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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Global existence of a radiative Euler system coupled to an electromagnetic field

Xavier Blanc
  • Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris, France
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/ Bernard Ducomet
  • Université Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, 61 avenue du Général de Gaulle, 94010 Créteil Cedex 10 Paris, France
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/ Šárka Nečasová
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  • Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 11567 Praha 1, Czech Republic
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Published Online: 2018-03-21 | DOI: https://doi.org/10.1515/anona-2017-0117

Abstract

We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely, the 3D radiative compressible Euler system coupled to an electromagnetic field. Assuming smallness hypotheses for the data, we prove that the problem admits a unique global smooth solution and study its asymptotics.

Keywords: Compressible; Euler; radiation hydrodynamics

MSC 2010: 35Q30; 76N10

1 Introduction

In [3], after the studies of Lowrie, Morel and Hittinger [15], and Buet and Després [5], we considered a singular limit for a compressible inviscid radiative flow, where the motion of the fluid is given by the Euler system for the evolution of the density ϱ=ϱ(t,x), the velocity field u=u(t,x) and the absolute temperature ϑ=ϑ(t,x), and where radiation is described in the limit by an extra temperature Tr=Tr(t,x). All of these quantities are functions of the time t and the Eulerian spatial coordinate x3.

In [3] we proved that the associated Cauchy problem admits a unique global smooth solution, provided that the data are small enough perturbations of a constant state.

In [4] we coupled the previous model to the electromagnetic field through the so-called magnetohydrodynamic (MHD) approximation, in presence of thermal and radiative dissipation. Hereafter, we consider the perfect non-isentropic Euler–Maxwell system and we also consider a radiative coupling through a pure convective transport equation for the radiation (without dissipation). Then we deal with a pure hyperbolic system with partial relaxation (damping on velocity).

More specifically the system of equations to be studied for the unknowns (ϱ,u,ϑ,Er,B,E) reads

tϱ+divx(ϱu)=0,(1.1)t(ϱu)+divx(ϱuu)+x(p+pr)=-ϱ(E+u×B)-νϱu,(1.2)t(ϱE)+divx((ϱE+p)u)+uxpr=-σa(aϑ4-Er)-ϱEu,(1.3)tEr+divx(Eru)+prdivxu=-σa(Er-aϑ4),(1.4)tB+curlxE=0,(1.5)tE-curlxB=ϱu,(1.6)

divxB=0,(1.7)divxE=ϱ¯-ϱ,(1.8)

where ϱ is the density, u the velocity, ϑ the temperature of matter, E=12|u|2+e(ϱ,ϑ) is the total mechanical energy, Er is the radiative energy related to the temperature of radiation Tr by Er=aTr4, and pr is the radiative pressure given by pr=13aTr4=13Er, with a>0. Finally, E is the electric field and B is the magnetic induction.

We assume that the pressure p(ϱ,ϑ) and the internal energy e(ϱ,ϑ) are positive smooth functions of their arguments, with

Cv:=eϑ>0,pϱ>0,

and we also suppose for simplicity that ν=1τ (where τ>0 is a momentum-relaxation time) and μ,σa and a are positive constants.

A simplification appears if one observes that, provided that equations (1.7) and (1.8) are satisfied at t=0, they are satisfied for any time t>0, and consequently they can be discarded from the analysis below.

Notice that the reduced system (1.1)–(1.4) is the non-equilibrium regime of radiation hydrodynamics, introduced by Lowrie, Morel and Hittinger [15] and, more recently, by Buet and Després [5], and studied mathematically by Blanc, Ducomet and Nečasová [3]. Extending this last analysis, our goal in this work is to prove global existence of solutions for system (1.1)–(1.8) when the data are sufficiently close to an equilibrium state, and study their large time behavior.

For the sake of completeness, we mention that related non-isentropic Euler–Maxwell systems have been the subject of a number of studies in the recent past. Let us quote the recent works [9, 10, 12, 14, 18, 21].

In the following, we show that the ideas used by Ueda, Wang and Kawashima in [20, 19] in the isentropic case can be extended to the (radiative) non-isentropic system (1.1)–(1.6). To this end, we follow the following plan. In Section 2 we present the main results and then, in Section 3, we prove the well-posedness of system (1.1)–(1.6). Finally, in Section 4, we prove the large time asymptotics of the solution.

2 Main results

We are going to prove that system (1.1)–(1.8) has a global smooth solution close to any equilibrium state. Namely, we have the following theorem.

Theorem 2.1.

