Emmrich E., Wittbold P. (editors) Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series

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Coupling of a scalar conservation law with a parabolic problem 145

Lemma 4.11. Let G : R → R be Hölder continuous with exponent α ∈ (0, 1). Then

G(v) ∈ W

αs,

(Ω) for any v ∈ W

s,p

(Ω), 0 < s < 1, 1 < p < ∞, and the mapping

G : W

s,p

(Ω) → W

αs,

(Ω) is bounded.

Proof of Proposition 4.10. Note ﬁrst that (4.13) implies that there exists a subsequence,

still denoted by (u

)

µ>0

, such that u

∗

* u in L

∞

(Q). The two convergence results

about λ

∇φ(u

) are an immediate consequence of (4.15). So we just have to prove the

strong convergence of (u

)

µ>0

in L

). To do so, we refer to the arguments used in

[9, Chapt. 2].

From (4.16), (∂

)

µ>0

is bounded in L

(0, T ; H

−1

(Ω

)). Moreover, due to (4.13)

and (4.15), the sequence (φ(u

))

µ>0

is bounded in L

(0, T ; V ). For any s ∈ (0, 1), we

also have

(0, T ; V ) ,→ L

(0, T ; H

(Ω

)) ,→ L

(0, T ; W

s,2

(Ω

)).

Since u

= φ

−1

(φ(u

)), by (4.17) and Lemma 4.11, we assert that u

is bounded

in L

2/τ

(0, T ; W

τs,2/τ

(Ω

)). The compactness of the embedding of W

τs,2/τ

(Ω

) in

2/τ

(Ω

) and the Lions–Aubin compactness theorem (see [14, p. 57]) ensure that

W := {v ∈ L

2/τ

(0, T ; W

τs,2/τ

(Ω

)); ∂

v ∈ L

(0, T ; H

−1

(Ω

))}

is compactly embedded in L

2/τ

(0, T ; L

2/τ

(Ω

)). Hence, the conclusion follows.

We are now able to state the main theorem of this section.

Theorem 4.12. Problem (1.1) has a weak entropy solution that is the limit of the se-

quence (u

)

µ>0

of solutions to (4.12) in L

(Q), 1 ≤ q < ∞, as µ → 0

Proof. We consider the function u given in Proposition 4.10. Because of (4.13), we

can apply Theorem 1.5 to assert that there exists a function π ∈ L

∞

((0, 1) × Q

) such

that for any continuous and bounded function ψ on Q

× [m, M] and ξ ∈ L

)

lim

µ→0

ψ(t, x, u

(t, x))ξdxdt =

ψ(t, x, π(α, t, x))ξdxdtdα. (4.18)

Our aim is ﬁrst to establish that, as in Section 3, the process π is reduced to u

and, second, to prove that u is a weak entropy solution to (1.1).

In order to do so, we come back to (4.14) and, for any real k, we choose the test

function v

= sgn

(φ(u

) − φ(k))ϕ

, where ϕ

∈ C

∞

(−T, T ), ϕ

∈ C

∞

(Ω) with

, ϕ

≥ 0. Thereby, we obtain

h∂

, v

i +

Ω

(λ

(x)∇φ(u

) − b(x)f(u

)) · ∇v

dx = 0.

We integrate with respect to the time variable and perform the following transforma-

tions in a same way as in Section 3. Thanks to a generalization of Lemma 1.1 (see

146 Julien Jimenez

[18]), the evolutionary term may be written as

h∂

, sgn

(φ(u

) − φ(k))ϕ

iϕ

= −

)ϕ

∂

dxdt −

Ω

)ϕ

(0)dx

with I

) =

sgn

(φ(τ) − φ(k))dτ . For the diffusion term, we obtain

(x)∇φ(u

) · ∇v

dxdt ≥

sgn

(φ(u

) − φ(k))∇φ(u

) · ∇ϕ

dxdt.

Moreover, we have

b(x)f(u

) · ∇v

dxdt = −

b(x)F

)∇ϕ

dxdt

with F

) =

(τ)sgn

(φ(τ) − φ(k))dτ .

We take the limit with respect to µ separately on the hyperbolic and parabolic area.

On Q

, we take advantage of (4.18), while on Q

, we use Proposition 4.10. In this

way, we obtain

(u)ϕ

∂

+ b(x)F

(u) · ∇ϕ

dxdt

(π)ϕ

∂

+ b(x)F

(π) · ∇ϕ

dxdtdα

−

sgn

(φ(u) − φ(k))∇(φ(u) − φ(k)) · ∇ϕ

dxdt

Ω

)ϕ

(0)dx ≥ 0.

