Radially Symmetric Stationary Wave for Two-dimensional Burgers Equation

We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that such a stationary wave satisfies nice decay estimates and is time-asymptotically nonlinear stable under radially symmetric perturbation.


Introduction and main results
In this paper, we consider the problem on the precise description of the large time behaviors of global smooth solutions to the following initial-boundary value problem of multidimensional Burgers equation in an exterior domain Ω := R n \B r0 (0) ⊂ R n for n ≥ 2: u t + (u · ∇)u = µ∆u, t > 0, x ∈ Ω, u(0, x) = u 0 (x), x ∈ Ω, (1.1) u(t, x) = b(t, x), t > 0, x ∈ ∂B r0 (0), and, as in [2,3,5], our main purpose is to understand how the space dimension n effect the large time behaviors of solutions of the initial-boundary value problem (1.1). Here u = (u 1 (t, x), · · · , u n (t, x)) is a vector-valued unknown function of t ∈ R + and x = (x 1 , · · · , x n ) ∈ R n , u · ∇ = n j=1 u j ∂ ∂xj , µ > 0 and r 0 > 0 are some given constants. u 0 (x) and b(t, x) are given initial and boundary values respectively satisfying the compatibility condition b(0, x) = u 0 (x) for all x ∈ ∂B r0 (0).
Throughout this paper, we will concentrate on the radially symmetric solutions for the initial-boundary value problem (1.1). For such a case, under the assumption that b(t, x) = x |x| v − , u 0 (x) = x |x| v 0 (|x|) satisfying lim |x|→+∞ v 0 (|x|) = v + and v 0 (r 0 ) = v − for some given constants v ± ∈ R with v 0 (|x|) being some given scalar function, then if we introduce a new unknown function v(t, r) by letting u(t, x) = x r v(t, r) with r = |x|, we can deduce that v(t, r) := v(t, |x|) solves the following initial-boundary value problem where the initial data v 0 (r) is assumed to satisfy the compatibility condition . * Email address: hhjjzhao@whu.edu.cn † Email address: qszhao@whu.edu.cn It is well-known that, cf. [7,8,9,11,12,13] and the references cited therein, to give a precise description of the large time behaviors of global solutions v(t, r) to the initial-boundary value problem (1.2), in addition to the rarefaction waves and viscous shock waves, which are sufficient to describe the asymptotics of the global solutions for the corresponding Cauchy problem of one-dimensional scalar viscous conservation laws, a new type of nonlinear wave, i.e. the so-called stationary wave φ(r) solving the following problem d dr should be introduced, which is due to the appearance of the boundary condition (1.2) 2 .
The main purpose of this paper focuses on the existence and time-asymptotically nonlinear stability of such a stationary wave φ(r) under radially symmetric perturbation. To this end, if one integrates (1.4) 1 with respect to r from r to ∞, then the problem (1.4) is rewritten as Moreover, if we set then we can get from (1.5) that ψ(r) solves Such a problem has been studied by I. Hashimoto and A. Matsumura in [2,3,5], what they found for the multidimensional Burgers equation is that, unlike the one-dimensional case, the stationary wave φ(r) (or ψ(r)) satisfying (1.5) (or (1.7)) is, generally speaking, no longer monotonic. The results obtained in [2,3,5] can be summarized as in the following: • When v + = 0, (1.5) 1 is Bernoulli type ordinary differential equation which can be solved explicitly, thus one can deduce that for n = 2, (1.5) admits the following unique nontrivial solution φ 2 1 (r) for r ≥ r 0 if and only if v − < 0, while for n ≥ 3, (1.5) possesses the following unique nontrivial solution . These stationary waves φ 2 1 (r) and φ n 1 (r)(n ≥ 3) are monotonic and satisfy |φ n 1 (r)| 1 r| ln r| , n = 2, r 1−n , n ≥ 3 (1.10) for r ≥ r 0 and are time-asymptotic nonlinear stability, cf. [2,3,5]. Here and in the rest of this paper f (r) g(r) means that there exists a generic positive constant C such that |f (r)| ≤ C|g(r)| holds for r ≥ r 0 . f (r) ∼ g(r) if f (r) g(r) and g(r) f (r); • When n = 3, (1.7) 1 is an autonomous ordinary differential equation and, similar to the one-dimensional case, it can be solved explicitly also. In fact, it is easy to see that (1.7) admits a unique nontrivial solution for r ≥ r 0 if and only if v + < 0, V − < |v + |. Such a stationary wave ψ S (r) satisfies and is also shown to be nonlinear stable under radially symmetric perturbation, cf. [5]; • For the case when one can not deduce an explicit formula for the solutions of (1.5) or (1.7), the result available up to now focuses on the case n ≥ 4. In such a case, it is shown in [5] then one can deduce that ψ S (r) is a upper bound of the solution ψ(r) of (1.7), while gives the lower bound, from which one yield the existence of stationary wave ψ(r) to (1.7), which satisfies for r ≥ r 0 . Although it is no longer monotonic, its nonlinear stability is justifies in [5] for v ± < 0, V − < v + and is later extended in [14] to cover the case when (1.12) holds.
