A discontinuous Galerkin Method based on POD model reduction for Euler equation

1.   Introduction

In this article, we prove the non-existence of solutions to the following quasilinear elliptic problem which has degenerate coercivity in their principal part by approximation,

where $1<p<N, q>1$ and $\lambda$ is a Radon measure. $\Omega$ is a bounded smooth subset of $\mathbb{R}^N(N>2)$. $a(x, t, \xi):\Omega\times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}^N$ is the Carathéodory function (i.e: $a(x, t, \xi)$ is measure on $\Omega$ for every $(t, \xi)$ in $\mathbb{R}\times \mathbb{R}^N$, and $a(\cdot, t, \xi)$ is continuous on $\mathbb{R}\times \mathbb{R}^N$ for almost every $x$ in $\Omega$), such that the following assumptions hold,

for almost every $x\in\Omega, t\in \mathbb{R}, \xi, \xi'\in \mathbb{R}^N$ with $\xi\neq\xi'$, where $0\leq\theta<1$, $c$ and $c_0$ are two positive constants, $b\in L^{p'}(\Omega)$ is a non-negative function, $p'$ is the conjugate Hölder exponent of $p$.

It is well-known that[3,9], problem $-\Delta u+|u|^{q-1}u = \delta_0$ has no distributional solution if $q\geq\frac{N}{N-2}$. On the other hand, if $q<\frac{N}{N-2}$, then there exists a unique solution to

In the famous work [9], Brezis proved that if $\{u_n\}$ is sequence of solution to the nonlinear elliptic problem

with $q>\frac{N}{N-2}$, and ${f_n}\in L^\infty (\Omega)$ is a sequence functions such that, for any $\varrho>0$,

Then $u_n$ converges to the unique solution $u$ to the following equation

This fact shows that $B_\varrho(0)$ is a removable singularity set of solution to equation (5) provided $q>\frac{N}{N-2}$. Orsina and Prignet[24] extended the result of [9] to more general operator $-\text{div}(a(x, u, \nabla u))$, where $a(x, u, \nabla u)$ satisfies (2)-(4) with $\theta = 0$. The main results of [24] shown that problem (1) with $\theta = 0$ has a solution for every given bounded measure $\lambda$ if $q <\frac{r(p-1)}{r-p}$. Some other related results see [12,6,10,8,14,23,26,27,21,19,16] and references therein.

The main goal of this paper is to study the non-existence of solutions to problem (1). More precisely, consider the limit of approximating equation (9)(see Theorem 1.2 below), our main task is to understand which is the limit of solutions to (9) and what equation it satisfies. A point worth emphasizing is that, even if $p = 2$, the convergence of solutions is not true if the right hand side are distributions weakly converging in $W^{-1, 2}(\Omega)$, see [5] for some counterexamples.

In order to state the main results of this paper, we need some definitions.

Let $K$ be a compact subset of $\Omega$, $r>1$ is a real number. The $r$ capacity of $K$ respect to $\Omega$ is defined as

where $\chi_K$ is the characteristic function of $K$.

Let $\lambda$ be a bounded measure on $\Omega$, we say that $\lambda$ is concentrated on a set $E$ if $\lambda(B) = \lambda(B\cap E)$ for every Borel subset $B$ of $\Omega$. Thanks to the Hahn decomposition, $\lambda$ can be decomposed as the difference of two nonnegative mutually singular measure, that is $\lambda = \lambda^+-\lambda^-$.

If $\lambda$ is concentrated on a set $E$, as a consequence of the fact that $\lambda^+$ and $\lambda^-$ are mutually singular, we have that $\lambda^+$ and $\lambda^-$ concentrated a set $E^+$ and $E^-$ respectively and ${E^+}\cap{E^-} = \emptyset$.