Then we take the limit with respect to η. We remark that

lim

η→0

(u) = sgn(φ(u) − φ(k))(f (u) − f(k)) = F (u, k) a.e. on Q

and Q

Therefore, we get in the limit

|u − k|ϕ

∂

+ b(x)F(u, k) · ∇ϕ

dxdt

|π − k|ϕ

∂

+ b(x)F(π, k) · ∇ϕ

dxdtdα

Ω

− k|ϕ

(0)dx −

∇|φ(u) − φ(k)| · ∇ϕ

dxdt ≥ 0. (4.19)

Coupling of a scalar conservation law with a parabolic problem 147

We justify now that π satisﬁes the initial condition (4.3) and the boundary condition

(4.4). When we consider test functions ϕ

that belong to C

∞

(Ω

), we deduce that

−

|π − k|ϕ

∂

+ b(x)F(π) · ∇ϕ

dxdtdα ≤

Ω

− k|ϕ

(0)dx.

Therefore, we can apply Lemma 3.13 (with Q = Q

) to state that

esslim

t→0

Ω

|π(α, t, x) − u

(x)|dxdα = 0. (4.20)

Hence, π fulﬁlls the initial condition (4.3) on Ω

Next we choose in (4.14) the test function v

= ∂

, k)ϕ

(see Example 3.4

for the deﬁnition of H

), where ϕ

∈ C

∞

(0, T ), ϕ

∈ C

∞

(Ω

) with ϕ

= 0 on Γ

, ϕ

≥ 0. We take the limit with respect to η and apply Lemma 3.14 to assert that,

for any nonnegative β ∈ L

(Σ

\ Σ

)

ess lim

s→0

−

\Σ

(π(α, σ + sν

), k)βb(σ) · ν

dσdα ≥ 0,

where J

is deﬁned in Example 3.4. As δ → 0

, we obtain that π satisﬁes (4.4).

Therefore, π fulﬁlls (4.2) for all ϕ ∈ C

∞

) as well as (4.3) and (4.4), where integrals

over Q

, Ω

and Σ

\ Σ

are replaced by integrals over (0, 1) × Q

, (0, 1) × Ω

and

(0, 1) × Σ

\ Σ

, respectively, with respect to the corresponding measure. Then, by

reasoning as in Theorem 4.5, if π

(α, ·) and π

(β, ·) are two process solutions for initial

data u

0,1

and u

0,2

, we obtain for almost all t ∈ (0, T )

Ω

|π

(α, t, x) − π

(β, t, x)|dxdαdβ ≤

Ω

0,1

− u

0,2

|dx.

Therefore, if u

0,1

= u

0,2

on Ω

, there exists a bounded function u

on Q

such that

a.e. on Q

(·) = π

(α, ·) = π

(β, ·) for almost all α and β in (0, 1).

Besides, u

= u

a.e. on Q

. Thus, from (4.19), it follows that u fulﬁlls (4.2) for all

ϕ ∈ C

∞

(0, T ) ⊗ C

∞

(Ω) and, by density, for all ϕ ∈ C

∞

(Q) as well as (4.4).

To complete the proof it remains to show that (4.3) holds for u. Due to (4.20), we

just have to focus on Ω

. We consider (4.19) for ϕ

∈ C

∞

(Ω

). This yields

−

Ω

|u − k|ϕ

dx + g(t)

(t)dt ≤

Ω

− k|ϕ

ϕ(0)dx

with

g(t) =

Ω

(

|φ(u(τ, x)) − φ(k)|∆ϕ

− sgn(u(τ, x) − k)(f (u(τ, x) − f(k))b(x) · ∇ϕ

)

dτdx.

148 Julien Jimenez

Therefore, the time-depending function

t 7→

Ω

|u − k|ϕ

dx + g(t)

can be identiﬁed a.e. with a nonincreasing and bounded function. Then it has an essen-

tial limit as t → 0

. Since g(t) → 0 as t → 0

, it follows

esslim

t→0

Ω

|u − k|ϕ

dx ≤

Ω

− k|ϕ

dx,

for all nonnegative ϕ

∈ C

∞

(Ω

), which implies (see [17]), for any k ∈ L

∞

(Ω

esslim

t→0

Ω

|u − k(x)|dx ≤

Ω

− k(x)|dx.

We choose k = u

to ensure that u satisﬁes (4.3). This concludes the proof.

