A NOTE ON THE INITIAL-BOUNDARY VALUE PROBLEM FOR CONTINUITY EQUATIONS WITH ROUGH COEFFICIENTS. Gianluca Crippa. Carlotta Donadello. Laura V.

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1 Manuscript submitted to Website: AIMS Journals Volume, Number, Xxxx XXXX pp. A NOTE ON THE INITIAL-BOUNDARY VALUE PROBLEM FOR CONTINUITY EQUATIONS WITH ROUGH COEFFICIENTS Gianluca Crippa Departement Mathematik und Informatik, Universität Basel, CH-451 Basel, Switzerland Carlotta Donadello Laboratoire de Mathématiques, Université de Franche-Comté, F-253 Besançon Cedex, France Laura V. Spinolo IMATI-CNR, I-271 Pavia, Italy Abstract. In this note we establish the existence of solutions of initial-boundary value problems for continuity equations with low regularity coefficients. We also announce a uniqueness result and some related counterexamples. 1. Introduction. We are concerned with the continuity equation t u + div bu = cu + f, 1 where b : ], T [ R d, c :], T [ R and f :], T [ R are given functions and the unknown is u :], T [ R. Finally, R d is an open set and div denotes the divergence computed with respect to the space variable only. The investigation of 1 in the case when b has low regularity is the object of several recent research papers. Here we only quote the two milestones provided by the works by DiPerna and Lions [11] and by Ambrosio [1], which deal with the case when b enjoys Sobolev and BV regularity, respectively. We refer to the lecture notes by Ambrosio and Crippa [2] for a more extended bibliography. Both [11] and [1] establish existence and uniqueness results for the Cauchy problem obtained by coupling 1 with an initial datum in the case when = R d. We also point out that these results are motivated by applications to different classes of nonlinear PDEs, see the lecture notes by De Lellis [1] and the informal overview by Crippa and Spinolo [9] for the applications concerning systems of conservation laws. This note aims at establishing existence of solutions of the initial-boundary value problem for 1 under weak regularity assumptions on b. More precisely, if all 2 Mathematics Subject Classification. Primary: 35F16. Key words and phrases. Continuity equation, initial-boundary value problem, transport equation, low regularity coefficients, existence. 1

2 2 G. CRIPPA, C. DONADELLO AND L.V. SPINOLO functions were smooth up to the boundary then the formulation of our problem would read as follows: t u + divbu = cu + f in ], T [ u = ḡ on Γ 2 u = ū on {}, where ū and ḡ are bounded smooth functions and Γ is the subset of ], T [ where the characteristics are entering the domain ], T [. Note, however, that if b and u are not sufficiently regular if, for instance, u is only an L function, then their values on zero-measure sets are not well defined. However, in [8] see also 2 in here we introduce a distributional formulation of 2 and we consequently provide a definition of distributional solution, see Definition 2.2 below. This is done by relying on the theory of normal traces for low regularity vector fields, see the works by Anzellotti [4], Chen and Frid [6], Chen, Torres and Ziemer [7] and Ambrosio, Crippa and Maniglia [3]. The main result of this note reads as follows. Theorem 1.1. Let R d be an open set with uniformly Lipschitz continuous boundary. Assume that the following conditions hold: b L ], T [ ; R d and div b L ], T [ ; c L ], T [ and f L ], T [. Then for every ū L and ḡ L Γ there is a distributional solution of problem 2. Three remarks are here in order. First, we refer to the book by Leoni [12, Definition 12.1] for the definition of open set with uniformly Lipschitz continuous boundary. Second, the proof of Theorem 1.1 closely follows an argument due to Boyer [5]. The main novelties of Theorem 1.1 compared to the analysis in [5] are: i we replace the condition div b with div b L and ii we remove the assumptions that c and that is bounded. Finally, in [8] we prove that the solution of 2 is unique provided that b enjoys BV regularity up to the boundary of. We also discuss some examples showing that, if BV regularity is violated, then 2 admits, in general, infinitely many solutions. In particular, this happens even if b enjoys BV regularity in every open set compactly contained in but the BV regularity deteriorates at the boundary. This note is organized as follows: in 2 we provide the distributional formulation of problem 2 and in 3 we give the proof of Theorem Notation. L n : the n-dimensional Lebesgue measure. H m : the m-dimensional Hausdorff measure. div b: the distributional divergence of the vector field b :], T [ R d, computed with respect to the space variable only. ϕ : the gradient of the Sobolev function ϕ :], T [ R, computed with respect to the space variable only. L p := L p, H d 1. L p ], T [ := L p ], T [, L 1 H d 1. : the closure of the set R d.

