I am trying to find the paper/book which studies problem (1),(3) given bellow using the vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.

More detailed explanation:

From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:

The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of $$(1) \hspace{1cm} u_t+A(u)\cdot u_x = 0 \hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}}, \hspace{1cm} u(x,0)=u_{0}(x)\\[2ex] $$ as the limit $\epsilon\rightarrow0$ of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx}.\\[2ex] $$

In literature, when solving problem (2), we use the same initial data $u_0$, if they are smooth enough. Otherwise, if they are not smooth enough, we take smooth enough version $u_0^{\epsilon}$ (of course here it could be some other paramether instead of $\epsilon$; but that would be more complicated case).

Riemann initial data are given with:

$$(3) \hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0. \end{cases}$$

I am interested in problems where we change initial data (3) to the smooth one, solve problem (2) with the smooth data (we get them for example, by convolution of Riemann data with a mollifier). At the end we connect it with the original problem (1), (3) by letting $\epsilon \rightarrow 0$.

So far I have found three works that study this problem but the solutions are not in the Sobolev spaces:

1."Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data" - D. Hoff, 1995

2."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005

3."On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017

I am trying to find the papers/books/notes that study problem (1),(3) given bellow using the vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.

More detailed explanation:

From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:

The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of $$(1) \hspace{1cm} u_t+A(u)\cdot u_x = 0, \hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}}, \hspace{1cm} u(x,0)=u_{0}(x),\\[2ex] $$ as the limit of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx},\\[2ex] $$ when $\epsilon\rightarrow0.$

In literature, when solving problem $(2)$, we use the same initial data $u_0$, if they are smooth enough. Otherwise, if they are not smooth enough, we take smooth enough version $u_0^{\epsilon}$ (of course here it could be some other paramether instead of $\epsilon$, but let's not complicate this more than it needs to be).

Riemann initial data are given with:

$$(3) \hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0. \end{cases}$$

I am interested in papers where the authors used this technique:

1. First they transform the initial data $(3)$ to the smooth one $u_0^{\epsilon}$ (this can be done for example by convolution of Riemann data $(3)$ with a mollifier).
2. Than they solve the problem $(2)$ with the smooth data $u_0^{\epsilon}$ using the vanishing viscosity method.
3. At the end they connect that solution to the solution of the original problem $(1), (3)$ by letting $\epsilon \rightarrow 0$.

The papers that I've found and that satisfy this conditions are given bellow (but none of them has solutions in the Sobolev spaces):

1."Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data" - D. Hoff, 1995

2."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005

3."On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017

Help with this would be nice (there are just too much references in the literature that use the vanishing viscosity method and I can't read them all).

I am trying to find the paper/book where they study problem (1),(3) given bellow using Vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.

A little bit of explanation: From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:

The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of $$(1) \hspace{1cm} u_t+A(u)\cdot u_x = 0 \hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}}, \hspace{1cm} u(x,0)=u_{0}(x)\\[2ex] $$ as the limit $\epsilon\rightarrow0$ of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx}.\\[2ex] $$

In literature, when solving problem (2), somethimes we have the same initial data $u_0$, sometimes smooth enough version $u_0^{\epsilon}$. But what about if data are not smooth? For example, if we have Riemann data:

$$(3) \hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0 \end{cases}$$

or some other discontinuous initial data.

I am interested in problems where we change initial data (3) to the smooth one, solve problem (2) with the smooth data (we get them for example, by convolution of Riemann data with a mollifier). At the end we connect it with the original problem (1), (3).

So far I have found three works that study this problem but the solutions are not in the Sobolev spaces:

1."Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data" - D. Hoff, 1995

2."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005

3."On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017

I am trying to find the paper/book which studies problem (1),(3) given bellow using the vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.

More detailed explanation:

From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:

The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of $$(1) \hspace{1cm} u_t+A(u)\cdot u_x = 0 \hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}}, \hspace{1cm} u(x,0)=u_{0}(x)\\[2ex] $$ as the limit $\epsilon\rightarrow0$ of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx}.\\[2ex] $$

In literature, when solving problem (2), we use the same initial data $u_0$, if they are smooth enough. Otherwise, if they are not smooth enough, we take smooth enough version $u_0^{\epsilon}$ (of course here it could be some other paramether instead of $\epsilon$; but that would be more complicated case).

Riemann initial data are given with:

$$(3) \hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0. \end{cases}$$

I am interested in problems where we change initial data (3) to the smooth one, solve problem (2) with the smooth data (we get them for example, by convolution of Riemann data with a mollifier). At the end we connect it with the original problem (1), (3) by letting $\epsilon \rightarrow 0$.

So far I have found three works that study this problem but the solutions are not in the Sobolev spaces:

1."Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data" - D. Hoff, 1995

2."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005

3."On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017

I am trying to find the paper/book where they study problem (1),(3) given bellow using Vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.

A little bit of explanation: From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:

The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of $$(1) \hspace{1cm} u_t+A(u)\cdot u_x = 0 \hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}}, \hspace{1cm} u(x,0)=u_{0}(x)\\[2ex] $$ as the limit $\epsilon\rightarrow0$ of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx}.\\[2ex] $$

In literature, when solving problem (2), somethimes we have the same initial data $u_0$, sometimes smooth enough version $u_0^{\epsilon}$. But what about if data are not smooth? For example, if we have Riemann data:

$$(3) \hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0 \end{cases}$$

or some other discontinuous initial data.

For me this would be ideal: In the reference paper authors started with Riemann data. Than they solve problem (2) with original piecewise continuous Riemann data OR with continuous/smooth approximation of the data (for example, convolve Riemann data with a mollifier; or just use other approximation of a initial data). They use that to solve problem (1),(3). So they would have some continuous/smooth/discontinuous data that goes to piecewise continuous/discontinuous data.

Of course I accept any other reference anyone knows. :)

So far I have found two works that study this problem but the solutions are not in the Sobolev spaces: 1."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005 2."On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017

I am trying to find the paper/book where they study problem (1),(3) given bellow using Vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.

A little bit of explanation: From Chapter 10 of "Hyperbolic partial differential equations", P. Lax, 2006:

The vanishing viscosity method for solving the initial value problem for hyperbolic systems of conservation laws consists of finding solutions of $$(1) \hspace{1cm} u_t+A(u)\cdot u_x = 0 \hspace{1cm} A_{ij}=\frac{\partial{f_i}}{\partial{u_j}}, \hspace{1cm} u(x,0)=u_{0}(x)\\[2ex] $$ as the limit $\epsilon\rightarrow0$ of solutions of the parabolic system $$(2) \hspace{1cm} u_t+A(u)\cdot u_x = \epsilon u_{xx}.\\[2ex] $$

In literature, when solving problem (2), somethimes we have the same initial data $u_0$, sometimes smooth enough version $u_0^{\epsilon}$. But what about if data are not smooth? For example, if we have Riemann data:

$$(3) \hspace{1cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x\geq 0 \end{cases}$$

or some other discontinuous initial data.

I am interested in problems where we change initial data (3) to the smooth one, solve problem (2) with the smooth data (we get them for example, by convolution of Riemann data with a mollifier). At the end we connect it with the original problem (1),  (3).

So far I have found three works that study this problem but the solutions are not in the Sobolev spaces:

1."Global Solutions of the Navier-Stokes Equations for Multidimensional Compressible Flow with Discontinuous Initial Data" - D. Hoff, 1995

2."Vanishing viscosity solutions of nonlinear hyperbolic systems" - S. Bianchini, A. Bressan, 2005

3."On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding" - W. Shen, 2017