5 Concluding remarks

We end these lecture notes by some remarks. We have studied a simpliﬁed coupling

problem. One could, for example, also assume, as in [1, 12], that the nonlinearities are

different on Q

and Q

. Indeed, we could consider the following problem: Find a mea-

surable and essentially bounded function u on Q such that in the sense of distributions











∂

u + ∇ · (b

(x)f

(u)) = 0 in Q

∂

u + ∇ · (b

(x)f

(u)) = ∆φ(u) in Q

u = 0 on (0, T ) × ∂Ω,

u(0, ·) = u

on Ω.

(5.1)

In this case, the notion of weak entropy solution in Deﬁnition 4.2 is not suitable

anymore. We have to add in the entropy formulation (4.2) an interface integral that

takes into account the discontinuity of the ﬂux function along Σ

. Indeed, setting

b(x)f(u) = b

(x)f

(u)χ

Ω

+ b

(x)f

(u)χ

Ω

, we propose the following deﬁnition of

a weak entropy solution.

Deﬁnition 5.1. A function u is said to be a weak entropy solution to (5.1) if

(i) u ∈ L

∞

(Q), φ(u) ∈ L

(0, T ; V ),

(ii) for all nonnegative ϕ ∈ C

∞

(Q)and all k ∈ R

(|u − k|∂

ϕ + (b(x)F (u, k) − ∇|φ(u) − φ(k)|) · ∇ϕ)dxdt

(σ)f

(k) − b

(σ)f

(k)) · ν

sgn(φ(u) − φ(k))dσ ≥ 0.

Coupling of a scalar conservation law with a parabolic problem 149

(iii) u satisﬁes the initial condition (4.3) and the boundary condition (4.4).

We refer to [11, Chap. 4] or [12] for the study of (5.1). When we use the vanishing

viscosity method, the major difﬁculty relies on the study of the viscous problem related

to (5.1). Roughly speaking, we are not able to prove the maximum principle without

additional assumption on the ﬂux. We also underline that the existence and uniqueness

results of [12] are obtained in the framework of outwards characteristics (see Remark

4.1). The case of inwards characteristics, i.e., the case where the characteristics are

entering the hyperbolic zone, is treated in [2] for (1.1). We emphasize the fact that the

general case (i.e., no assumption on the characteristics) is still an open problem.

Acknowledgments. I warmly thank Petra Wittbold and Etienne Emmrich for their kind

invitation at the Technische Universität Berlin. Moreover, their remarks, suggestions

and comments greatly improved these lecture notes. I also would like to thank the

Stiftung Luftbrückendank for supporting my stay in Berlin.

References

[1] G. Aguilar, L. Lévi and M. Madaune-Tort, Coupling of multidimensional parabolic and hyper-

bolic equations, J. Hyperbolic Differ. Equ. 3 (2006), pp. 53–80.

[2] , Nonlinear multidimensional parabolic-hyperbolic equations, 2006 International con-

ference in honor of J. Fleckinger, Electron. J. Diff. Eqns. Conf. 16 (2007), pp. 15–28.

[3] G. Aguilar, F. Lisbona and M. Madaune-Tort, Analysis of a nonlinear parabolic-hyperbolic

problem, Adv. Math. Sci. Appl. 7 (1997), pp. 165–181.

[4] C. Bardos, A. Y. LeRoux and J. C. Nédélec, First order quasilinear equations with boundary

conditions, Commun. Partial Differ. Equations 4 (1979), pp. 1017–1034.

[5] R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique pour les sciences et

techniques, vol. 8, Masson, Paris, 1980.

[6] , Mathematical analysis and numerical methods for science and technology, vol. 5,

Springer, Berlin, 1992.

[7] L. C. Evans and R. F. Gariepy, Measure theory and ﬁne properties of functions, CRC Press,

London, 1992.

[8] R. Eymard, T. Gallouët and R. Herbin, Existence and uniqueness of the nonlinear hyperbolic

equation, Chin. Ann. Math. Ser. B 16 (1995), pp. 1–14.

[9] G. Gagneux and M. Madaune-Tort, Analyse mathématiques de modèles non linéaires de

l’ingénierie pétrolière, Springer, Berlin, 1996.

[10] E. Godlewski and P. A. Raviart, Hyperbolic systems of conservation laws, Mathématiques et

Applications, S.M.A.I., Ellipses, Paris, 1991.

[11] J. Jimenez, Modèles non linéaires de transport dans un milieu poreux hétérogène, Ph.D. thesis,

Univ. Pau, 2007.

[12] J. Jimenez and L. Lévi, Entropy formulations for a class of scalar conservation laws with space-

discontinuous ﬂux functions in a bounded domain, J. Engrg. Math. 60 (2008), pp. 319–335.

[13] S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb.

