Propagation of discontinuities in solutions of First Order Partial Differential Equations

Transcription

1 Propagation of discontinuities in solutions of First Order Partial Differential Equations Phoolan Prasad Department of Mathematics Indian Institute of Science, Bangalore URL: prasad/ ABSTRACT: Certain type of singularities in a solution of a hyperbolic equation in general and a first order PDE in particular exist on a characteristic curve. For a linear first order PDE, all types of singularities in the solution propagate along characteristic curves. However, the discussion would require using concept of a weak solution in a more general sense than what we use here. In this section we shall restrict ourselves to a special class of weak solutions in domain D, which are continuous but have singularities along a curve Ω D. These singularities are discontinuities with finite jumps in the first derivatives for linear and quasilinear equations and in the second derivatives for nonlinear equations. 1 Existence and propagation of singularities on a characteristic curve We remark here that the coefficients of all transport equations 1.18), 1.26) and 1.27) are C 1 functions of σ. Hence their solutions exist locally in a neighbourhood of σ = Kinematical conditions on a curve of discontinuity Consider a smooth curve Ω : φx, y) = 0 in a domain D in x, y)-plane such that Ω divides D into two subdomains D and D + on two sides of Ω, i.e., D = D Ω D +. Remark 1.1. The operators φ x + φ x y and φ y y φ x x at a point on Ω represent y derivatives in normal and tangent directions with respect to the curve Ω. 1

2 Definition 1.2. Let DΩ m, where m is a non-negative integer, be the space of functions u : D R belonging to C m D\Ω) such that the limiting values of the partial derivatives of u of order up to m, as we approach Ω from the two sides, exist. Definition 1.3. For a smooth function f, except for a finite jump discontinuity on Ω, and with limiting values f and f + as we approach Ω from D and D + respectively, i.e., for f DΩ 0, the jump [f] across Ω is defined by [f] = f f ) Remark 1.4. If u DΩ 1 CD), jump in the tangential derivative of u on Ω is zero, i.e., [φ y u x φ x u y ] = 0 1.2) but, if there is a nonzero jump in any first derivative of u on Ω, the jump in the normal derivative is nonzero, i.e., [φ x u x + φ y u y ] ) A transversal derivative with respect to the curve Ω, say, i.e., a differentiation in ζ a direction other than the tangential direction is a linear combination of two operators in remark 1.1). Thus ) ) ζ = λ φ x x + φ y + µ φ y y x φ x, λ ) y Hence 1.3) implies that the jump in the derivative of u in a transversal direction, namely [ ] u ) ζ Problem Examine discontinuities in first order derivatives of u = x t. Theorem 1.5. If u DΩ 1 CD), then [u x ] = φ x Ω )ω 1, [u y ] = φ y Ω )ω 1, 1.6) where ω 1 : Ω R is a measure of the strength of the discontinuity in the first derivatives of u. Proof. This follows from 1.2). 2

3 Theorem 1.6. If u DΩ 2 C1 D), then [u xx ] = φ x Ω ) 2 ω 2, [u xy ] = φ x Ω φ y Ω ) ω 2, [u yy ] = φ y Ω ) 2 ω 2, 1.7) where ω 2 : Ω R is a measure of the strength of discontinuities in the second derivatives of u. Proof. Hint: The tangential derivatives of u x and u y on Ω are continuous. 1.2 Linear and semilinear equations Consider the PDE??), namely ax, y)u x + bx, y)u y = cx, y, u) 1.8) and a generalised or weak solution u of this equation in a domain D such that u DΩ 1 CD). Note we have not defined a generalised solution but for this particular equation let us accept that u DΩ 1 CD) is a generalised solution if u is a genuine solution in D 1 and D 2. Unlike the classical solution which must be a C 1 D) function, the first derivatives of a generalised solution may suffer finite jump across Ω : φx, y) = 0. Remark 1.7. Consider a generalised solution of 1.8). If ω 1, appearing in 1.6), vanishes then u is a genuine solution in D. Remark 1.8. If u D 2 Ω CD) and u satisfies 1.8) in D and D +, then also u is a generalised solution. Now we prove a theorem. Theorem 1.9. If Ω is a curve of discontinuity of the first order derivatives of the generalised solution u DΩ 1 CD) then Ω is a characteristic curve. Proof. Let us make a transformation of independent variables x, y) to φ, ψ), where ψ is another smooth function such that J := φ, ψ)/x, y) 0. The equation 1.8) transforms to aφ x + bφ y )u φ + aψ x + bψ y )u ψ = c. 1.9) ) Since /ψ) = 1/J) φ y φ x x, u y ψ is a derivative of u in the direction of tangent to Ω and hence it is continuous on Ω. The coefficients of u φ, u ψ and the function c are also continuous on Ω. Taking jump of the equation 1.9) across Ω, we get aφ x + bφ y ) Ω [u φ ] = ) 3