Let $\lambda = \lambda^+-\lambda^-$ be a measure, $f_n = f^+_n-f^-_n$ approximations of $\lambda$ in the following way:

for every function $\varphi$, which is continuous and bounded on $\Omega$, where $\{f^+_n\}$ and $\{f^-_n\}$ are sequences of nonnegative $L^\infty(\Omega)$ functions. We not assume that $f^+_n$ and $f^-_n$ are the positive and negative part of $f_n$. Observe that choosing $\varphi\equiv1$ in (6), we obtain

For all $k>0, s\in \mathbb{R}$, define

Firstly we stale the existence result.

Theorem 1.1. Let $\Omega$ be a bounded smooth subset of $\mathbb{R}^N(N>2)$, $1<p<N$, $g\in L^1(\Omega)$ and (2)-(4) hold. Then there exists a unique entropy solution $u\in W_0^{1, p}(\Omega)$ to problem

if

Moreover,

where $M^{p_1}, M^{p_2}$ represents the Marcinkiewicz space with exponent

Remark 1. The previous result gives existence and uniqueness of the entropy solution $u\in W_0^{1, p}(\Omega)$ to (8) for every $1<p<N$ and $0<\theta<1$. If $\theta = 0$, the same result for (8) can be proved by the same techniques of [2].

Our main results are following:

Theorem 1.2. Let $1<p<r\leq N$ and $\lambda = \lambda^+-\lambda^-$ be a bounded Radon measure which is concentrated on a set $E$ with zero $r$ capacity. Let $f_n = f^+_n-f^-_n$ be a sequence of $L^\infty(\Omega)$ functions which converge to $\lambda$ in the sense of (6). $g\in L^1(\Omega)$ and let $g_n$ is a sequence of $L^\infty(\Omega)$ functions which converge to $g$ weakly in $L^1(\Omega)$. Suppose $u_n\in W_0^{1, p}(\Omega)$ is the solution to problem:

Then $|\nabla u_n|^{p-1}$ strong converges to $|\nabla u|^{p-1}$ in $L^\sigma(\Omega)$ as $n\to \infty$ for

if

where $u$ is unique solution of (8). Moreover,

Remark 2. The above theorem shows that there is not a solution to problem (1) can be obtained by approximation, if $q$ is large enough and the measure $\lambda$ is concentrated on a set with zero $r$ capacity.

Remark 3. Boccardo et.al [7] considered the non-existence result to the following problem

where $\gamma>1$ and $\mu$ is a non-negative Radon measure, concentrated on a set $E$ with zero harmonic capacity, $a(x, \xi)$ satisfies (2)-(4) with $\theta = 0$, $p = 2$ and $b(x) = 0$. While in Theorem 1.2, $\lambda$ is a bounded Radon measure concentrated on a set $E$ with zero $r$ capacity with $p<r\leq N$, instead of $p$ capacity. Therefore Theorem 1.2 is not a triviality extend the results of Theorem 4.1 of [7]. Furthermore, in Theorem 1.2, $\theta(p-1)\in (0, p-1)$ since $\theta\in (0, 1)$. Note that, in problem (12), they required that $\gamma>1$. It is worth pointing out that different ranges of $\gamma$ have an important impact on the behavior of solutions to problem (12), more details see [25,18,1,17,13,4].

The structure of this paper is as follows: Section 2 mainly gives some lemmas which play a important role in the process of proof of the main theorem. The proof of theorem 1.1 and 1.2 are given in Section 3.

2.   Useful tools and function setting

In the following, $C$ is a constant and its value may changes from line to line.

In order to prove Theorem 1.1 and 1.2, the following basic lemmas and definitions are required.

Lemma 2.1. (see Lemma 2.1 of [22]) Let $K^+$ and $K^-$ be two disjoin compact subsets of $\Omega$ with zero $r$ capacity, $\lambda = \lambda^+-\lambda^-$ be a measure which is concentrated on a set with zero $r$ capacity with $1<r\leq N$, Then there exist two functions $\psi_\delta^+$ and $\psi_\delta^-$ in $C_c^\infty(\Omega)$, such that

for every $\delta>0$.

Definition 2.2. Let $u$ be an measurable function on $\Omega$ such that $T_k(u)\in W_0^{1, p}(\Omega)$ for every $k>0$. Then there exist a unique measurable function $v:\Omega\rightarrow \mathbb{R}^N$ such that

Define the gradient of $u$ as the function $v$ and denote it by $v = \nabla u$.