3 A NOTE ON THE IBVP FOR CONTINUITY EQUATIONS 3 2. Distributional formulation of problem 2. We first observe that by interpreting the first and the last line of 2 in the sense of distributions we obtain T u t η + b η dxdt + ū η, dx + η Cc [, T [. T c u η + f η dxdt = The following results is proven in [8] and provides a distributional interpretation of 2 under the solely assumptions that b, div b, c and f are all bounded. Lemma 2.1. Let R d be an open set with uniformly Lipschitz continuous boundary. Assume that b L ], T [ ; R d satisfies div b L ], T [. Then the following implications hold: T i there is a unique function, which we denote by Tr b, that belongs to the space L ], T [ and satisfies, for every ϕ C c [, T [ R d, T T ϕ div b dxdt + t ϕ + b ϕ dxdt = ϕ Tr b dh d 1 dt ϕ, dx. 4 Also, the function Tr b satisfies the inequality Tr b L b L. ii Assume moreover that c, f L [, T [ and that u L ], T [ satisfies 3. Then there is a unique function, which we denote by Trbu, that belongs to L ], T [ and satisfies, for every ϕ Cc [, T [ R d, T u t ϕ+b ϕ dxdt + = T T c u + f ϕ dxdt ϕ Trbu dh d 1 dt ū ϕ, dx. Also, the function Trbu satisfies the inequality Trbu L b L u L. Some remarks are here in order. First, if the functions are smooth up to the boundary, then the Gauss-Green formula implies that Tr b = b n and Trbu = ub n, where n is the outward pointing unit normal vector to. Second, in [8] we exhibit an example where Tr b but Trbu 1. The vector field b in the example enjoys BV regularity in every open set compactly contained in, but the BV regularity deteriorates at the boundary. Finally, based on Lemma 2.1, we can rigorously define the sets Γ and Γ + by setting Γ := {t, x ], T [ : Tr b < }, Γ + := {t, x ], T [ : Tr b }. 6 We can now introduce the definition of distributional solution of 2. Definition 2.2. Let R d be an open set with uniformly Lipschitz continuous boundary and assume that b L ], T [ ; R d, div b L ], T [, c L ], T [ and f L ], T [. A distributional solution of problem 2 is a function u L ], T [ satisfying 3 such that the equality Trbu = ḡ Tr b holds on Γ. 3 5

4 4 G. CRIPPA, C. DONADELLO AND L.V. SPINOLO 3. Proof of Theorem 1.1. In the following, we establish the existence of functions u L ], T [ and β L Γ + such that, for every ϕ Cc [, T [ R d, T u t ϕ + b ϕ T dx + c u ϕ + f ϕ dxdt 7 = ϕ ḡ Tr b dh d 1 dt + ϕβ dh d 1 dt ū ϕ, dx. Γ Γ + By comparing the previous expression with 3 and 5 and by recalling Definition 2.2, we infer that u is a distributional solution of problem 2 and that { ḡ Tr b t, x Γ Trbut, x = β t, x Γ +. The proof of Theorem 1.1 is divided into three steps: in 3.1 we introduce a second order approximation and we state an existence result for the approximate problem. In 3.2 we establish a priori bounds on the family of approximate solutions and in 3.3 we pass to the limit and obtain a distributional solution of Second order approximation. We introduce a family of approximate problems, whose classical formulation is the following: t u ε + divbu ε = ε u ε + c u ε + f on ], T [ ε u ε n + u ε ḡ [ Tr b ] = on ], T [ 8 u ε = ū on {}. In the previous expression, ε > is a parameter, n as usual denotes the outward pointing unit normal vector to and [Tr b] is the negative part of the function Tr b. In this section we assume that ū, ḡ and f, besides being bounded, are also square integrable see the statement of Lemma 3.2. First, we provide the definition of weak solution of 8. To this end, we introduce the following notation: V : the Sobolev space W 1,2. V : the dual space of V, endowed with the standard dual norm. F, u : the duality between F V and u V. The bilinear form B ε t, : V V R is defined for L 1 -a.e. t ], T [ as B ε t, u, v := u bt, v dx + ε ut, v dx c ut, v dx + u v [ Tr b ] + t, dh d 1, where [ Tr b ] + denotes the positive part of the function Tr b. The functional Ft V is defined for L 1 -a.e. t ], T [ by setting Ft, v := vḡ [ Tr b ] t, dh d 1 + ft, v dx 1 Note that continuity of v Ft, v follows from the square integrability of f and from the fact that, under the regularity assumptions we impose on, the trace map is continuous V L 2, see for example Leoni [12, Theorem 15.23]. The following definition is classical, see for example Salsa [14, 9.3.1]. 9