Let (ϱ¯,0,ϑ¯,Er¯,B¯,0) be a constant state, with ϱ¯>0, ϑ¯>0 and Er¯>0, and compatibility condition Er¯=aϑ¯4, and suppose that d3. There exists ε>0 such that for any initial state (ϱ0,u0,ϑ0,Er0,B0,E0) satisfying

divxB0=ϱ0-ϱ¯,divxB0=0,(ϱ0-ϱ¯,u0,ϑ0-ϑ¯,Er0-Er¯,B0-B¯,E0)Hd

and

(ϱ0,u0,ϑ0,Er0,B0,E0)-(ϱ¯,0,ϑ¯,Er¯,B¯,0)Hdε,

there exists a unique global solution (ϱ,u,ϑ,Er,B,E) to (1.1)–(1.8) such that

(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-Er¯,B-B¯,E)C([0,+);Hd)C1g([0,+);Hd-1).

In addition, this solution satisfies the following energy inequality:

(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-Er¯,B-B¯,E)(t)Hd+0t((ϱ-ϱ¯,u,ϑ-ϑ¯,Er-Er¯)(τ)Hd2+xB(τ)Hd-22+E(τ)Hd-12)𝑑τC(ϱ0-ϱ¯,0,ϑ0-ϑ¯,Er0-Er¯,B0-B¯,E0)Hd2(2.1)

for some constant C>0 which does not depend on t.

The large time behavior of the solution is described as follows.

Theorem 2.2.

Let d3. The unique global solution (ϱ,u,ϑ,Er,B,E) to (1.1)–(1.8), defined in Theorem 2.1, converges to the constant state (ϱ¯,0,ϑ¯,Er¯,B¯,0) uniformly in xR3 as t. More precisely,

(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-Er¯,E)(t)Wd-2,0as t.

Moreover, if d4, then

(B-B¯)(t)Wd-4,0as t,

Remark 2.3.

Note that, due to lack of dissipation by viscous, thermal and radiative fluxes, the Kawashima–Shizuta stability criterion (see [17] and [1]) is not satisfied for the system under study, and the techniques of [13] relying on the existence of a compensating matrix do not apply. However, we will check that radiative sources play the role of relaxation terms for the temperature and radiative energy and this will lead to global existence for the system.

3 Global existence

3.1 A priori estimates

Multiplying (1.2) by u, (1.5) by B, (1.6) by E and adding the result to equations (1.3) and (1.4), we get the total energy conservation law

t(12ϱ|u|2+ϱe+Er+12(|B|2+|E|2))+divx((ϱE+Er)u+(p+pr)u+E×B)=0.(3.1)

Introducing the entropy s of the fluid by the Gibbs law ϑds=de+pd(1ϱ) and denoting by Sr:=43aTr3 the radiative entropy, equation (1.4) is rewritten as

tSr+divx(Sru)=-σaEr-aϑ4Tr.(3.2)

The internal energy equation is

t(ϱe)+divx(ϱeu)+pdivxu-νϱ|u|2=-σa(aϑ4-Er),

and by dividing it by ϑ, we get the entropy equation for matter

t(ϱs)+divx(ϱsu)-νϑ|u|2=-σaaϑ4-Erϑ.(3.3)

So adding (3.3) and (3.2), we obtain

t(ϱs+Sr)+divx((ϱs+Sr)u)=aσaϑTr(ϑ-Tr)2(ϑ+Tr)(ϑ2+Tr2)+νϑ|u|2.(3.4)

By subtracting (3.4) from (3.1) and using the conservation of mass, we get

t(12ϱ|u|2+Hϑ¯(ϱ,ϑ)-(ϱ-ϱ¯)ϱHϑ¯(ϱ¯,ϑ¯)-Hϑ¯(ϱ¯,ϑ¯)+Hr,ϑ¯(Tr)+12(|B-B¯|2+|E|2))=divx((ϱE+Er)u+(p+pr)u+ϑ¯(ϱs+Sr)u)-ϑ¯aσaϑTr(ϑ-Tr)2(ϑ+Tr)(ϑ2+Tr2)-νϑ|u|2.

By introducing the Helmholtz functions

Hϑ¯(ϱ,ϑ):=ϱ(e-ϑ¯s)andHr,ϑ¯(Tr):=Er-ϑ¯Sr,

we check that the quantities Hϑ¯(ϱ,ϑ)-(ϱ-ϱ¯)ϱHϑ¯(ϱ¯,ϑ¯)-Hϑ¯(ϱ¯,ϑ¯) and Hr,ϑ¯(Tr)-Hr,ϑ¯(T¯r) are non-negative and strictly coercive functions reaching zero minima at the equilibrium state (ϱ¯,ϑ¯,E¯r).

