The pseudospectrum of the convention-diffusion operator with a variable reaction term

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In this paper, we study the spectrum of non-self-adjoint convection-diffusion operator with a variable reaction term defined on an unbounded open set Ω of Rn . Our idea is to build a family of operators that have the same convection-diffusion-reaction formula, but which will be defined on bounded open sets { Ω η } η ∈]0,1[ of Rn . Based on the relationships that link this family to Ω , we obtain relations between the spectrum and the pseudospectrum. We use the notion of the pseudospectrum to build relationships between convection-diffusion operator and its restrictions to bounded domains. Using these relationships we are able to find the spectrum of our operator in R+ . Also, the techniques developed to obtain the spectrum allow us to study the properties of the spectrum of this operator when we go to the limit as the reaction term tends to zero. Indeed, we show a spectral localization result for the same convection-diffusion-reaction operator when a perturbation is carried on the reaction term and no longer on the definition domain.

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Introduction The study of the spectrum of convection-diffusion operator is one of the most complicated problems in functional analysis. In this paper, we study the spectrum of the following operator: Au = 4u + rh( Yn i=1 xi) ru + V u; (0.1) where h 2 C2 (R;R) such that h00 is positive and V = V1 + V2; V1(x) = Xn i=1 h0( Yn j=1 xj) Yn j=1 j6=i xi 2 : Non-negative potential V2 is considered as a reaction to the convection-diffusion phenomena represented by A0 = + rh( Yn i=1 xi) r + V1: The operator A in the case h(x) = x was studied in [1]. We recall that, for an unbounded operator T defined on D(T) H to H and for " > 0; the pseudospectrum is given by (see [2]) sp"(T) = z 2 re(T) : (zI T) 1 > " 1 [ sp(T): The resolvent set is given by re(T) = z 2 C : (zI T) 1 exists and bounded ; sp(T) denotes the spectrum of T and is defined as sp(T) = C n re(T): An equivalent definition of the pseudospectrum has been given in [3]: sp" (T) = [ D:H!H; linear and kDk<" sp (T + D) : 356 H. Guebbai, S. Segni, M. Ghiat, W. Merchela Pseudospectrum is easier to calculate and more efficient than spectrum when dealing with unbounded operators [4, 5]. In fact, it has been established that an approximation of the spectrum of differential operators may be unstable when going to the limit, unlike the pseudospectrum which shows to be stable (see [6-8]). For example, if T is a normal operator, its pseudospectrum is equal to the " -neighborhood of its spectrum. The " -neighborhood of S C is given by N" (S) = fs + z : s 2 S; jzj < "g : It is clear that, for all S C; \\ ">0 N" (S) = S: Moreover we take advantage of the fact that the spectrum of an operator is divided into two sets: the pointwise spectrum, spp (T) ; which consists of all the eigenvalues of T ; the essential spectrum, spess (T) which consists of all 2 C such that the operator ( I T) is injective, but not surjective. In addition, we define the limit of a sequence of sets as follows: for all > 0; S C; lim !0 S = s 2 C : 9 fs g >0 ; s 2 S ; lim !0 s = s : In this article, we study in detail the pseudospectrum and the spectrum of the operator A to establish that its spectrum is real positive. We conclude this work with a result on the stability of the spectrum obtained by the pseudospectral theory. 