5 A NOTE ON THE IBVP FOR CONTINUITY EQUATIONS 5 Definition 3.1. A weak solution of 8 is a function such that u ε : [, T ] V 1. u ε L 2 ], T [; V and u ε ], T [; V, where u ε denotes the distributional derivative of u ε. 2. For L 1 -a.e. t ], T [, 3. u ε = ū u ε t, v + B ε t, u ε t, v = Ft, v v V. 11 Note that requirement 3 above makes sense since by using requirement 1 we infer that u ε C [, T ]; L 2, see for example Salsa [14, Theorem 7.22]. Also, note that by the bold letter u ε we denote the function taking values in V, while u ε is the real-valued function u ε t, = u ε t. Remark 1. By relying on standard arguments, we get that any weak solution of 8 satisfies, for every ϕ Cc [, T [ R d, T u ε t ϕ + b ϕ T T dxdt ε u ε ϕ dxdt + c u ϕ + f ϕ dxdt = T ϕ, xū dx ḡϕ [ Tr b ] dh d 1 dt + T u ε ϕ [ Tr b ] + dh d 1 dt. The following lemma provides an existence and uniqueness result for 8. Lemma 3.2. Assume that b and c verify the same assumptions as in the statement of Theorem 1.1 and assume moreover that ḡ L Γ L 2 Γ, ū L L 2 and f L 2 ], T [ L ], T [. Then for any given ε > problem 8 admits a unique weak solution, in the sense of Definition 3.1. The proof of Lemma 3.2 follows by a classical Faedo-Galerkin method see for instance [14, 9.3.2] A priori estimates. In this section we establish the estimates we need to study the convergence ε + of the family u ε solving 8. Lemma 3.3. Let u ε be the weak solution of problem 8. Then u ε t 2 L 2 ū 2 L Γ + ḡ 2 Trb dh d 1 ds + f 2 2 L exp div b 2 L + 2 c L + 1t for every t ], T [. Also, T ε u ε t, x 2 dxdt C, 14 where C is a constant only depending on T, b L, divb L, ḡ L 2, ū L 2, c L and f L 2. Proof. First, we recall see e.g. [14, Theorem 7.22] that t u ε s, u ε s ds = 1 u ε t 2 L 2 2 u ε 2 L 2 t [, T ]. 15

6 6 G. CRIPPA, C. DONADELLO AND L.V. SPINOLO Next, we suitably choose the test functions in 4 and we use the density of the space Cc R d in W 1,1 and the continuity of the trace operator from W 1,1 onto L 1, obtaining that for L 1 -a.e. t ], T [ w div bt, dx+ bt, w dx = w Tr bt, dh d 1 w W 1,1. 16 We now choose v := u ε s as a test function in 11, we integrate over ], t[ and we use 16 with w = u 2 εs,. After straightforward computations, we arrive at t t u ε t 2 L + 2ε u 2 ε 2 dxds + u 2 ] +dh ε[ Tr b d 1 ds + t t = ū 2 L 2 + t u ε ḡ 2[ Tr b ] dh d 1 ds u 2 εdiv b dxds + 2 t ḡ 2[ Tr b ] dh d 1 ds + 2 t c u 2 εdxds f u ε dxds t ū 2 L + div b 2 L + 2 c L + 1 u ε s 2 L 2ds + t ḡ 2[ Tr b ] dh d 1 ds + f 2 L 2. Hence, by relying on the Gronwall Lemma we get 13 and then 14. We now establish a maximum principle Lemma 3.4. Let u ε be the weak solution of problem 8, then u ε L M + f L T exp div b L + c L T, 17 where Also, u ε [Tr b] + L M := max{ ḡ L, ū L }. 18 M + f L T b L exp div b L + c L T. 19 Proof. The argument is divided into four steps. Step 1: we introduce some preliminary notation and remarks. i We set B := div b L + c L. ii We define the function m ε by setting m ε t, x := [uε t, x + M + f L te Bt] 2. By a slight abuse of notation, in the following we denote by m ε the function 2 [ u ε t, x + M + f L te Bt]. Note that, for L 1 -a.e. t ], T [, m ε L 1 and m ε V = W 1,2 since [ uε + M + f L te Bt] max{ uε, }. In the following we also use the formula u ε m ε = u ε + M + f L te Bt m ε Me Bt m ε f L t e Bt m ε = 2m ε Me Bt m ε f L t e Bt m ε. 2