Definition 2.3. Let $f\in L^1(\Omega)$, $q>0$ and (2)-(4) hold. A measurable function $u$ is an entropy solution to problem (8), if $T_k(u)\in W_0^{1, p}(\Omega)$ for every $k>0$, $|u|^q\in L^1(\Omega)$ and

for every $\varphi\in W_0^{1, p}(\Omega)\cap L^1(\Omega)$.

Definition 2.4. Marcinkiewicz space $M^s(\Omega)(s>0)$ is the space composed of all the measurable functions $v$ that satisfy

for any $k>0$, where the constant $C>0$.

If $|\Omega|$ is bounded and $0<\varepsilon<s-1$, then the following embedding relationship hold:

Lemma 2.5. Let $u\in M^s(\Omega)$ with $s>0$. If there exist a constant $\rho>0$, such that for any $k>0$,

for some positive constant $C$. Then

Proof. Let $\sigma$ be a fixed positive real number, for every $k>0$,

Moreover,

Since $u\in M^s(\Omega)$, by Definition 2.4, there exist a constant $C$ such that

Combining (14)-(16), we have

Therefore, by Definition 2.4, we get $|\nabla u|\in M^{\frac{ps}{s+\rho}}$.

Lemma 2.6. Let $\{u_n\}$ be a sequence in $W_0^{1, p}(\Omega)$ and assume that there exist positive constants $\rho$ and $C$ with $p>\rho$, such that

for any $k$ and $n$. Then there exists a subsequence, still denoted by $\{u_n\}$, which converges to a measurable function $v$ almost everywhere in $\Omega$.

Lemma 2.7. Let $u$ be an entropy solution to (8), then

Proof. For any given $h$ and $k>0, s\in \mathbb{R}$, define

Take $T_{k, h}(u)$ as test function in (8), we have

Since $uT_{k, h}(u)\geq0$, we find

and

According to the assumption (2) and (17)-(19), we get,

Proposition 1. Let $u\in W_0^{1, p}(\Omega)$ be an entropy solution to (8) and satisfy

for every $k>0$ and $p>\rho$. Then $u\in M^{p_1}(\Omega)$, where $p_1 = {N(p-\rho)}/{(N-p)}$. More precisely, there exists $C = C(N, p, \theta)>0$ such that

Proof. For every $k>0$, by the Sobolev embedding theorem and (20),

where $p^* = \frac{Np}{N-p}$. For $0<\eta\leq k$, we have

Hence

Setting $\eta = k$, we obtain

This fact shows that $u\in M^{p_1}(\Omega)$ with $p_1 = {N(p-\rho)}/{(N-p)}$.

Proposition 2. Assume that $u\in W_0^{1, p}(\Omega)$ is an entropy solution to (8), which satisfies (20) for every $k$. Then $\nabla u\in M^{p_2}(\Omega)$, where $p_2 = {N(p-\rho)}/{(N-\rho)}$, that is there exists $C = C(N, p, \theta)>0$ such that

for every $h>0$.

Proof. For $k, \lambda>0$, set

Using the fact that the function $\lambda\mapsto\psi(k, \lambda)$ is nonincreasing, we get, for $k, \lambda>0$,

By Proposition 1,

where $p_1 = {N(p-\rho)}/{(N-p)}$. Since $\psi(0, s)-\psi(k, s) = |\{|\nabla u|^p>s, |u|<k\}|$, thanks to (20), we have

Combining (21)-(23), we arrive at

Let $\frac{Ck^{\rho}}{\lambda} = Ck^{-p_1}$ and $\lambda = h^p$, (24) implies that

That is $\nabla u\in M^{p_2}(\Omega)$ with $p_2 = {N(p-\rho)}/{(N-\rho)}$.

3.   Proof of main theorem

In this section we prove Theorem 1.1 and 1.2 combining the results of Sections 2.