1. Convection-diffusion operator Let ( Rn be an unbounded open set. Let h 2 C2(R;R) be such that h00 is positive. Let A be the convection-diffusion operator (see [9]) defined on L2( ;C) into itself by (0.1). We define the hermitian form ' on L2( ;C) as '(f; g) = Z rf rgdx + Z rh( Yn i=1 xi) - rfgdx + Z V fgdx; where the quadratic form associated with ' is given by Q(u) = kruk2 L2( ) + Z rh( Yn i=1 xi) - ruudx + Z V juj2dx: For u 2 C1 c ( ); we set z = R ( rh( Qn i=1 xi)) u rudx; so z = Z @ rh( Yn i=1 xi) - juj2dx | {z } =0 + Z h( Yn i=1 xi) - juj2dx + Z rh( Yn i=1 xi) - ruudx = Z h( Yn i=1 xi) - juj2dx z: Using z + z = 2Re(z) = R h( Qn i=1 xi) juj2dx yields Re(Q(u)) = kruk2 L2( ) + Z 1 2 h( Yn i=1 xi) + krh( Yn i=1 xi)k2 L2( ) + V2(x) - juj2dx 0; THE PSEUDOSPECTRUM OF THE CONVENTION-DIFFUSION OPERATOR 357 and jIm(Q(u))j = Im Z rh( Yn i=1 xi) - u rudx - = z Re(z) i 1 2 kruk2 L2( ) + Z krh( Yn i=1 xi)k2 L2( )juj2dx - + 1 2 Z h( Yn i=1 xi) - juj2dx : Hence, ' is a sectorial form defined on the vector space R given by the following expression: R = H1 0( ;C) \\ u 2 L2( ;C) : V u 2 L2( ;C) : We recall that A is the operator associated with ' [10, Theorem 2.1, p. 322], and the domain of A is given by D(A) = H2( ;C) T R: Our goal is to determine the spectrum of A: We notice that D(A) is a Dirichlet integral boundary condition. Consider the eigenvalue problem: find 2 C and u 2 D(A) n f0g such that u + rh( Yn i=1 xi) - ru + V u = u on ; u = 0 on @ : Let f -g-2]0;1[ be a sequence of bounded open sets such that - -; for all - -; and S -2]0;1[ - = : For all - 2]0; 1[; we identify the hermitian form '- on L2( -;C) by '-(f; g) = Z - rf rgdx + Z - rh( Yn i=1 xi) ! rfgdx + Z - V fgdx: We recall that '- is a sectorial form defined on R- = H1 0( -;C): We denote by A- the differential operator associated with '- [10, Theorem 2.1, p. 322]. The domain of A- is given by D(A-) = H2( -;C) T H1 0( -;C): We notice that A- is defined by the same formula as A: Let B- be the differential operator which is defined by the same formula as A; but is given on D(B-) = H2 0( -;C): The extension of each function in D(B-0) to - by zero belongs to D(B-) [11, Lemma 3.22, p. 57]. Thus D(B-0) D(B-): To achieve our goal we will define the spectrum of A-; after that we will establish a relation between the pseudospectrum and spectrum of A-; B- and A: 2. Spectrum of A- and B- In this section, we will explain some characteristics of the operator A- which allow us to locate the spectrum of A: 2.1. Spectrum of A- Define the following scalar product on L2( -) : 8(u; v) 2 L2( -) - L2( -); hu; vi- = Z - exp h( Yn i=1 xi) - uvdx: Theorem 2.1. For all - 2]0; 1[; A- is self-adjoint with respect to h ; i-: 358 H. Guebbai, S. Segni, M. Ghiat, W. Merchela P r o o f. For all u 2 D(A-); we define eu = exp 1 2 h( Yn i=1 xi) - u: Hence, we get eu = u + rh( Yn i=1 xi)) ru - + 1 4 krh( Yn i=1 xi)k2 L2( -)u - exp 1 2 h( Yn i=1 xi) - : Then, for all (u; v) 2 D(A-) - D(A-); hA-u; vi- = Z - exp h( Yn i=1 xi) - A-uvdx = Z - u + rh( Yn i=1 xi) ru + + krh( Yn i=1 xi)k2 L2( -) + V2(x) - u - exp 1 2 h( Yn i=1 xi) ! evdx = Z - euevdx + Z - 5 4 krh( Yn i=1 xi)k2 L2( -) + V2(x) - euevdx = Z - reurevdx Z @ - @eu @ |{vez} =0 d + Z - 5 4 krh( Yn i=1 xi)k2 