7 A NOTE ON THE IBVP FOR CONTINUITY EQUATIONS 7 iii We choose a sequence of smooth cut-off functions {ψ n } n N. More precisely, we require that, for every n N, ψ n Cc R d and ψ n x 1, ψ n x 1/n x R d and ψ n 1 on B n. In the previous expression, B n denotes the ball of radius n and center at in R d. iv Finally, we observe that for every n N and L 1 a.e. t ], T [ t u ε s + B M + f L s e Bs + f L e Bs, m εψ n ds [ū ] 2ψn = m ε t, xψ n x dx + M x dx. This formula can be established by relying on an approximation argument, see for example the analysis in the book by Lions and Magenes [13, Section 2.2]. Step 2: we use equation 11. First, we observe that by applying 16 with w = vu ε t, we infer that equation 11 implies u ε, v + u ε v div b dx + v b u ε dx + ε u ε v dx + u ε v[tr b] dh d 1 c u vdx = ḡv[tr b] dh d 1 + f v dx v V and L 1 a.e. t ], T [. Next, we fix n N and we apply 16 with w = m ε t, ψ n, obtaining m ε ψ n div b dx + m εψ n b u ε dx = m ε ψ n Tr b dh d 1 m ε b ψ n dx. 22 Hence, by plugging v := m εt, ψ n as a test function in 21 and by using 2 and 22 we obtain u ε, m εψ n + m ε ψ n div b dx Me Bt m εψ n div b dx f L t e Bt m ε ψ n div b dx + m ε ψ n [Tr b] + dx + 2ε χ ε u ε 2 ψ n dx 2m ε ψ n c dx + Me Bt m εψ n c dx + f L t e Bt m ε ψ n c dx + R n = ψ n [Tr b] m ε + m ḡ ε + M + f L te Bt u ε + M + f L te Bt dh d 1 + f m εψ n, where χ ε is the characteristic function of the set where u ε + M + f L te Bt and R n := m ε b ψ n dx + ε m ε u ε ψ n dx

8 8 G. CRIPPA, C. DONADELLO AND L.V. SPINOLO Step 3: we establish an estimate on the time integral of the first and third term in 23. We fix τ ], T [ and we integrate in time to get τ τ τ u ε, m εψ n dt m εme Bt ψ n div b dxdt f L t e Bt m εψ n div b dx dt τ τ τ + M e Bt m ε ψ n c dxdt + f L t e Bt m ε ψ n c dx f m εψ n dx τ = u ε + BMe Bt + f L t + f L e Bt, m εψ n dt τ m εψ n Me Bt B + div b c dxdt τ f L t e Bt B + div b cm εψ n τ τ f L e Bt m ε ψ n dx dt f m εψ n dx = m ε τ, xψ n x dx [ ū + M ] 2ψn dx τ m εψ n Me Bt B + div b c dxdt τ f L t e Bt B + div b c m ε ψ n dx dt τ f L e Bt + f m ε ψ n dx dt m ε τ, xψ n x dx. To get the last inequality, we have used that ū + M, B + div b c, f L e Bt + f and that m ε. Step 4: we conclude the proof of Lemma 3.4. First, we point out that the convexity of the function z [z] 2 implies that m ε +m ε ḡ + M + f L te Bt u ε + M + f L te Bt [ḡ + M + f L te Bt] 2 =. Next, we observe that m ε ψ n [Tr b] + dh d 1 + 2ε χ ε u ε 2 ψ n dx. Hence, by time integrating 23 and combining 24 with the above observations we obtain τ τ m ε τ, xψ n xdx + m ε ψ n div b 2c dxdt + R n dt. 25 Next, we observe that, by recalling estimates 13, 14 and the bounds on ψ n, we get τ lim R n dt. n + 24