Lemma 3.1.

Let ϱ¯ and ϑ¯=T¯r be given positive constants. Let O1 and O2 be the sets defined by

𝒪1:={(ϱ,ϑ)2:ϱ¯2<ϱ<2ϱ¯,ϑ¯2<ϑ<2ϑ¯},𝒪2:={Tr:T¯r2<Tr<2T¯r}.

Then there exist positive constants C1,2(ϱ¯,ϑ¯) and C3,4(T¯r) such that

C1(|ϱ-ϱ¯|2+|ϑ-ϑ¯|2)Hϑ¯(ϱ,ϑ)-(ϱ-ϱ¯)ϱHϑ¯(ϱ¯,ϑ¯)-Hϑ¯(ϱ¯,ϑ¯)C2(|ϱ-ϱ¯|2+|ϑ-ϑ¯|2)(3.5)

for all (ϱ,ϑ)O1, and

C3|Tr-T¯r|2Hr,ϑ¯(Tr)-Hr,ϑ¯(T¯r)C4|Tr-T¯r|2

for all TrO2.

Proof.

The first assertion is proved in [8], and we only sketch the proof for convenience. According to the decomposition

ϱHϑ¯(ϱ,ϑ)-(ϱ-ϱ¯)ϱHϑ¯(ϱ¯,ϑ¯)-Hϑ¯(ϱ¯,ϑ¯)=(ϱ)+𝒢(ϱ),

where

(ϱ)=Hϑ¯(ϱ,ϑ¯)-(ϱ-ϱ¯)ϱHϑ¯(ϱ¯,ϑ¯)-Hϑ¯(ϱ¯,ϑ¯)and𝒢(ϱ)=Hϑ¯(ϱ,ϑ)-Hϑ¯(ϱ,ϑ¯),

one checks that is strictly convex and reaches a zero minimum at ϱ¯, while 𝒢 is strictly decreasing for ϑ<ϑ¯ and strictly increasing for ϑ>ϑ¯, according to the standard thermodynamic stability properties, see [8]. Computing the derivatives of Hϑ¯ leads directly to estimate (3.5).

The second assertion follows from the properties of

xHr,ϑ¯(x)-Hr,ϑ¯(Tr)=ax3(x-43ϑ¯)+a3ϑ¯4.

Using the previous entropy properties, we have the following energy estimate.

Proposition 3.2.

Let the assumptions of Theorem 2.1 be satisfied with

V=(ϱ,u,ϑ,Er,B,E)𝑎𝑛𝑑V¯=(ϱ¯,0,ϑ¯,Er¯,B¯,0).

Consider a solution (ϱ,u,ϑ,Er,B,E) of system (1.1)–(1.3) on [0,t], for some t>0. Then, for a constant C0>0, one gets

V(t)-V¯L22+0tu(τ)L22dτC0V0-V¯L22.(3.6)

Proof.

We define

η(t,x)=Hϑ¯(ϱ,ϑ)-(ϱ-ϱ¯)ϱHϑ¯(ϱ¯,ϑ¯)-Hϑ¯(ϱ¯,ϑ¯)+Hr,ϑ¯(Tr),

multiply (3.4) by ϑ¯, and subtract the result to (3.1). By integrating over [0,t]×3, we find

312ϱ|u|2+η(t,x)+12|B-B¯|2+12|E|2dx+0t3ϑ¯ϑν|u|2312ϱ0|u0|2(t)+η(0,x)+12|B0-B¯|2+12|E0|2dx.

Applying Lemma 3.1 yields (3.6). ∎

By defining, for any d3, the auxiliary quantities

E(t):=sup0τt(ϱ-ϱ¯,u,B-B¯,E)(τ)W1,,F(t):=sup0τt(V-V¯)(τ)Hd,I2(t):=0t(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L2𝑑τ

and

D2(t):=0t((ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(τ)Hd2+E(τ)Hd-12+xB(τ)Hd-22)𝑑τ,

we can bound the spatial derivatives as follows.

Proposition 3.3.

Assume that the hypotheses of Theorem 2.1 are satisfied. Then, for C0>0, we have

xV(t)Hd-12+0txu(τ)Hd-12dτC0xV0Hd-12+C0(E(t)D(t)2+F(t)I(t)D(t)).