In the proofs of Theorem 1.1 and 1.2, $\omega(n, m, \delta)$ will denote any quantity (depending on $n, m$ and $\delta$) such that

If the quantity does not depend on one or more of the three parameters $n, m$ and $\delta$, we will omit the dependence from it in $\omega$. For example, $\omega(n, \delta)$ is any quantity such that

3.1.   Proof of Theorem 1.1

The proof of Theorem 1.1 will be divided in several steps.

Proof. (1)Uniqueness: Let $u_1$ and $u_2$ be two entropy solutions to equation (8). The proof of the fact that $u_1 = u_2$ will follow from the following four steps.

Step 1. Assume that $g_i\in L^1(\Omega), (i = 1, 2)$. Choosing $T_k(u_1-T_hu_2)$ and $T_k(u_2-T_hu_1)$ as test function in (8) respectively, we get

Step 2. Denote

For $x\in A_0$,

and

Thus, for every $x\in A_0$,

For $x\in A_1$, $\nabla T_k(u_1-T_hu_2) = \nabla (u_1-h) = \nabla u_1$. By (2), we get

For $x\in A_2$, $\nabla T_k(u_1-T_hu_2) = \nabla (u_1-u_2)$. Thus

Similarly, denote

Then for $x\in A_1^*$, $\nabla T_k(u_2-T_hu_1) = \nabla (u_2-h) = \nabla u_2$. By (2), we get

For $x\in A_2^*$, $\nabla T_k(u_2-T_hu_1) = \nabla (u_2-u_1)$. Thus

Summing up (26)-(30) in the form $I\geq I_0-I_1$, where

Now, we estimate $I_{11}$. By the Hölder inequality and (3), we have

Therefore, by Lemma 2.7 and Proposition 2, $I_{11}\rightarrow0$ as $h\rightarrow\infty$ for every $k>0$. $I_{12}\rightarrow0$ as $h\rightarrow\infty$ for every $k>0$ can be obtained in the same way.

Hence, we find

Step 3. Now estimate the terms on the right hand side of (25). Denote

For $x\in B_0$, since $T_k(u_1-T_hu_2) = T_k(u_1-u_2)$ and $T_k(u_2-T_hu_1) = T_k(u_2-u_1)$, we arrive at

and

For $x\in B_1$, since $T_k(u_2-T_hu_1) = T_k(u_2-h)$. Then

and

For $x\in B_2$, since $T_k(u_1-T_hu_2) = T_k(u_1-h)$, we get

and

According to $|B_1|\rightarrow0, |B_2|\rightarrow0$ as $h\rightarrow\infty$ and $|u|^q\in L^1(\Omega)$ for fixed $k>0$, we get

Step 4. Combining (25) and (31)-(34), we have

where $\varepsilon(h)\rightarrow0$ as $h\rightarrow\infty$. Since $A_0$ converges to $\{x\in\Omega:|u_1-u_2|<k\}$ by measure as $h\rightarrow\infty$ for fixed $k>0$, we conclude that

for all $k>0$. This fact, combine with (4), implies that $\nabla u_1 = \nabla u_2$ a.e in $\Omega$. Then we get $u_1 = u_2$ a.e in $\Omega$.

(2) Existence:

Step 1. Let

where $\beta(u) = |u|^{q-1}u$, which is continuous with respect to $u$. Then $g(x) = F(x, 0)\in L^1(\mathbb{R}^N)$ and $\beta$ is monotonous nondecreasing with respect to $u$ with $\beta(0) = 0$ and $\beta(u)u\geq0$.

Let $g_n\in C_0^\infty$, such that $g_n$ converges to $g$ in $L^1(\Omega)$, with $\|g_n\|_{L^1(\Omega)}\leq\|g\|_{L^1(\Omega)}$ for every $n\geq1$. Define $\beta_n(s) = T_n(\beta)$. In this way, $|\beta_n(s)|\leq|\beta(s)|$ for every $s\in \mathbb{R}$ and $x\in \Omega$. Finally we take

Then by [20], there exists $u_n\in W_0^{1, p}(\Omega)$ such that

holds in the sense of distributions in $\Omega$.