L2( -) + V2(x) - euevdx: So, for all - 2]0; 1[; hA- ; i- is also a scalar product, which means that A- is self-adjoint [10, Theorem 2.7]. As a consequence, sp(A-) is real for all - 2]0; 1[: Since we cannot extend the scalar product h ; i- over L2( ); we cannot guarantee that A is self-adjoint. We define K = inf 1 2 h( Yn i=1 xi) + krh( Yn i=1 xi)k2 L2 - + V2(x) : (x1; ::::; xn) 2 - ; K1 = inf krh( Yn i=1 xi)k2 L2 - : (x1; ::::; xn) 2 - ; M = C 2 PF + K 5K1 4 ; E = f- 2 ]0; 1] : M 0g ; where CPF is the Poincarre-Friedrichs constant (see [10]). Theorem 2.2. For all - 2]0; 1[; If - =2 E; the essential spectrum spess(A-) is included in ] 5 4K1;C 2 PF + K[; and the point spectrum spp(A-) is included in [C 2 PF + K;+1[; If - 2 E; A- has no essential spectrum, and the point spectrum is included in [C 2 PF + K;+1[: P r o o f. For all - 2]0; 1[ and all u 2 D(A-); Re(hA-u; ui) = 1 2 (hA-u; ui + hu;A-ui) = 1 2 (hA-u; ui + hu;A-ui): THE PSEUDOSPECTRUM OF THE CONVENTION-DIFFUSION OPERATOR 359 However Z - uudx = Z @ - @u @ |{uz} =0 d + Z - rurudx = Z - jruj2dx: Z - rh( Yn i=1 xi) ruudx = Z - rh( Yn i=1 xi) ruudx Z - h( Yn i=1 xi) juj2 dx: Then Re(hA-u; ui) = kruk2 L2( -) + Z - 1 2 h( Yn i=1 xi) + krh( Yn i=1 xi)k2 L2( -) + V2(x) ! juj2dx kruk2 L2( -) + Kkuk2 L2( -): By the theorem of Poincarre-Friedrich (see [10]), Re(hA-u; ui) C 2 PF kuk2 L2( -) + Kkuk2 L2( -) (C 2 PF + K)kuk2 L2( -): Thus, for all 2 R; k(A- I)ukL2( -) (C 2 PF + K )kukL2( -): Hence (A- I) is injective for all < C 2 PF + K; and thus sp(A-) is included in [C 2 PF +K;+1[: Let H = H1 0( -) and 2 [ 1; 5 4K1[: The sesquilinear form is defined on H by ' (u; v) = Z - (rurv + ( 5 4 V1 + V2 )uv)dx; which verifies j' (u; v)j krukL2( -)krvkL2( -) + CkukL2( -)kvkL2( -); where C = sup 5 4 V1 (x) + V2 (x) : x 2 - + j j; and j' (u; v)j min 1; ( 5 4 K1 ) kuk2 H: Since, for all g 2 L2( -); the semilinear form L : H ! C; v 7! R - gvdx is continuous, it follows from the Lax-Miligram theorem that the equation ' (u; v) = L(v) has a unique solution u in H for all v 2 H: Take into consideration the problem (p) 8< : for g 2 L2( -), find u 2 L2( -) such that A-u u = g on -, u = 0 on @ -. 360 H. Guebbai, S. Segni, M. Ghiat, W. Merchela We use the same variable change as in the previous theorem: we multiply the equation by exp(h( Qn i=1 xi) 2 ) and set eg = g exp(h( Qn i=1 xi) 2 ); eu = u exp(h( Qn i=1 xi) 2 ): We see that (p) is equivalent to (ep) 8< : for eg 2 L2( -), find eu 2 L2( -) such that eu + ( 5 4V1 + V2)eu eu = eg on -, eu = 0 on @ -. Therefore, the sesquilinear form ' (u; v) = R - rurv + ( 5 4K1 )uv) is an inner product in L2( -) for < 5 4K1: Hence, (ep) has a unique solution eu; and (p) has a unique solution u defined by u = eu exp h Qn i=1 xi : 2.2. Relation between A- and B- It is known that, for every - 2]0; 1[; B- A-; i. e. D(B-) D(A-) and, for all f 2 D(B-); B-f = A-f: Furthermore, for every - 2]0; 1[; B- A: In fact, for any f 2 D(B-) = H2 0( -); extending f to by 0 gives f 2 D(A): This proves that [ 0<-<1 spp(B-) spp(A-): To determine the spectrum of A the study of the difference spp(A-)nspp(B-) is required. In this section, we prove that this difference is empty for - 2]0; 1[: Lemma 2.1. For all " > 0 and for all - 2]0; 1[; sp"(B-) = sp"(A-): P r o o f. Let 2 sp"(B-): Then there exists f 2 D(B-) such that kB-f fkL2( -) kfkL2( -) < ": But f 2 D(A-); so kA-f fkL2( -) kfkL2( -) < "; and 2 sp"(A-): Inversely, let 2 sp"(A-): Then there is f 2 D(A-) such that kA-f fkL2( -) kfkL2( -) < ": Since the space of infinitely differentiable functions with compact support C1 c ( -) is dense in D(A) with respect to the graph norm defined by k kA = kA kL2( ) + k kL2( ) ; for all THE PSEUDOSPECTRUM OF THE CONVENTION-DIFFUSION OPERATOR 361 f 2 D(A-); there is a sequence (fn)n2N in C1 c ( -) such that limn!