4 ) Since /φ) = 1/J) ψ y ψ x x, u y φ is a derivative in the direction of tangent to the curve ψ =constant and hence a transversal derivative of u with respect to the curve Ω. Hence [u φ ] 0 and from the equation 1.10) we get aφ x + bφ y = 0 on Ω. 1.11) 1.11) implies that the tangent direction of Ω : φx, y) = 0 satisfies dy dx = b a. Hence the curve Ω is a characteristic curve. Example is a solution of gx, y) = { x + y, x > 0 x + y) 2, x < ) u x u y = ) in any domain in x, y)-plane which does not intersect the y-axis. But note that this function is discontinuous on on a non-characteristic line x = 0 and hence is not a generalised solution in R 2. A generalised solution of 1.13) is { x + y, x + y > 0 hx, y) = 1.14) x + y) 2, x + y < 0. Note that the first derivatives of h are not continuous on Ω, which is a characteristic curve. Here, [h x ] = 1 = [h y ] on Ω. For a generalised solution u, the curve Ω of discontinuity of the first derivatives of u is a characteristic curve, which is given by the equation dx dσ = ax, y), dy dσ = bx, y) 1.15) Since both vectors a, b) and φ y, φ x ) are in tangent direction of Ω, we can replace the relation 1.6) by [u x ] = b Ω W 1, [u y ] = a Ω W ) where W 1 = ϕ y ) a Ωω 1, a function of σ, is a new measure of the strength of the discontinuity in the first derivatives of u. Theorem If the strength of the discontinuity in the first derivatives of a generalised solution u DΩ 2 CD) of 1.8) is known at one point of the curve of discontinuity Ω, then it is uniquely determined at all points of Ω. Remark In the proof of this theorem we need the continuity of the second derivatives of u in D 1 and D 2 in order to use u yx = u xy in both subdomains. 4

5 Proof. Differentiating 1.8) with respect to x and then taking jump across Ω, we get a x + b ) [u x ] = c u a x )[u x ] b x [u y ]. 1.17) y Substituting 1.16) in this we get b dw 1 dσ = { bc u a x ) db dσ + ab x } W 1 on Ω 1.18) where d = a + b and a, b, a dσ x y x, b x, c x and c u are functions of σ on Ω. This equation contains the variable u. The variation of u along the characteristic curve is given by??). Let us assume that the amplitude W 10 of the discontinuity in derivative is known at a point x 0, y 0 ), where the value of the solution is u 0. Equations 1.15) give the characteristic Ω through x 0, y 0 ). Then equation??) gives the value of the solution on Ω. Finally the transport equation 1.18) gives the value of W 1 on Ω and shows that if W 1 is known at some point of the characteristic curve Ω, it is determined at all points of Ω. All these results are local. Remark For a linear PDE, the characteristic curves carry the values of u, [u x ] and [u y ] throughout the domain where characteristics are defined. For a semilinear equation, since the compatibility condition du = cx, y, u) is nonlinear the solution dσ may blow up even if characteristics are well defined. Then, of course, u x and u y are not defined. It is simple to verify this statement in 4iii) in Problem set Quasilinear equations Consider the PDE??), namely ax, y, u)u x + bx, y, u)u y = cx, y, u). 1.19) Take a generalised solution ux, y) DΩ 1 CD). Since u CD), for this known solution, the functions Ax, y) = ax, y, ux, y)), Bx, y) = bx, y, ux, y)), Cx, y) = cx, y, ux, y)) 1.20) are continuous functions in D. As in the case of semilinear equations, we can prove the following theorem. Theorem The curve Ω of discontinuity of the first derivatives of a generalised solution u DΩ 1 CD) of 1.19) is a characteristic curve given by dx dσ = ax, y, ux, y)), dy dσ 5 = bx, y, ux, y)) 1.21)

6 Proof. The proof is similar to that of the theorem 1.11). We could proceed to derive the transport equation for the measure W 1 of the strength of the discontinuity satisfying 1.16) but we leave it as a simple exercise. Instead we derive transport equation for [u x ] assuming as explained in 1.8) that u DΩ 2 CD). We differentiate 1.19) with respect to x to get a x + b ) u x = c x + c u u x a x u x a u u x ) 2 b x u y b u u x u y ). 1.22) y Taking jump of this equation across Ω, we get d[u x ] dσ = c u a x )[u x ] b x [u y ] a u [u x ) 2 ] b u [u x u y ]. 1.23) Now two new terms appear, which we deal as follow [u x ) 2 ] = u 2 x u 2 x+ = 2u x+ + [u x ])[u x ], 1.24) [u x u y ] = u x u y u x+ u y+ = [u x ][u y ] + u x+ [u y ] + u y+ [u x ]. 1.25) Substituting 1.24) and 1.25) in 1.23) and using [u y ] = a b Ω[u x ], we get { d[u x ] dσ = c u + ab } x ba x bu y+ au x+ b u 2a u u x+ [u x ] + ab u ba u [u x ] ) b b b which is the required transport equation. This is again an ODE for variation of the jump [u x ] but involves the variables u, u x+ and u y+. Only one of u x+ and u y+ is independent since they satisfy the PDE 1.19). The equation for u is the compatibility condition??). We can easily derive the equations for u x+ from 1.22) in the form du x+ dσ = c x + c u a x )u x+ a u u x+ ) 2 b x u y+ b u u x+ u y+ ). 1.27) In addition to the value u 0 and strength [u x ] 0 of the discontinuity at the point x 0, y 0 ), we also need value of u x+ ) 0. The characteristic equations??), the compatibility condition??) give the characteristic curve Ω and the value of u on it. The transport equation 1.27) gives u x+ on Ω. Finally the transport equation 1.26) gives the strength of the discontinuity [u x ] on Ω. Therefore, if the strength [u x ] 0 of the initial discontinuity is given at a point x 0, y 0 ), it is uniquely determined at all points of the characteristic. As in case of a semilinear equation, the the theorem 1.10) remains true also for a quasilinear equation. Since the equation 1.26) is nonlinear, the initial discontinuity [u x ] of finite value may tend to infinity in finite time t c, say, and at that time the continuity of the 6