9 A NOTE ON THE IBVP FOR CONTINUITY EQUATIONS 9 Hence, by letting n + in 25 and applying the Gronwall Lemma, we conclude that u ε τ, x + e Bτ M + f L τ for L d+1 -a.e. τ, x. By using an analogous argument, we obtain that u ε τ, x e Bτ M + f L τ for L d+1 -a.e. τ, x ], T [ and this concludes the proof of 17. Finally, by combining 17 with 23 we get Conclusion of the proof of Theorem 1.1. We proceed in two steps. Step A: we prove Theorem 1.1 under the additional assumptions ḡ L 2 Γ, ū L 2 and f L 2 ], T [. By applying Lemma 3.2 and recalling Remark 1, we deduce that, for every ε >, there is a function {u ε } satisfying 12. By relying on Lemma 3.4, we infer that the families {u ε } and {u ε [Tr b] + } are weakly- compact in L ], T [ and in L Γ +, respectively. By recalling the a priori estimate 14, we can pass to the limit in 12 and obtain a couple u, β satisfying 7. Step B: we remove the assumptions ḡ L 2 Γ, ū L 2 and f L 2 ], T [. Given a function ū L, we can construct a sequence {ū k } k N such that ū k L ū L ; ū k L 2 for every k; ū k ū for L d -a.e. x. 26 For example, we can take ū k := ū1 k, where { k } k N is a sequence of open bounded sets invading. We analogously construct sequences {ḡ k } k N and {f k } k N approaching ḡ and f, respectively. We term {u k, β k } k N the corresponding sequence of functions satisfying 7, constructed as in Step A. Note that by recalling 17, 19 and 26 we infer that the sequences {u k } k N and {β k } k N are weakly- compact in L ], T [ and in L Γ +, respectively. Hence, by extracting converging subsequences we conclude the proof of Theorem 1.1. Acknowledgments. The authors would like to thank Wladimir Neves for pointing out reference [5]. Donadello and Spinolo also thank the University of Basel for the kind hospitality during their visits. Crippa s research is partially supported by the Swiss National Science Foundation grant Donadello acknowledges partial support from the ANR grant CoToCoLa. Spinolo s participation to HYP212 was supported by the ERC Starting Grant ConsLaw P.I. Stefano Bianchini. REFERENCES [1] L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. Invent. Math., 1582:227 26, 24. [2] L. Ambrosio and G. Crippa. Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In Transport equations and multi-d hyperbolic conservation laws, volume 5 of Lect. Notes Unione Mat. Ital., pages Springer, Berlin, 28. [3] L. Ambrosio, G. Crippa, and S. Maniglia. Traces and fine properties of a BD class of vector fields and applications. Ann. Fac. Sci. Toulouse Math. 6, 144: , 25. [4] G. Anzellotti. Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 4, 135: , [5] F. Boyer. Trace theorems and spatial continuity properties for the solutions of the transport equation. Differential Integral Equations, 188: , 25. [6] G.-Q. Chen and H. Frid. Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal., 1472:89 118, [7] G.-Q. Chen, M. Torres, and W. P. Ziemer. Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Comm. Pure Appl. Math., 622:242 34, 29. [8] G. Crippa, C. Donadello, and L. V. Spinolo. Initial-boundary value problems for continuity equations with BV coefficients. Submitted. Also arxiv: /.

10 1 G. CRIPPA, C. DONADELLO AND L.V. SPINOLO [9] G. Crippa and L. V. Spinolo. An overview on some results concerning the transport equation and its applications to conservation laws. Commun. Pure Appl. Anal., 95: , 21. [1] C. De Lellis. Notes on hyperbolic systems of conservation laws and transport equations. In Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., pages Elsevier/North-Holland, Amsterdam, 27. [11] R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 983: , [12] G. Leoni. A first course in Sobolev spaces, volume 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 29. [13] J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. [14] S. Salsa. Partial differential equations in action. From modelling to theory. Universitext. Springer-Verlag Italia, Milan, 28. Received xxxx 2xx; revised xxxx 2xx. address: gianluca.crippa@unibas.ch address: Carlotta.Donadello@univ-fcomte.fr address: spinolo@imati.cnr.it