+1 kfn fkA = 0: So, for all > 0; there exists N 2 N such that, for all n N; we have kA-fn fnkL2( ) kfnkL2( ) kA-f fkL2( ) kfkL2( ) < : We set = " kA-f fkL2( -) kfkL2( -) > 0: Then there exists n0 2 N such that kA-fn0 fn0kL2( ) kfn0kL2( ) < ": However fn0 2 D(B-); then 2 sp"(Bn): Corollary 2.1. For all " > 0; if 0 < - -0 < 1; then sp"(A-0) sp"(A-): P r o o f. Let 2 sp"(B-0): Then there exists f 2 D(B-0) such that kB-0f fkL2( -0 ) kfkL2( -0 ) < ": Extending f to - by 0; kB-f fkL2( -) kfkL2( -) < ": It follows that belongs to sp"(B-): Now we can apply Lemma 2.1 to complete the proof. We proved that the family fsp"(A-)g0<-<1 is decreasing, and this makes us to say that the family fsp(A-)g0<-<1 is decreasing with respect to inclusion. In fact, for all 0 < - -0 < 1; sp(A-0) sp"(A-0) sp"(A-): But A- is self-adjoint, i. e. sp"(A-) = N" (sp(A-)) ; for all - 2]0; 1[: Then sp(A-0) \\ ">0 N" (sp(A-)) = sp(A-): Theorem 2.3. For all - 2 E; sp(B-) = sp(A-): P r o o f. Since A- is self-adjoint for all - 2 E; we have \\ ">0 sp"(A-) = sp(A-): 362 H. Guebbai, S. Segni, M. Ghiat, W. Merchela Then, by using Lemma 2.1, we find sp(B-) sp"(B-) = sp"(A-) ) sp(B-) \\ ">0 sp"(A-) = sp(A-): Reciprocally, let 2 sp(A-) for some - 2 E: According to Theorem 2.2, there exists f 2 D(A); f 6= 0 such that A-f = f: Since C1 c ( -) is dense in D(A-) with respect to the graph norm, there is a sequence (fn)n2N in C1 c ( -) which converges to f in the graph norm. We define the sequence gn = fn kfnkL2( -) ; n 2 N: For all n 2 N; gn 2 D(B-); kgnkL2( -) = 1; and lim n!+1 kB-gn gnkL2( -) = kA-f fkL2( -) kfkL2( -) = 0: Then 2 sp(B-): In fact, if (B- I) 1 exists and is bounded, then 1 = kgnkL2( -) (B- I) 1 kB-gn gnkL2( -) ! 0: Therefor, the operator (B- I) 1; if it exists, can not be bounded, which means that B- I can not be surjective. 3. Pseudospectrum and spectrum of A The pseudospectrum has better stability than the spectrum. Pseudospectrum is easier to be controlled and can be considered as the finest stable for the passage to the limit. 3.1. Pseudospectrum In this subsection, we establish a relation between the spectrum and the pseudospectrum of A seen as limits of sp (A-) and sp (B-) respectively. Theorem 3.1. For all " > 0; sp"(A) = [ -2E sp"(A-) = [ -2E sp"(B-): P r o o f. Let 2 S 0<-<1 sp"(B-): Then there exist -1 2 E and f 2 D(B-1) such that kB-1f fkL2( -1 ) kfkL2( -) < ": THE PSEUDOSPECTRUM OF THE CONVENTION-DIFFUSION OPERATOR 363 Extending f to by 0, kAf fkL2( ) kfkL2( ) < ": So, it follows that belongs to sp"(A); and [ -2E sp"(B-) sp"(A): Reciprocally, let 2 sp"(A): Then there is f 2 D(A) such that kAf fkL2( ) kfkL2( ) < ": Since C1 c ( ) is dense in D(A) with respect to the graph norm, for all f 2 D(A); there is a sequence (fn)n2N in C1 c ( ) such that lim n!