Even so, for the two-dimensional case, to the best of our knowledge, the only result available up to now is on the case when v + = 0 and in such a case, the unique solution φ 2 1 (r) to (1.5) is given by (1.8). Thus it is an interesting problem to see what happens when v + < 0 and the main purpose of this paper is concentrated on such a problem.
Throughout the rest of this paper, we will focus on the case n = 2 and in such a case, (1.7) can be rewritten as For the solvability of the problem (1.15), we have the following result Theorem 1.1. There exists a constant a * satisfying Moreover such a solution ψ(r) satisfies Here φ(r) is the corresponding solution of the following problem dφ dr Remark 1.1. From the proof of Theorem 1.1, it is easy to see that, generally speaking, the unique solution ψ(r) to the problem (1.15) is no longer monotonic. In fact, we can deduce from the proof of Theorem 1.1 that is firstly strictly monotonic increasing to its maximum, then is strictly monotonic decreasing and tends to v + as r → +∞.
For the time-asymptotically nonlinear stability of the stationary wave φ(r) constructed in Theorem 1.1, unlike the one-dimensional case, the main trouble is caused by the fact that such a stationary wave φ(r) is no longer monotonic and to overcome such a difficulty, as in [14], we use the anti-derivative method by introducing the new unknown function With the above preparations in hand, we now turn to state our result on the nonlinear stability of the stationary wave φ(r) constructed in Theorem 1.1. In fact, motivated by [14], if we introduce the weight function and by employing the weighted energy method as in [14] with a slight modification, we can get that Theorem 1.2. Suppose that the condition listed in Theorem 1.1 holds and w 0 ∈ H 2 satisfying then the initial-boundary value problem (1.21) admits a unique global solution w(t, r) satisfying (i). From the assumption (1.23) we imposed on the initial data w 0 (r), one can deduce that w 0 L 2 ([r0,+∞)) should be small, while w 0r L 2 ([r0,+∞)) can be large. Even so, such a stability result is essentially a stability result with small initial perturbation. It would be interesting to see whether the radially symmetric stationary wave ψ(r) constructed in Theorem 1.1 is nonlinear stable for large initial perturbation or not; (ii). Theorem 1.2 shows that the radially symmetric stationary wave ψ(r) constructed in Theorem 1.1 is timeasymptotically nonlinear stable under radially symmetric perturbation. An interesting problem is to see whether it is nonlinear stable or not under general multidimensional perturbation. For some recent progress on this problem for the case when n ≥ 3, v , those interested is referred to [4].
For the temporal convergence rates of the unique global solution v(t, r) of the initial-boundary value problem (1.2) toward the stationary wave φ(r), since φ(r) satisfies (1.18), we have by repeating the argument used in [14] that Theorem 1.3. Under the assumptions stated in Theorem 1.2, we can get that • For any β and γ satisfying holds for all t ≥ 0.
Now we outline the main ideas used to prove our main results. For the existence of radially symmetric stationary wave ψ(r) to (1.15), as in [5], one first considers the following Cauchy problem Let ψ n (r) be the unique local solution of (1.28) defined on the interval [r 0 , R), if one can find suitable lower bound ψ S (r) and upper bound ψ S (r) for ψ(r) such that • both ψ S (r) and ψ S (r) are well-defined for r ≥ r 0 ; then one can easily deduce by the continuation argument that the Cauchy problem (1.28) admits a unique solution ψ n (r) which is defined on r ≥ r 0 and belongs to C ∞ ([r 0 , +∞)). Moreover such a ψ n (r) satisfies |ψ n (r) − v + | r −2 and thus it is indeed the desired solution of (1.7). The difference between the case n ≥ 4 and the case n = 2 lies in the way to construct the desired upper bound ψ S (r) for the solution ψ n (r) of (1.28). In fact, for each n ≥ 2, ψ n S (r) defined by (1.13) always gives the desired lower bound for the solution ψ n (r) of (1.28), while for n = 2, ψ S (r) given by (1.11) is no longer a upper bound for the solution ψ n (r) of (1.28) with n = 2. To overcome such a difficulty, for each r 1 ≥ max r 0 , µ |v+| , we first consider the following auxiliary Cauchy problem and we can show that (i). the Cauchy problem (1.29) admits a unique solution η(r; r 1 ) on the interval [r 0 , +∞); (ii). η(r; r 1 ) satisfies |η(r; is an increasing function of r 1 and is bounded from above by |v + |, thus the limit a * = lim r1→+∞ a(r 1 ) exists; (iv). Based on the existence of such a constant a * , we can then construct the desired upper bound ψ S (r) for the solution ψ n (r) of (1.28) with n = 2 and to show that a * is indeed the threshold value to guarantee the existence of the stationary wave ψ(r) to (1.15).