By density arguments, we can take $T_h(u_n-T_k(u_n))$ and $T_k(u_n)$ as the test function in (35) respectively, we have

and

Combine (36) with (2) (fix the ellipticity constant $c = 1$) and $\gamma_nT_h(u_n-T_k(u_n))\geq0$, we get,

Since $a(x, u_n, \nabla u_n)\cdot\nabla u_n\geq 0$ by (2), we have

Combine (37) with $\gamma_nT_k(u_n)\geq0$, we have

Step 2. Convergence. Using (38) and Proposition 1, we have $|\{|u_n|>k\}|$ is bounded uniformly for every $k>0$. Thanks to (40), we see that $\{\nabla T_k(u_n)\}$ is bounded in $L^p_{\text loc}(\Omega)$ for every $k>0$.

Next we prove that $u_n\rightarrow u$ locally in measure.

For $t, \epsilon>0$, we have

Thus

Choosing $k$ large enough such that $|\{|u_n|>k\}|<\epsilon$ and $|\{|u_m|>k\}|<\epsilon$. Since $\{\nabla T_k(u_n)\}_n$ is bounded in $L^p(\Omega)$ and $T_k(u_n)\in W_0^{1, p}(\Omega)$ for every $k>0$. Assume that $\{T_k(u_n)\}$ is a Cauchy sequence in $L^q(\Omega\cap B_R)$ for any $q<pN/(N-p)$ and any $R>0$,

Then

for all $n, m\geq n_0(k, t, R)$. This show that $\{u_n\}$ is a Cauchy sequence in $B_R$. Hence that $u_n\rightarrow u$ locally.

Now to prove that $\nabla u_n$ converges to some function $v$ locally. We need to prove that $\{\nabla u_n\}$ is a Cauchy sequence in any ball $B_R$. Let $t, \epsilon>0$ again, then

Choose $l$ large enough such that $|\{|\nabla u_n|>l\}|\leq\epsilon$ for all $n\in \mathbb{N}$. If $a$ is a continuous function independent of $x$, then by (4), there exists a $\mu>0$, such that $|\xi|<l, |\xi'|<l$ and $|\xi-\xi'|>t$ means

This is a consequence of continuity and strict monotonicity of $a$. Set

Taking $T_k(u_n-u_m)$ as the test function of (35) and by (37), (41), we have

Then

if $k$ is small enough such that $k^{1+\theta(p-1)}\leq\mu\epsilon/C$.

Since $l$ and $k$ have been confirmed, if $n_0$ large enough, we have $|(\{|u_n-u_m|>k\}\cap B_R)|\leq\epsilon$ for $n, m\geq n_0$. Then we get $|\{|\nabla u_n-\nabla u_m|>t\}\cap B_R|\leq4\epsilon$. This prove that $\nabla u_n$ converges to some function $v$ locally.

Finally, since $\{\nabla T_k(u_n)\}_n\in L^p(\Omega)$ for every $k>0$, it converges weakly to $\{\nabla T_k(u)\}$ in $L^p_{\text loc}(\Omega)$. We have $u\in W_0^{1, p}(\Omega)$ and $\nabla u = v$ a.e in $\Omega$.

Step 3. In order to prove the existence of the solution completely, we still need to prove that sequence $\{a(x, u, \nabla u)\}_n$ is bounded in $L^q_{\text loc}(\Omega)$ for all

Indeed, by Proposition 2, $|\nabla u_n|^{p-1}\in M^{\frac{N(1-\theta)}{N-(1+\theta(p-1))}}\subset L^q_{\text loc}(\Omega)$. And by (3), we have $|a(x, u_n, \nabla u_n)|\in L^{p'}(\Omega)\subset L^q_{\text loc}(\Omega)$. According to the Nemitskii's theorem, $\nabla u_n\rightarrow\nabla u$ implies that

It follows that

for all $q\in\left(1, \frac{N(1-\theta)}{N-(1+\theta(p-1))}\right)$.