+1 kAfn fnkL2( ) kfnkL2( ) = kAf fkL2( ) kfkL2( ) : Like in the proof of Lemma 2.1, we choose n0 such that kAfn0 fn0kL2( ) kfn0kL2( ) < ": There is - small enough for which the support of g = fn0 is included in -: It follows that belongs to sp"(A-): Thus, sp"(A) [ -2E sp"(A-): Now, we use Lemma 2.1 to conclude the proof. From the previous theorem, we deduce that sp"(A) N"(R+) = fz 2 C : Rez > 0; jImzj < "g [ fz 2 C : Rez < 0; jzj < "g : In fact, for all - 2 E; A- is self-adjoint. Then sp"(A-) = N"(sp(A-)): But, sp(A-) R+: We obtain [ -2E sp(A-) R+: 3.2. Spectra In this part, we will set a new relation between the spectrum of A and that of A-: First, we begin with a topological result that will allow us to obtain the desired property. P r o p o s i t i o n 3.1. For all " > 0; [ -2E sp"(A-) = N" [ -2E sp(A-) ! : 364 H. Guebbai, S. Segni, M. Ghiat, W. Merchela P r o o f. Let 2 S -2E sp"(A-): There is -1 2 E such that 2 sp"(A-1) = N" (sp(A-1)) : So, = s + z; where s 2 sp(A-1) and jzj < ": But s 2 S -2E sp(A-) implies 2 N" [ -2E sp(A-) ! : Reciprocally, let 2 N" S -2E sp(A-) ! : Then = s+z; where s 2 S -2E sp(A-) and jzj < ": So, there is -1 2 E such that = s + z 2 N"(sp(A-1)) = sp"(A-1): Thus, 2 S -2E sp"(A-): Theorem 3.2. sp(A) = [ -2E sp(A-): P r o o f. Let 2 S -2E sp(A-): There is -1 2 E such that 2 sp(A-1): Then there is f 2 D(A-1); where A-1f f = 0: Since C1 c ( -1) is dense in D(A-1) with respect to the graph norm, there is a sequence (fn)n2N in C1 c ( -1) which converges to f in the graph norm. We define the sequence gn = ( fn kfnkL2( -) on -1 ; n 2 N; 0 on = -1 ; n 2 N: We have, for all n 2 N; gn 2 D(A); kgnkL2( -) = 1; then lim n!+1 kAgn gnkL2( -) = 0: Thus 2 sp(A): By Theorem 3.1, we have sp(A) sp"(A) = [ -2E sp"(A-); and by Proposition 3.1, sp(A) N" [ -2E sp(A-) ! : So, we obtain sp(A) [ -2E sp(A-) as " tends to 0. THE PSEUDOSPECTRUM OF THE CONVENTION-DIFFUSION OPERATOR 365 4. Formula perturbation This section is devoted to the study of pseudospectrum stability when the perturbation is applied directly to the operator’s formula. For this purpose, we define for > 0; A = A0 + V3; where V3 is a continuous function over such that K2 = sup x2 jV3(x)j < 1;K3 = inf x2 jV3(x)j ; which means that, for all > 0; D(A ) = D(A0) : Our aim is to compare sp (A ) and sp (A0) : For - > 0; A ;-; A0;- are defined in the same way as A-: It is clear, that for " > 0; 8 0; sp" (A ) = [ -2E sp"(A ;-); where E = - 2 ]0; 1] : C 2 PF + K 5K1 4 K3 0 ; and 8 0; sp (A ) = [ -2E sp(A ;-): Theorem 4.1. lim !0 sp (A ) sp (A0) lim !0 sp K2 (A ) : P r o o f. Let " > 0; - > 0 and 2 sp" (A0;-) : Then there exists f 2 D(A0;-) such that kA0;-f fkL2( -) < " kfkL2( -) : Therefore, for > 0; kA ;-f fkL2( -) < (" + K2) kfkL2( -) ; which means that 2 sp"+ K2 (A ;-) : However, A0;- and A ;- are self-adjoint operators, then sp (A0;-) sp K2 (A ;-) ) sp (A0) [ -2E0 sp K2 (A ;-) : We use the fact that E0 E for all > 0 to get sp (A0) sp K2 (A ) ) sp (A0) lim !0 sp K2 (A ) : Inversely, it is clear that, for all > 0 and all - > 0; sp (A ;-) sp K2 (A0;-) : 366 H. Guebbai, S. Segni, M. Ghiat, W. Merchela Then sp (A ) = [ -2E sp (A ;-) [ -2E sp K2 (A0;-) ) lim !0 sp (A ) lim !0 [ -2E sp K2 (A0;-) ; but [ -2E sp K2 (A0;-) = N K2 [ -2E sp (A0;-) ! : We use lim !0 E = E0 to get lim !0 sp (A ) sp (A0) : Acknowledgements. We are very grateful to the editor and reviewer for their remarks proposed to improve our paper. We thank Mr. Ammar Khellaf for his effort and help.
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About the authors