The basis of the above analysis is that • suppose that the Cauchy problem (1.29) possesses a solution η(r; r 1 ) defined on the interval (R 1 , R 2 ) with r 0 ≤ R 1 ≤ r 1 < R 2 ≤ +∞, then η(r; r 1 ) is monotonic increasing for R 1 ≤ r ≤ r 1 and monotonic decreasing for r 1 ≤ r < R 2 .
For the time-asymptotically nonlinear stability of the stationary wave constructed in Theorem 1.1, the main difficulty, as pointed out in [2,3,5], is caused by the fact that such a stationary wave is no longer monotonic. Motivated by [14], our main idea to overcome the above difficulty lies in the following: • the first is to use the anti-derivative method by introducing the new unknown function w(t, r) defined by (1.20); • the second is to use a space weighted energy method to deduce the desired nonlinear stability result. The key point is to introduce the weight function χ(r) given by (1.22) to overcome the difficulties induced by the non-monotonicity of the stationary wave and the boundary condition.
Compared with that of [14], we use a refined continuation argument so that we can get a nonlinear stability result only under the assumption (1.23). The rest of this paper is organized as follows. In Section 2, we prove Theorem 1.1 and the proof of Theorem 1.2 will be given in Section 3.

Notations:
We denote the usual Lebesgue space of square integrable functions over [r 0 , ∞) by L 2 = L 2 ([r 0 , ∞)) with norm · and for each non-negative integer k, we use H k to denote the corresponding kth-order Sobolev space H k ([r 0 , ∞)) with norm · H k .
For α ∈ R, we denote the algebraic weighted Sobolev space, that is the space of functions f satisfying r α/2 f ∈ H k , by H k,α with norm f k,α := r α/2 f H k . For k = 0, we denote · 0,α by | · | α for simplicity. We also denote the exponential weighted Sobolev space, that is, the space of functions f satisfying e αr/2 f ∈ H k for some α ∈ R, by H k,α exp . For k = 0, we denote · 0,α exp by | · | α,exp for simplicity. For an interval I ⊂ R and a Banach space X, C(I; X) denotes the space of continuous X-valued functions on I, C k (I; X) the space of k-times continuously differentiable X-valued functions.
2 The proof of Theorem 1.1 This section is devoted to proving Theorem 1.1. To this end, we first consider the following Cauchy problem The local existence of smooth solution ψ(r) to the Cauchy problem (2.1) is well-established and suppose that such a local solution ψ(r) has been extended to the interval [r 0 , R) for some R > r 0 , to show that such a ψ(r) is indeed a solution of (1.15), we only need to show that • ψ(r) can be extended step by step to the interval [r 0 , +∞); For this purpose, by exploiting the standard continuation argument, we only need to deduce certain lower and upper bounds for ψ(r) on the interval [r 0 , R). To yield the desired lower bound for ψ(r) is relatively easy. In fact, we can get from the inequality 1 2µ From which and (2.1) 2 , we can easily deduce that Moreover, one can find that ψ 2 S (r) given by (1.13) with n = 2 is well-defined for r ≥ r 0 and satisfies Now we turn to find an upper bound for ψ(r) on the interval [r 0 , R). Since the last term − µ(n−1)(n−3) 2r 2 in the right hand side of (1.7) 1 has different sign for n ≥ 4 and for n = 2, the method used in [5], which has been proven to be effective for n ≥ 4, cannot be applied to the two-dimensional case any more. More precisely, even for the case when v + < 0, V − < |v + |, the function ψ S (r) defined by (1.11) is not an upper bound for ψ(r) on the interval [r 0 , R). Even so, one can easily deduce that for the case when v − < 0, the function ψ 2 1 (r) defined by with φ 2 1 (r) being defined by (1.8), indeed gives a upper bound for ψ(r), that is ψ(r) ≤ ψ 2 1 (r), r 0 ≤ r < R. (2.5) Noticing that both ψ 2 S (r) defined by (1.13) and ψ 2 1 (r) given by (2.4) are defined on r ≥ r 0 and are uniformly bounded on [r 0 , +∞) which follows from the estimates (2.3) and (1.10),one can thus deduce that the above local solution ψ(r) can indeed be extended step by step to the interval [r 0 , +∞). Even so, the problem is that for the case v + < 0, since lim To overcome such a difficulty, for each r 1 ≥ max r 0 , µ |v+| (the existence of such a r 1 is guaranteed by the assumption that v + < 0), we first consider the following auxiliary Cauchy problem and we want to show that the Cauchy problem (2.4) admits a unique solution η(r; r 1 ) on the interval [r 0 , +∞).
