transmitted 2-shock, and a transmitted 3-wave. The reected 1-wave then interacts with the left 2-shock, and so on. If we put v l = V > 0, v m = v M =0,andv r = ;V, then it follows by convexity of; that the 2-shocks will approach each other and meet in nite time t = T. Since each interaction yields a reected wave it follows that there are an innite number of interactions in nite time. The proof of the theorem is completed by suitably choosing V and a(v). To obtain the behavior described in part (a) of the theorem we let a(v) bealinear function. One can then show that the strength of the reected and transmitted waves in each interaction are larger than the strength of the incoming wave provided V is large enough. Since there is an innite number of interactions in nite time one thus get blowup in sup-norm by choosing V suciently large. To have the behavior described in part (b) we leta(v) be a quadratic function. For a suitable choice of V one can prove that the corresponding solution is periodic in state space. Again, since there is an innite number of interactions in nite time one conclude that while the sup-norm remains uniformly bounded the total variation of the solution tends to innity ast " T.We refer to [5] for the details of the computations. Acknowledgments. We thank A. Bressan for suggesting the problem of blowup for hyperbolic equations and for his kind interest. The rst author is indebted to A. Bressan, B. Piccoli, H. Holden, and N. H. Risebro for discussion on blowup for systems. References [1] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhauser, Boston, 1995.

10 302 Helge Kristian Jenssen and Carlo Sinestrari [2] A. Bressan, Blowup Asymptotics for the Reactive-Euler Gas Model, SIAM J. Math. Anal., 22 (1992), 587{601. [3] C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26 (1977), 1097{1119. [4] A. Jeffrey, Breakdown of the Solution to a Completely Exceptional System of Hyperbolic Equations, J. Math. Anal. Appl., 45 (1974), 375{381. [5] H. K. Jenssen, Blowup for Systems of Conservation Laws, preprint 1998, available at [6] H. K. Jenssen, C. Sinestrari Blowup asymptotics for scalar conservation laws with a source, submitted, available at [7] J. L. Joly, G. Metivier, J. Rauch, Resonant one-dimensional nonlineargeometric optics, J. Funct. Anal., 114 (1993), no. 1, 106{231. [8] J. L. Joly, G. Metivier, J. Rauch, A nonlinear instability for 3 3 systems of conservation laws, Comm. Math. Phys., 162 (1994), no. 1, 47{59. [9] A. Majda, R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves. I. A single space variable, Stud. Appl. Math., 71 (1984), no. 2, 149{179. [10] R. Natalini, C. Sinestrari, A. Tesei, Incomplete blow-up of solutions of quasilinear balance laws, Arch. Rat. Mech. Anal., 135 (1996), 259{296. [11] R. Natalini, A. Tesei, Blow{up of solutions for a class of balance laws, Comm. Partial Dierential Equations, 19 (1994), 417{453. [12] R. Young, Exact Solutions to Degenerate Conservation Laws, preprint Department of Mathematical Sciences, Norwegian University of Science and Technology, NTNU, N-7034 Trondheim, Norway address: helgekj@math.ntnu.no Dipartimento di Matematica, Universita di Roma \Tor Vergata", Via della Ricerca Scientica, Roma, Italy. address: sinestra@axp.mat.utovrm.it

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