Hamza Guebbai

Universit´e 8 Mai 1945

Email: guebaihamza@yahoo.fr; guebbai.hamza@univ-guelma.dz
Associate Professor of Mathematics Department B.P. 401, Guelma, Alg´erie

Sami Segni

Universit´e 8 Mai 1945

Email: segnianis@gmail.com; segni.sami@univ-guelma.dz
Associate Professor of Mathematics Department B.P. 401, Guelma, Alg´erie

Mourad Ghiat

Universit´e 8 Mai 1945

Email: mourad.ghi24@gmail.com; ghiat.mourad@univ-guelma.dz
Associate Professor of Mathematics Department B.P. 401, Guelma, Alg´erie

Wassim Merchela

Derzhavin Tambov State University

Email: merchela.wassim@gmail.com
PhD Student of Mathematics 33 Internatsionalnaya St., Tambov 392000, Russian Federation

References

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  2. E. Shargorodsky, “On the definition of pseudospectra”, Bull. London Math. Soc., 41:2 (2009), 524-534.
  3. S. Roch, B. Silbermann, “C*-algebra techniques in numerical analysis”, J. Operator Theory, 35 (1996), 241-280.
  4. L.N. Trefethen, Pseudospectra of matrices, Longman Sci. Tech. Publ., Harlow, 1992.
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  6. E.B. Davies, “Pseudospectra of Differential Operators”, J. Operator Theory, 43:3 (2000), 243-262.
  7. E.B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, New York, 1995.
  8. L. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra, Maths.SP/9909179, London, 1999.
  9. S.C. Reddy, L.N. Trefethen, “Pseudospectra of the convection-diffusion operator”, SIAM J. Appl. Math., 54:1 (1994), 1634-1649.
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  11. R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

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