To do so, one can first deduce from the well-known local solvability result for the Cauchy problem of ordinary differential equations again that (2.6) admits a unique smooth solution η(r; r 1 ) which are defined on the interval (R 1 , R 2 ) with r 0 ≤ R 1 < r 1 < R 2 ≤ +∞ (For the case when r 1 = r 0 , the modification is straightforward, we only need to replace the interval (R 1 , R 2 ) by [r 0 , R 2 )). Our next lemma tells that r = r 1 is the global maximum point of η(r; r 1 ) on the interval (R 1 , R 2 ). Lemma 2.1. Suppose that η(r; r 1 ) is a smooth solution of the Cauchy problem (2.6) defined on the interval (R 1 , R 2 ), then η(r; r 1 ) is monotonic increasing for R 1 < r ≤ r 1 and monotonic decreasing for r 1 ≤ r < R 2 , thus sup R1<r<R2 η(r; r 1 ) = η(r 1 ; Proof. We only prove that η(r; r 1 ) is monotonic decreasing for r 1 ≤ r < R 2 . For this purpose, we first get from (2.6) that dη(r; r 1 ) dr (2.8) tells us that dη(r;r1) dr < 0 holds for all r 1 < r ≤ r 1 + ε. Here ε > 0 is a suitably chosen sufficiently small positive constant. Now if we set from which one can further deduce that there exists a sufficiently small positive constant r * > 0 such that dη(r;r1) dr < 0 holds for all r ∈ (R * , R * + r * ), but this fact contradicts the definition of R * and consequently R * = R 2 . This completes the proof of Lemma 2.1.
From Lemma 2.1, one can deduce the following estimate on the upper bound of the solution η(r; r 1 ) of the Cauchy problem (2.6) To get an estimate on the lower bound of η(r; r 1 ), we need the following comparison principle for the Cauchy problem (2.6), which will be frequently used in this section.
Since the proof of Lemma 2.2 is standard, we omit the details for brevity. Now we turn to deduce the desired lower bound estimate for η(r; r 1 ). To this end, we define η(r; r 1 ) = v + + (η(r 1 ; and η(r; r 1 ) = 1 r which solve the following Cauchy problems respectively.
Moreover, since lim r1→+∞ η(r 1 ; r 1 ) = lim one can show that η * (r; a * ) satisfies With the above preparations in hand, we now turn to prove our main result Theorem 1.1. Our main idea is to deduce a suitable upper bound on the solution ψ(r) of the Cauchy problem (2.1) on the interval [r 0 , R) in which ψ(r) is defined.
For the case when V − ≥ a * , we first consider the Cauchy problem (2.1) and suppose that its unique solution ψ(r) is defined on the interval [r 0 , R) for some R > r 0 , since V − ≥ a * , we can get from Lemma 2.2 that the following estimate ψ(r) ≥ η * (r; a * ), r 0 ≤ r < R (2.31) holds. In such a case, even if such a ψ(r) can be extended to the interval [r 0 , +∞) (consequently the estimate (2.13) also holds for r ≥ r 0 ), we can only deduce that lim r→+∞ ψ(r) ≥ |v + | since lim r→+∞ η * (r; a * ) = |v + |. Thus such a ψ(r) can not satisfy (1.15).
Having obtained Proposition 3.1 and Proposition 3.2, we now turn to prove Theorem 1.2 by the continuation argument. In fact, the assumption (1.23) imposed on the initial data w 0 (r) together with Sobolev's inequality imply that which guarantees the local existence of solution w(t, r) ∈ X µ (0, T ) for some T > 0. On the other hand, from the a priori estimates (3.2), (3.3) and (3.4) obtained in Proposition 3.2, one can get that holds for 0 ≤ t ≤ T provided that the assumption (1.23) holds and consequently, by employing Proposition 3.1 again, w(t, r) can be extended to the time interval t = T + t 1 and w(t, r) ∈ X µ (0, T + t 1 ). Repeating the above procedure and by the continuation argument, one can thus extend w(t, r) step by step to a global one. This completes the proof of Theorem 1.2. Since Theorem 1.3 can be proved by repeating the argument used in [14] to prove Theorem 2.2 and Theorem 2.3 there, we thus omit for brevity.