81:123 (1970), pp. 228–255.

150 Julien Jimenez

[14] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod,

Gauthier-Villars, Paris, 1969.

[15] J. Málek, J. Ne

cas, M. Rokyta and M. R˚uži

cka, Weak and measure-valued solutions to evolu-

tionary PDEs, Chapman & Hall, London, 1996.

[16] M. Marcus and V. J. Mizel, Every superposition mapping one Sobolev space into another is

continuous, J. Funct. Anal. 33 (1979), pp. 217–229.

[17] F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris,

Sér. I 322 (1996), pp. 729–734.

[18] , L

-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Dif-

fer. Equations 131 (1996), pp. 20–38.

[19] J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987.

Author information

Julien Jimenez, Laboratory of applied mathematics, University of Pau, BP 1155, 64013 Pau Cedex,

France.

E-mail:

de Gruyter 2009

Standing waves in nonlinear Schrödinger equations

tefan Le Coz

Abstract. In the theory of nonlinear Schrödinger equations, it is expected that the solutions will ei-

ther spread out because of the dispersive effect of the linear part of the equation or concentrate at one

or several points because of nonlinear effects. In some remarkable cases, these effects counterbal-

ance, and special solutions that neither disperse nor focus appear, the so-called standing waves. For

the physical applications as well as for the mathematical properties of the equation, a fundamental

issue is the stability of waves with respect to perturbations. Our purpose in these notes is to present

various methods developed to study the existence and stability of standing waves. We prove the

existence of standing waves by using a variational approach. When stability holds, it is obtained by

proving a coercivity property for a linearized operator. Another approach based on variational and

compactness arguments is also presented. When instability holds, we show by a method combining

a virial identity and variational arguments that the standing waves are unstable by blow-up.

Keywords. Nonlinear Schrödinger equation, standing waves, orbital stability, instability, blow-up,

variational methods.

AMS classiﬁcation. 35Q55, 35Q51, 35B35, 35A15.

1 Introduction

In these lecture notes, we consider the nonlinear Schrödinger equation

i∂

u + ∆u + |u|

p−1

u = 0. (1.1)

Equation (1.1) arises in various physical and biological contexts, for example in non-

linear optics, for Bose–Einstein condensates, in the modelling of the DNA structure,

etc. We refer the reader to [14, 73] for more details on the physical background and

references.

Here, the unknown u is a complex valued function of t ∈ R and x ∈ R

for N > 1,

u : R ×R

→ C.

In most of the applications, t stands for the time and x for the space variable, but

sometimes it can be the converse, for example in nonlinear optics. The number i is the

imaginary unit (i

= −1), ∂

is the ﬁrst derivative with respect to the time t, ∆ denotes

the Laplacian with respect to the space variable x (∆ =

j=1

∂

∂x

). Finally, p ∈ R is

such that p > 1, which means that the equation is superlinear.

To simplify the exposition, we have restricted ourselves to the study of nonlinear

Schrödinger equations with a power-type nonlinearity, but one can also consider more

general versions of these equations, see [14, 73] and the references cited therein.

152 S

tefan Le Coz

The purpose of these notes is to study the properties of particular solutions of (1.1)

f the form e

iωt

ϕ(x) with ω ∈ R and ϕ satisfying

−∆ϕ + ωϕ −|ϕ|

p−1

ϕ = 0. (1.2)

Such solutions are called standing waves. They are part of a more general class of so-

lutions arising for various nonlinear equations like Korteweg–de Vries, Klein–Gordon

or Kadomtsev–Petviashvili equations. These equations enjoy special solutions whose

proﬁle remains unchanged under the evolution in time. These special solutions are

called solitary waves or solitons (see [14, 18, 20, 73] for an overview of physical and

mathematical questions around solitary waves).

This kind of phenomena was discovered in 1834 by John Scott Russel. The Scottish

engineer was supervising work in a canal near Edinburgh when he observed that the

brutal stop of a boat in the canal was creating a wave that does not seem to vanish.

Indeed, he was able to follow this wave horseback on a distance of several miles. His

life long, he studied this surprising phenomenon but without being able to give it a

theoretical justiﬁcation. In fact, most of the scientists of that time did believe that such

a wave, which does not disperse, could not exist. The ﬁrst theoretical justiﬁcation of the

existence of solitary waves was given by Korteweg and de Vries in 1895. They derived

an equation for the motion of water admitting solitary wave solutions. But one had to

wait until the 1950’s to see the beginning of an intensive research from mathematicians

and physicists about solitary waves.