Proof.

By rewriting system (1.1)–(1.6) in the form

{tϱ+uxϱ+ϱdivxu=0,tu+(ux)u+pϱϱxϱ+pϑϱxϑ+13aϱxEr+E+u×B¯+νu=-u×(B-B¯),tϑ+(ux)ϑ+ϑpϑϱCvdivxu=-σaϱCv(aϑ4-Er),tEr+(ux)Er+43Erdivxu=-σa(Er-aϑ4),tB+curlxE=0,tE-curlxB-ϱ¯u=(ϱ-ϱ¯)u,(3.7)

and applying x to this system, we get

t(xϱ)+(ux)xϱ+ϱdivxxu=F1,t(xu)+(ux)xu+pϱϱxxϱ+pϑϱxxϑ+13aϱxxEr+xE+xu×B¯+νxu=-x[u×(B-B¯)]+F2,t(xϑ)+(ux)xϑ+ϑpϑϱCvdivxxu=-x[σaϱCv(aϑ4-Er)]+F3,t(xEr)+(ux)xEr+43Erdivxxu=-x[σa(Er-aϑ4)]+F4,t(xB)+curlxxE=0,t(xE)-curlxxB-ϱ¯xu=x[(ϱ-ϱ¯)u],

where

F1:=-[x,ux]u-[x,ϱdivx]u,F2:=-[x,ux]u-[x,pϱϱx]ϱ-[x,pϑϱx]ϑ-[x,13aϱx]Er,F3:=-[x,ux]ϑ-[x,ϑpϑϱCvdivx]u,F4:=-[x,ux]Er-[x,43Erdivx]u.

Then, by taking the scalar product of each of the previous equations, respectively, by

pϱϱ2xϱ,xu,Cvϑxϑ,14aϱErxEr,xBandxE,

and adding the resulting equations, we get

t+divx+ν(xu)2=+𝒮,(3.8)

where

:=12(xu)2+12pϱϱ(xϱ)2+12Cvϑ(xϑ)2+1214aϱEr(xEr)2+12(xE)2+12(xB)2,:=(pϱϱxϱ+pϑϱxϑ+13aϱxEr)xu+12((xu)2+pϱϱ(xϱ)2+Cvϑ(xϑ)2+14aϱEr(xEr)2)u,:=12[pϱϱ2]t(xϱ)2+12[Cvϑ]t(xϑ)2+12[14aϱEr]t(xEr)2+12divx(pϱϱ2u)(xϱ)2+12divxu(xu)2+12divx(Cvϑu)(xϑ)2+12divx(14aϱEru)(xEr)2+x(pϱϱ)xϱxu+x(pϑϱ)xϑxu+x(13aϱ)xErxu+pϱϱ2xϱF1+xuF2+CvϑxϑF3+xErF4+ϱ¯xExu,𝒮:=-xux[u×(B-B¯)]-Cvϑxϑx[σaϱCv(aϑ4-Er)]-14aϱErxErx[σa(Er-aϑ4)]+xEx[(ϱ-ϱ¯)u].

By integrating (3.8) on space, one gets

t3𝑑x+xuL223(||+|𝒮|)𝑑x.

Integrating now with respect to t and summing on , with ||d, yields

xV(t)Hd-12+0txu(τ)Hd-12dτC0xV0Hd-12+C0||=1d3(||+|𝒮|)dx.

By observing that

|tϱ|C|xϱ|,|tϑ|C(|xϱ|+|xϑ|+|xEr||Δϑ|)and|tEr|C(|xϱ|+|xϑ|+|xEr|),

and using the commutator estimates (see the Moser-type calculus inequalities in [16])

(F1,F2,F3,F4)L2x(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L22,

we see that

||C(xϱL+xuL+xϑL+xErL)x(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L22.

Then integrating with respect to time gives

0t|(τ)|dτCsup0τt{xϱL+xuL+xϑL+xErL}0tx(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L22dτCE(t)D2(t)

for any ||d. Similarly, we estimate

|𝒮|CxuL22x[u×(B-B¯)]L22+CxϑL22x[σaϱCv(aϑ4-Er)]L22+CxErL22x[σa(Er-aϑ4)]L22+CxEL22x[(ϱ-ϱ¯)u]L22.

Then we get

|𝒮|CB-B¯LxuL22+C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L2x(B,E)L+C(xϱL+xuL+xϑL+xErL)x(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L22.