7 solution u CD) breaks down. The limiting values of the solution u from two sides of Ω do remain finite and a shock appears in the solution at t c and at the place, where [u x ]. We do not wish to spend much time on the equation 1.26) but an example of its application to Burgers equation is given in [8], see also [9]. The phenomena of break down of continuity of the solution and appearance of a shock is very important in the theory of hyperbolic conservation laws with genuine nonlinearity. It now forms an important part of a basic course in PDE. 1.4 Nonlinear PDE For a nonlinear PDE??), the expressions F p and F q on the right hand sides of the characteristic equations??) contain the partial derivatives u x and u y. Hence a curve of discontinuity in u x and u y can not coincide with the characteristic curves on both sides of it. Analysis of such curves of discontinuities will be for more complicated. Hence we consider a genuine solution u D 2 Ω C1 D) of??), for which we assume that the second order derivatives are discontinuous, i.e, ω 2 in 1.7) is nonzero. We can easily prove that Ω is a characteristic of??). For the derivation of the transport equation of the discontinuities in second order derivatives, we shall have to take u D 3 Ω C1 D) as explained in 1.8)) and differentiate??) or??) in the characteristic direction, use 1.7) and then derive the transport equation for ω 2 or one of the second derivatives. The resultant equation will be quite long. It will be nice to derive this transport equation for a particular equation such as u 2 x + u y + u = 0. We leave it as an exercise to the reader to deduce d[u xx ] dy = 1 + 4u xx+ )[u xx ] 2[u xx ] ) along the characteristic curves given by the Charpit s equations dx dy = 2u x, du dy = 2u x) 2, du x dy = u x, du y dy = u y. 1.29) The evolution of u xx+ along the characteristic curve is given by du xx+ dy = u xx+ 2u 2 xx ) For a nonlinear equation in general, and for the equation u 2 x +u y +u = 0 in particular, there is a relation between u, u x and u y and hence only two, say u and u x, can be prescribed at any point x 0, y 0 ). Now the four equations in 1.29) are to be solved simultaneously with initial data u 0, u x ) 0, u y ) 0 at a point x 0, y 0 ). This will give the characteristic curve Ω through x 0, y 0 ) and the values of u, u x and u y on it. Then, if the strength [u xx+ ] 0 of the discontinuity is prescribed at x 0, y 0 ), we can solve 1.30) to get the value of [u xx+ ] on Ω. 7

8 Therefore, if the strength [u xx ] 0, of the initial discontinuity is given at a point x 0, y 0 ), it exist along the characteristic curve through the point x 0, y 0 ) and is uniquely determined at all points of the characteristics in a neibourhood of x 0, y 0 ). References [1] R. Courant and D. Hilbert. Methods of Mathematical Physics, vol 2: Partial Differential Equations. Interscience Publishers, [2] S. S. Demidov. The Study of Partial Differential Equations of the First Order in the 18 th and 19 th Centuries. Archive for History of Exact Sceinces, 1982, 26, [3] L. C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol 19, American Mathematical Society, [4] P. R. Garabedian. Partial Differential Equations. John Wiley and Sons, [5] E. Goursat. A Course in Mathematical Analysis, Vol II, Part Two: Partial Differential Equations. Dover Publication of the book published in [6] F. John. Partial Differential Equations. Springer-Verlag, [7] L. Perko. Differential Equations and Dynamical Systems. Springer, [8] Phoolan Prasad. A theory of first order PDE through propagation of discontinuities. Ramanujan Mathematical Society News Letter, 2000, 10, [9] Phoolan Prasad. Nonlinear Hyperbolic Waves in Multi-dimensions. Monographs and Surveys in Pure and Applied Mathematics, Chapman and Hall/CRC, 121, [10] Phoolan Prasad and Renuka Ravindran. Partial Differential Equations. Wiley Eastern Ltd, [11] I. N. Sneddon. Elements of Partial Differential Equations., McGraw-Hill,