Heuristically, such solutions appear because of the balance of two contradictory

effects: the dispersive effect of the linear part, which tends to ﬂatten the solution as

time goes on (see Figure 1), and the focusing effect of the nonlinearity, which tends to

concentrate the solution, provided the initial datum is large enough (see Figure 2). For

Figure 1. D

ispersive effect.

some special initial data, each effect compensates the other and the general proﬁle of

the initial data remains unchanged (see Figure 3).

In the study of standing waves, two types of questions arise naturally.

(i) Do standing waves exist, i.e., does (1.2) admit nontrivial solutions? If yes, what

kind of properties for the solutions of (1.2) can be shown: Are they regular, what

is their decay at inﬁnity, can we ﬁnd variational characterizations for at least some

of these solutions?

Standing waves in nonlinear Schrödinger equations 1

Figure 2. F

ocusing effect.

Figure 3. B

alance between dispersion and focusing.

(ii) If standing waves exist, are they stable (in a sense to be made precise) as solutions

of (1.1)? If they are unstable, what is the nature of instability?

The ﬁrst type of questions is related to the study of existence of solutions for semili-

near elliptic problems. The methods used in this context are often of variational nature

and solutions are obtained by minimization under constraint or with min-max argu-

ments. The second type of questions concerns the dynamics of the evolution equation.

Nevertheless, both problems are intimately related. Indeed, informations of variational

nature on the solutions of the stationary equation are essential to derive stability or

instability results.

The rest of these notes is divided into four sections and an appendix. We ﬁrst give

basic results on the Cauchy problem for (1.1) in Section 2. In Section 3, we study the

existence of standing waves. Section 4 is devoted to stability whereas Section 5 deals

with instability. The proof of a stability criterion is given in the appendix.

Notation. The space of complex measurable functions whose r-th power is integrable

will be denoted by L

) and its standard norm by k · k

)

. When r = 2, the

space L

) will be endowed with the real inner product

(

u, v

)

= Re

u ¯vdx for u, v ∈ L

The space of functions from L

) whose distributional derivatives of order less than

or equal to m are elements of L

) will be denoted by W

m,r

). If m = 1 and

154 S

tefan Le Coz

r = 2

, we shall denote W

1,2

) by H

), its usual norm by k · k

)

and the

duality product between the dual space H

−1

) and H

) by h·, ·i. If m = r = 2,

we denote W

2,2

) by H

For notational convenience, we shall sometimes identify a function and its value at

some point. For example, we shall write e

iωt

ϕ(x) for the function (t, x) 7→ e

iωt

ϕ(x).

Similarly, we shall say that |x|v ∈ L

) if the function x 7→ |x|v(x) belongs to

). For a solution u of (1.1), we shall denote by u(t) the function x 7→ u(t, x).

Therefore, depending on the context, u may denote either a mapping from R to some

function space or a mapping from R ×R

to C.

We make the convention that when we take a subsequence of a sequence (v

) we

denote it again by (v

The letter C will denote various positive constants whose exact values may change

from line to line but are not essential in the course of the analysis.

2 The Cauchy problem

For the physical properties of the model as well as for the mathematical study of the

equation, it is interesting to look for quantities conserved along the time. At least

formally, equation (1.1) admits three conserved quantities. The ﬁrst one is the mass or

charge: If u satisﬁes (1.1) with initial datum u(0) = u

then

Q(u(t)) :=

ku(t

)

≡ Q(u

). (2.1)

The conservation of mass is obtained from multiplying (1.1) with ¯u, integrating over

and taking the imaginary part. The second conserved quantity is the energy (mul-

tiply (1.1) by ∂

¯u, integrate over R

and take the real part),

E(u(t)) :=

∇u(t)k

)

−

p + 1

ku(t

p+1

)

≡ E(u

). (2.2)

Finally, multiplying (1.1) by ∇¯u, integrating over R

and taking the real part, we

obtain the conservation of momentum,

M(u(t)) := Im

u(t)∇¯u(t)dx ≡ M (u

Among these three conserved quantities, the mass and the energy are real-valued, but

the momentum may be complex-valued, which makes it less convenient to use.

To beneﬁt from the conserved quantities, it is natural to search solutions of (1.1)

in function spaces where these quantities are well-deﬁned. Consequently, we look

for solutions of (1.1) in H

) and restrict the range of p to be subcritical for the

Sobolev embedding of H

) into L

p+1

), i.e., 1 < p < 1 +

N−2

if N > 3 and

1 < p < +∞ if N = 1, 2. Then we have the following result concerning the well-

posedness of the Cauchy problem for (1.1) (see [14] and the references cited therein).