Then integrating with respect to time yields

0t|𝒮(τ)|dτCsup0τt(B-B¯)(τ)L0txu(τ)L22dτ+Csup0τtx(B,E)(τ)L2×0t(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(τ)Lx(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(τ)L2dτ+Csup0τt{xϱL+xuL+xϑL+xErL(τ)}0tx(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)L22dτC(E(t)D2(t)+F(t)I(t)D(t))

for any ||d. ∎

The above results, together with (3.6), allow us to derive the following energy bound.

Corollary 3.4.

Assume that the assumptions of Proposition 3.2 are satisfied. Then

(V-V¯)(t)Hd2+0tu(τ)Hd2dτC(V-V¯)(0)Hd2+C(E(t)D(t)2+F(t)I(t)D(t)).(3.9)

Our goal is now to derive bounds for the integrals in the right- and left-hand sides of equation (3.9). For this purpose we adapt the results of Ueda, Wang and Kawashima [20].

Lemma 3.5.

Under the assumptions of Theorem 2.1, and supposing that d3, we have the following estimate for any ε>0:

0t((ϱ-ϱ¯,ϑ-ϑ¯,Er-E¯r)(τ)Hd2+E(τ)Hd-12)𝑑τε0txB(τ)Hd-22dτ+Cε{V0-V¯Hd-12+E(t)D(t)2+F(t)I(t)D(t)}.(3.10)

Proof.

We linearize the principal part of system (1.1)–(1.3) as follows:

tϱ+ϱ¯divxu=g1,(3.11)tu+a¯1xϱ+a¯2xϑ+a¯3xEr+E+u×B¯+νu=g2,(3.12)tϑ+b¯1divxu+b¯2(ϑ-ϑ¯)=g3,(3.13)tEr+c¯1divxu+c¯3(Er-E¯r)=g4,(3.14)tB+curlxE=0,(3.15)tE-curlxB-ϱ¯u=g5,(3.16)

with coefficients

a1(ϱ,ϑ)=pϱϱ,a2(ϱ,ϑ)=pϑϱ,a3(ϱ,ϑ)=13ϱ,a¯j=aj(ϱ¯,ϑ¯),b1(ϱ,ϑ)=ϑpϑϱCv,b2(ϱ,ϑ,Er)=aσaϱCv(ϑ2+ϑ¯2)(ϑ+ϑ¯),b3(ϱ,ϑ,Er)=aσaϱCv,b¯j=bj(ϱ¯,ϑ¯),c1(ϱ,ϑ,Er)=43Er,c2(ϱ,ϑ,Er)=aσa(ϑ2+ϑ¯2)(ϑ+ϑ¯),c3(ϱ,ϑ,Er)=σa,c¯j=cj(ϱ¯,ϑ¯),

and sources

g1:=-{uxϱ+(ϱ-ϱ¯)divxu},g2:=-{(ux)u+(a1-a¯1)xϱ+(a2-a¯2)xϑ+(a3-a¯3)xEr+u×(B-B¯)},g3:=-{(ux)ϑ+(b1-b¯1)divxu+(b2-b¯2)(ϑ-ϑ¯)+b3(Er-E¯r)},g4:=-{(ux)Er+(c¯1-c1)divxu+c2(ϑ-ϑ¯)+(c3-c¯3)(Er-E¯r)}

and

g5=(ϱ-ϱ¯)u.

By multiplying (3.11) by -a¯1divxu, (3.12) by a¯1xϱ+a¯2xϑ+a¯3xEr+E, (3.13) by -a¯2divxu+ϑ-ϑ¯, (3.14) by -a¯3divxu+Er-E¯r, (3.15) by 1, (3.16) by u and summing up, we get

a¯1(xϱut-ϱtdivxu)+a¯2(xϑut-ϑtdivxu)+a¯3(xErut-(Er)tdivxu)+Eut+Etu+{12[(ϑ-ϑ¯)2+(Er-E¯r)2]}t+(a¯1xϱ+a¯2xϑ+a¯3xEr+E)2+(a¯1xϱ+a¯2xϑ+a¯3xEr+E)(u×B¯+νu)+b¯2(ϑ-ϑ¯)2+c¯3(Er-E¯r)2+b¯1(ϑ-ϑ¯)divxu+c¯1(Er-E¯r)divxu+(a¯3c¯2-a¯2b¯2)(ϑ-ϑ¯)divxu+(a¯2b¯3-a¯3c¯3)(Er-E¯r)divxu-ucurlxB-ϱ¯u2-(divxu)2[a¯1+a¯2+a¯3]=G10,(3.17)

where

G10:=-a¯1g1divxu+[a¯1xϱ+a¯2xϑ+a¯3xEr+E]g2-[a¯2+ϑ-ϑ¯]divxug3-[a¯3+Er-E¯r]divxug4+g5u.