3.2.   Proof of Theorem 1.2

In this subsection, we give the proof of Theorem 1.2 following some ideas in [11,22].

Proof. Step 1 (A priori estimates). Firstly, choosing $T_k(u_n)(1-\varphi_\delta)^s$ as test function in the weak formulation of (9), where $s = \frac{\eta}{\eta-p+1}$ and $\eta$ will be given in (48), we have

By (2), we get

here $d\mu: = (1-\varphi_\delta)^sdx$.

Since $u_nT_k(u_n)\geq0$,

Using (3) and the Young inequality, we find

Combine (42)-(45), by (7) and $\{g_n\}\in L^1(\Omega)$, $b\in L^{p'}(\Omega)$, we have

For a fixed $\sigma\geq0$, thanks to (46), we get

which implies

Let

Clearly, such $\eta$ exists by (10). In view of (47)-(48), we have

By the Holder's inequality,

By Lemma 2.1, $1-\varphi_\delta$ is zero both on a neighbourhood of $K^+$ and $K^-$. Hence

Using (46) and (49), we conclude that

According to Lemma 2.5, we have $|\nabla u_n|\in M^s(\Omega)$, where $s = \frac{pq}{q+1+\theta(p-1)}$.

By (50) and Lemma 2.6, there exists a subsequence, still denoted by $u_n$, which converges to a measurable function $u$ almost everywhere in $\Omega$. So $T_k(u_n)\rightarrow T_k(u)$ in $\Omega$ for every $k>0$.

Since $T_k(u_n)\in W_0^{1, p}(\Omega)$, by the weak lower semi-continuity of the norm, $T_k(u)\in W_0^{1, p}(\Omega)$ for every $k>0$. Thus $u$ has an gradient $\nabla u$ in the sense of Definition 2.2, as a consequence of the a priori estimates on $\nabla u_n$ and (4), we have

for every $s<\frac{pq}{(q+1+\theta(p-1))(p-1)}$.

Step 2 (Energy estimates). Let $\psi_\delta = \psi_\delta^++\psi_\delta^-$, where $\psi_\delta^+$ and $\psi_\delta^-$ are as in Lemma 2.1. Then

and

Choose $\beta_m(u_n)(1-\psi_\delta)$ as test function in the weak formulation of (9), where

We obtain

Since $(A)$ and $-(E)$ are non-negative, we can get rid of them. And since $\beta_m(u_m)$ converges to $\beta_m(u)$ almost everywhere in $\Omega$ and in the weak$\ast$topology of $L^\infty(\Omega)$, $\beta_m(u_n)$ converges to zero in the weak$\ast$topology of $L^\infty(\Omega)$ as $m\rightarrow \infty$, we have

and

By $\psi_\delta = \psi_\delta^++\psi_\delta^-$ and (6),

and

We get (52), the proof of (53) is identical.

Step 3 (Passing to the limit). Now we show that $u$ is an entropy solution to (8) with datum $g$. Choose $T_k(u_n-\varphi)(1-\psi_\delta)$ as test function in the weak formulation of (9), we get

By (13),

while

The Fatou lemma implies

Using (13), (51), we have

While

and

So that we only need to deal with $(C)$. Let $m>k+\|\varphi\|_{L^\infty(\Omega)}$ be fixed,

By (52) and (53), we get

and

Summing up the result of (A)-(H), we have

Thus $u$ is the entropy solution of (8).

Finally we prove (10). Choose $\varphi\in C_c^\infty(\Omega)$ as test function in the weak formulation of (9), we get

Thanks to the assumptions of $f_n$, $g_n$ and by (51),

Since the entropy solution of (8) is also a distributional solution of the same problem, for the same $\varphi$,

Together with (54) and (55), we find

Thus (11) holds for every $\varphi\in C_c^\infty(\Omega)$. Since $|u_n|^{q-1}u_n$ is bounded in $L^1(\Omega)$, (11) can be extended by density to the functions in $C_c(\Omega)$.

Acknowledgments

The authors also would like to thank the anonymous referees for their valuable comments which has helped to improve the paper.