By rearranging the left-hand side of (3.17), we get

{H10}t+divxF10+D10=M10+G10,(3.18)

where

H10=-[a¯1(ϱ-ϱ¯)+a¯2(ϑ-ϑ¯)+a¯3(Er-E¯r)]divxu+Eu+12[(ϑ-ϑ¯)2+(Er-E¯r)2],F10=[a¯1(ϱ-ϱ¯)+a¯2(ϑ-ϑ¯)+a¯3(Er-E¯r)]ut-2[a¯1(ϱ-ϱ¯)+a¯2(ϑ-ϑ¯)+a¯3(Er-E¯r)]E+(a¯3c¯2-a¯2b¯2+b¯1)(ϑ-ϑ¯)u+(a¯2b¯3-a¯3c¯3+c¯1)(Er-E¯r)u,D10=a¯12|xϱ|2+a¯22|xϑ|2+a¯32|xEr|2+|E|2+2a¯1(ϱ-ϱ¯)2+b¯2(ϑ-ϑ¯)2+c¯3(Er-E¯r)2,M10=-{2a¯1a¯2xϱxϑ+2a¯1a¯3xϱxEr+2a¯2a¯3xϑxEr+2a¯2(ϱ-ϱ¯)(ϑ-ϑ¯)+2a¯2(ϱ-ϱ¯)(Er-E¯r)+(a¯1xϱ+a¯2xϑ+a¯3xEr+E)(u×B¯+νu)-ucurlxB-ϱ¯u2-(divxu)2[a¯1+a¯2+a¯3]-(a¯3c¯2-a¯2b¯2+b¯1)xϑu-(a¯2b¯3-a¯3c¯3+c¯1)xEru}.

Integrating (3.18) over space and using Young’s inequality yields

ddt3H10𝑑x+C(ϱL22+xϑL22+xErL22+EL22+ϱ-ϱ¯L22)εxBL22+Cε(uH12+ϑ-ϑ¯H12+Er-E¯rH12)+3|G10|dx.

In fact, in the same way one obtains estimates for the derivatives of V. Namely, applying x to system (3.11)–(3.16), we get

{H1}t+divxF1+D1=M1+G1,(3.19)

where

H1=-[a¯1x(ϱ-ϱ¯)+a¯2x(ϑ-ϑ¯)+a¯3x(Er-E¯r)]divxxu+xExu+12[(xϑ)2+(xEr)2],F1=[a¯1x(ϱ-ϱ¯)+a¯2x(ϑ-ϑ¯)+a¯3x(Er-E¯r)]ut+(a¯3c¯2-a¯2b¯2+b¯1)xϑxu+(a¯2b¯3-a¯3c¯3+c¯1)xErxu-2[a¯1x(ϱ-ϱ¯)+a¯2x(ϑ-ϑ¯)+a¯3x(Er-E¯r)]xE+xu×x(B-B¯),D1=a¯12|xxϱ|2+a¯22|xxϑ|2+a¯32|xxEr|2+|xE|2+2a¯1(x(ϱ-ϱ¯))2+b¯2(xϑ)2+c¯3(xEr)2,M1=-{2a¯1a¯2xxϱxxϑ+2a¯1a¯3xxϱxxEr+2a¯2a¯3xxϑxxEr+2a¯2x(ϱ-ϱ¯)x(ϑ-ϑ¯)+2a¯2x(ϱ-ϱ¯)x(Er-E¯r)+(a¯1xxϱ+a¯2xxϑ+a¯3xxEr+xE)(xu×B¯+νxu)-(a¯3c¯2-a¯2b¯2+b¯1)xxϑxu-(a¯2b¯3-a¯3c¯3+c¯1)xxErxu-curlxxux(B-B¯)-ϱ¯(xu)2-(divxxu)2[a¯1+a¯2+a¯3]},G1=-a¯1xg1divxxu+[a¯1xxϱ+a¯2xxϑ+a¯3xxEr+xE]xg2-a¯2xg3divxxu-a¯3xg4divxxu+xg5xu+xg3xϑ+xg4xEr.

Integrating (3.19) over space and time yields

3H1(t)𝑑x-3H1(0)𝑑x+C0t(xxϱL22+xxϑL22+xxErL22+xEL22)𝑑τ+C0t(x(ϱ-ϱ¯)L22+x(ϑ-ϑ¯)L22+x(Er-E¯r)L22)𝑑τε0tx(B-B¯)L22dτ+Cε0t(xuH12+x(ϑ-ϑ¯)H12+x(Er-E¯r)H12)dτ+0t3|G1|dxdτ.(3.20)

By observing that

|3H1(t)𝑑x|C(x(ϱ-ϱ¯)L22+x(ϑ-ϑ¯)L22+x(Er-E¯r)L22+xuH12),

and summing (3.20) on for 1d-1, we get

0t((ϱ-ϱ¯,ϑ-ϑ¯,Er-E¯r)(τ)Hd2+E(τ)Hd-12)dτCε(V-V¯)(0)Hd2+ε0txB(τ)Hd-22dτ+Cε(E(t)D2(t)+F(t)I(t)D(t))+||=1d-10t3|G1(τ)|dxdτ,

where we used Corollary 3.4.

Let us estimate the last integral in (3.20). We have

{xg1L2C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx+1(ϱ,u)L2,xg2L2C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx+1(ϱ,u)L2+CB-B¯LxuL2+Cx(B-B¯)L2uL,xg3L2C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx+1(ϱ,u)L2+C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx+2(ϑ,Er)L2,xg4L2C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx+1(ϱ,u)L2+C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx+2(ϑ,Er)L2,xg5L2C(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)Lx(ϱ,u)L2(3.21)

for 1||d-1. Then

0t3|G1(τ)|dxdτCx+1uL2xg1L2+C(x+1ϱL2+x+1ϑL2+x+1ErL2+xEL2)xg2L2+Cx+1uL2xg3L2+Cx+1uL2xg4L2+CxuL2xg5L2.

Plugging bounds (3.21) into the last inequality gives

||=1d-10t3|G1(τ)|dxdτCE(t)D2(t),

which completes the proof of Lemma 3.5. ∎

Finally, we check from [20, Lemma 4.4] that the following result for the Maxwell’s system holds true for our system with a similar proof.

Lemma 3.6.

Under the assumptions of Theorem 2.1, and supposing that d3, for any ε>0, the following estimate (here, we set V=(ϱ,u,ϑ,Er,B,E)T) holds:

0txB(τ)Hs-22dτCV0-V¯Hs-12+C0txE(τ)Hs-22dτ+C(E(t)D(t)2+F(t)I(t)D(t)).(3.22)

Proof.

By applying x to (1.5) and (1.6), multiplying, respectively, by -curlxxB and curlxxE, and adding the resulting equations, we get

-(xEcurlxxB)t+|curlxxB|2-divx(xE×xBt)=M2+G2,

where

M2=-ϱ¯xucurlxxB+|curlxxE|2

and

G2=-x((ϱ-ϱ¯)u)curlxxB.

Integrating in space gives

-ddt3xEcurlxxBdx+CcurlxxBL22curlxxEL22+xuL22+3|G2|dx.

By integrating on time and summing for 1||d-2, we have

0txBHd-22dtC(V-V¯)(t)Hd-1+C(V-V¯)(0)Hd-1+C0txEHd-22dt+C0tuHd-22dt+C||=0d-23|G2(τ)|dxdτC(V-V¯)(0)Hd-1+C0txE|Hd-22dt+C(E(t)D(t)2+F(t)I(t)D(t)),

where we used the bound

||=1d-10t3|G2(τ)|dxdτCE(t)D2(t),

obtained in the same way as in the proof of Lemma 3.5. The proof of Lemma 3.6 is completed. ∎

We are now in position to conclude with the proofs of Theorems 2.1 and 2.2.

3.2 Proof of Theorem 2.1

We first point out that local existence for the hyperbolic system (1.1)–(1.6) may be proved using standard fixed-point methods. We refer to [16] for the proof.

Now, by plugging (3.22) into (3.10) with ε small enough, we get

0t((ϱ-ϱ¯,ϑ-ϑ¯,Er-E¯r)Hd2+E(τ)Hd-12)𝑑τC{V0-V¯Hd-12+E(t)D(t)2+F(t)I(t)D(t)}.(3.23)

Putting this last estimate into (3.22) yields

0txB(τ)Hs-22dτCV0-V¯Hs2+C(E(t)D(t)2+F(t)I(t)D(t)).(3.24)

Then, from (3.10), (3.23) and (3.24), we get

(V-V¯)(t)Hd2+0t((ϱ-ϱ¯,ϑ-ϑ¯,Er-E¯r)(τ)Hd2+E(τ)Hd-12+xB(τ)Hd-22)dτCV0-V¯Hd2+C(E(t)D(t)2+F(t)I(t)D(t))

or, equivalently,

F(t)2+D(t)2CV0-V¯Hd2+C(E(t)D(t)2+F(t)I(t)D(t)).

Now, by observing that, provided d3, one has (V-V¯)(t)HdE(t)CF(t), and, provided d2, one has I(t)CD(t) for some positive constant C, we see that

F(t)2+D(t)2CV0-V¯Hd2+CF(t)D(t)2.

In order to prove global existence, we argue by contradiction, and assume that Tc>0 is the maximum time existence. Then we necessarily have

limtTcN(t)=+,

where N(t) is defined by

N(t):=(F(t)2+D(t)2)1/2.

Thus, we are left to prove that N is bounded. For this purpose, we use the argument used in [3]. After the previous calculation, we have

N(t)2C(V0-V¯Hd2+N(t)3)for all T[0,Tc].(3.25)

Hence, setting V0-V¯Hd=ε, we have

N(t)2ε2+N(t)3C.

By studying the variation of ϕ(N)=N2/(ε2+N3), we see that ϕ(0)=0, and that ϕ is increasing on the interval [0,(2ε2)1/3] and decreasing on the interval [(2ε2)1/3,+). Hence,

maxϕ=ϕ((2ε2)1/3)=13(2ε)2/3.

Hence, we can choose ε small enough to have ϕ(N)C for all N[0,N*], where N*>0, and we see that NN*, which contradicts (3.25).

4 Large time behavior

We have the following analogue of Proposition 3.2 for time derivatives.

Corollary 4.1.

Let the assumptions of Theorem 2.1 be satisfied, and consider the solution V:=(ϱ,u,ϑ,Er,B,E) of system (1.1)–(1.3) on [0,t], for some t>0. Then, for a constant C0>0, one gets

tV(t)Hd-12+0t(t(ϱ,u,ϑ,Er)(τ)Hd-12+t(B,E)(τ)Hd-22)𝑑τC0V0-V¯Hd2.(4.1)

Proof.

By using system (3.7), we see that

tVHd-1CV-V¯Hd,t(ϱ,u,ϑ,Er)Hd-1x(ϱ,u,ϑ,Er,B,E)Hd-1+C(ϱ,u,ϑ,Er,B,E)Hd-1

and

t(B,E)Hd-2x(B,E)Hd-2+CuHd-1.

Then, for d3, using the uniform estimate V-V¯Hd2C of Theorem 2.1, we get estimate (4.1). ∎

4.1 Proof of Theorem 2.2

By using Corollary 4.1, we get

0|ddt(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-1|dt20(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-1t(ϱ,u,ϑ,Er)(t)Hd-1dt0(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-12+t(ϱ,u,ϑ,Er)(t)Hd-12dtC0V0-V¯Hd2.

This implies that

t(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-12L1(0,)

and

tddt(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-1L1(0,),

and then

(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-10when t.

Now, by applying the Gagliardo–Nirenberg inequality and (2.1), we get

(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Wd-2,(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Hd-21/4x2(ϱ,u,ϑ,Er)(t)Hd-23/4.

So

(ϱ-ϱ¯,u,ϑ-ϑ¯,Er-E¯r)(t)Wd-2,0when t.

Similarly,

tE(t)Hd-12L1(0,)andtddtE(t)Hd-1L1(0,),

and then

E(t)Wd-1,0when t.

Finally,

txB(t)Hd-32L1(0,)andtddtxB(t)Hd-3L1(0,).

Then, arguing as before,

(B-B¯)(t)Wd-4,(B-B¯)(t)Hd-41/4x2B(t)Hd-33/4.

So

(B-B¯)Wd-4,0when t,

which completes the proof.

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About the article

Received: 2017-05-22

Accepted: 2018-02-28

Published Online: 2018-03-21


Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-15-CE40-0011

Funding Source: Grantová Agentura České Republiky

Award identifier / Grant number: 201-16-03230S

Šárka Nečasová acknowledges the support of the GAČR (Czech Science Foundation) project 16-03230S in the framework of RVO: 67985840. Bernard Ducomet is partially supported by the ANR project INFAMIE (ANR-15-CE40-0011).


Citation Information: Advances in Nonlinear Analysis, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0117.

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